hypothesis between two means

Module 10: Inference for Means

Hypothesis test for a difference in two population means (1 of 2), learning outcomes.

Under appropriate conditions, conduct a hypothesis test about a difference between two population means. State a conclusion in context.

Using the Hypothesis Test for a Difference in Two Population Means

The general steps of this hypothesis test are the same as always. As expected, the details of the conditions for use of the test and the test statistic are unique to this test (but similar in many ways to what we have seen before.)

Step 1: Determine the hypotheses.

The hypotheses for a difference in two population means are similar to those for a difference in two population proportions. The null hypothesis, H 0 , is again a statement of “no effect” or “no difference.”

H 0 : μ 1 – μ 2 = 0, which is the same as H 0 : μ 1 = μ 2

The alternative hypothesis, H a , can be any one of the following.

H a : μ 1 – μ 2 < 0, which is the same as H a : μ 1 < μ 2
H a : μ 1 – μ 2 > 0, which is the same as H a : μ 1 > μ 2
H a : μ 1 – μ 2 ≠ 0, which is the same as H a : μ 1 ≠ μ 2

Step 2: Collect the data.

As usual, how we collect the data determines whether we can use it in the inference procedure. We have our usual two requirements for data collection.

Samples must be random to remove or minimize bias.
Samples must be representative of the populations in question.

We use this hypothesis test when the data meets the following conditions.

The two random samples are independent .
The variable is normally distributed in both populations . If this variable is not known, samples of more than 30 will have a difference in sample means that can be modeled adequately by the t-distribution. As we discussed in “Hypothesis Test for a Population Mean,” t-procedures are robust even when the variable is not normally distributed in the population. If checking normality in the populations is impossible, then we look at the distribution in the samples. If a histogram or dotplot of the data does not show extreme skew or outliers, we take it as a sign that the variable is not heavily skewed in the populations, and we use the inference procedure. (Note: This is the same condition we used for the one-sample t-test in “Hypothesis Test for a Population Mean.”)

Step 3: Assess the evidence.

If the conditions are met, then we calculate the t-test statistic. The t-test statistic has a familiar form.

[latex]T\text{}=\text{}\frac{(\mathrm{Observed}\text{}\mathrm{difference}\text{}\mathrm{in}\text{}\mathrm{sample}\text{}\mathrm{means})-(\mathrm{Hypothesized}\text{}\mathrm{difference}\text{}\mathrm{in}\text{}\mathrm{population}\text{}\mathrm{means})}{\mathrm{Standard}\text{}\mathrm{error}}[/latex]

[latex]T\text{}=\text{}\frac{({\stackrel{¯}{x}}_{1}-{\stackrel{¯}{x}}_{2})-({μ}_{1}-{μ}_{2})}{\sqrt{\frac{{{s}_{1}}^{2}}{{n}_{1}}+\frac{{{s}_{2}}^{2}}{{n}_{2}}}}[/latex]

Since the null hypothesis assumes there is no difference in the population means, the expression (μ 1 – μ 2 ) is always zero.

As we learned in “Estimating a Population Mean,” the t-distribution depends on the degrees of freedom (df) . In the one-sample and matched-pair cases df = n – 1. For the two-sample t-test, determining the correct df is based on a complicated formula that we do not cover in this course. We will either give the df or use technology to find the df . With the t-test statistic and the degrees of freedom, we can use the appropriate t-model to find the P-value, just as we did in “Hypothesis Test for a Population Mean.” We can even use the same simulation.

Step 4: State a conclusion.

To state a conclusion, we follow what we have done with other hypothesis tests. We compare our P-value to a stated level of significance.

If the P-value ≤ α, we reject the null hypothesis in favor of the alternative hypothesis.
If the P-value > α, we fail to reject the null hypothesis. We do not have enough evidence to support the alternative hypothesis.

As always, we state our conclusion in context, usually by referring to the alternative hypothesis.

“Context and Calories”

Does the company you keep impact what you eat? This example comes from an article titled “Impact of Group Settings and Gender on Meals Purchased by College Students” (Allen-O’Donnell, M., T. C. Nowak, K. A. Snyder, and M. D. Cottingham, Journal of Applied Social Psychology 49(9), 2011, onlinelibrary.wiley.com/doi/10.1111/j.1559-1816.2011.00804.x/full) . In this study, researchers examined this issue in the context of gender-related theories in their field. For our purposes, we look at this research more narrowly.

Step 1: Stating the hypotheses.

In the article, the authors make the following hypothesis. “The attempt to appear feminine will be empirically demonstrated by the purchase of fewer calories by women in mixed-gender groups than by women in same-gender groups.” We translate this into a simpler and narrower research question: Do women purchase fewer calories when they eat with men compared to when they eat with women?

Here the two populations are “women eating with women” (population 1) and “women eating with men” (population 2). The variable is the calories in the meal. We test the following hypotheses at the 5% level of significance.

The null hypothesis is always H 0 : μ 1 – μ 2 = 0, which is the same as H 0 : μ 1 = μ 2 .

The alternative hypothesis H a : μ 1 – μ 2 > 0, which is the same as H a : μ 1 > μ 2 .

Here μ 1 represents the mean number of calories ordered by women when they were eating with other women, and μ 2 represents the mean number of calories ordered by women when they were eating with men.

Note: It does not matter which population we label as 1 or 2, but once we decide, we have to stay consistent throughout the hypothesis test. Since we expect the number of calories to be greater for the women eating with other women, the difference is positive if “women eating with women” is population 1. If you prefer to work with positive numbers, choose the group with the larger expected mean as population 1. This is a good general tip.

Step 2: Collect Data.

As usual, there are two major things to keep in mind when considering the collection of data.

Samples need to be representative of the population in question.
Samples need to be random in order to remove or minimize bias.

Representative Samples?

The researchers state their hypothesis in terms of “women.” We did the same. But the researchers gathered data by watching people eat at the HUB Rock Café II on the campus of Indiana University of Pennsylvania during the Spring semester of 2006. Almost all of the women in the data set were white undergraduates between the ages of 18 and 24, so there are some definite limitations on the scope of this study. These limitations will affect our conclusion (and the specific definition of the population means in our hypotheses.)

Random Samples?

The observations were collected on February 13, 2006, through February 22, 2006, between 11 a.m. and 7 p.m. We can see that the researchers included both lunch and dinner. They also made observations on all days of the week to ensure that weekly customer patterns did not confound their findings. The authors state that “since the time period for observations and the place where [they] observed students were limited, the sample was a convenience sample.” Despite these limitations, the researchers conducted inference procedures with the data, and the results were published in a reputable journal. We will also conduct inference with this data, but we also include a discussion of the limitations of the study with our conclusion. The authors did this, also.

Do the data met the conditions for use of a t-test?

The researchers reported the following sample statistics.

In a sample of 45 women dining with other women, the average number of calories ordered was 850, and the standard deviation was 252.
In a sample of 27 women dining with men, the average number of calories ordered was 719, and the standard deviation was 322.

One of the samples has fewer than 30 women. We need to make sure the distribution of calories in this sample is not heavily skewed and has no outliers, but we do not have access to a spreadsheet of the actual data. Since the researchers conducted a t-test with this data, we will assume that the conditions are met. This includes the assumption that the samples are independent.

As noted previously, the researchers reported the following sample statistics.

To compute the t-test statistic, make sure sample 1 corresponds to population 1. Here our population 1 is “women eating with other women.” So x 1 = 850, s 1 = 252, n 1 =45, and so on.

[latex]T\text{}=\text{}\frac{{\stackrel{¯}{x}}_{1}\text{}\text{−}\text{}{\stackrel{¯}{x}}_{2}}{\sqrt{\frac{{{s}_{1}}^{2}}{{n}_{1}}+\frac{{{s}_{2}}^{2}}{{n}_{2}}}}\text{}=\text{}\frac{850\text{}\text{−}\text{}719}{\sqrt{\frac{{252}^{2}}{45}+\frac{{322}^{2}}{27}}}\text{}\approx \text{}\frac{131}{72.47}\text{}\approx \text{}1.81[/latex]

Using technology, we determined that the degrees of freedom are about 45 for this data. To find the P-value, we use our familiar simulation of the t-distribution. Since the alternative hypothesis is a “greater than” statement, we look for the area to the right of T = 1.81. The P-value is 0.0385.

The green area to the left of the t value = 0.9615. The blue area to the right of the T value = 0.0385.

Generic Conclusion

The hypotheses for this test are H 0 : μ 1 – μ 2 = 0 and H a : μ 1 – μ 2 > 0. Since the P-value is less than the significance level (0.0385 < 0.05), we reject H 0 and accept H a .

Conclusion in context

At Indiana University of Pennsylvania, the mean number of calories ordered by undergraduate women eating with other women is greater than the mean number of calories ordered by undergraduate women eating with men (P-value = 0.0385).

Comment about Conclusions

In the conclusion above, we did not generalize the findings to all women. Since the samples included only undergraduate women at one university, we included this information in our conclusion. But our conclusion is a cautious statement of the findings. The authors see the results more broadly in the context of theories in the field of social psychology. In the context of these theories, they write, “Our findings support the assertion that meal size is a tool for influencing the impressions of others. For traditional-age, predominantly White college women, diminished meal size appears to be an attempt to assert femininity in groups that include men.” This viewpoint is echoed in the following summary of the study for the general public on National Public Radio (npr.org).

Both men and women appear to choose larger portions when they eat with women, and both men and women choose smaller portions when they eat in the company of men, according to new research published in the Journal of Applied Social Psychology . The study, conducted among a sample of 127 college students, suggests that both men and women are influenced by unconscious scripts about how to behave in each other’s company. And these scripts change the way men and women eat when they eat together and when they eat apart.

Should we be concerned that the findings of this study are generalized in this way? Perhaps. But the authors of the article address this concern by including the following disclaimer with their findings: “While the results of our research are suggestive, they should be replicated with larger, representative samples. Studies should be done not only with primarily White, middle-class college students, but also with students who differ in terms of race/ethnicity, social class, age, sexual orientation, and so forth.” This is an example of good statistical practice. It is often very difficult to select truly random samples from the populations of interest. Researchers therefore discuss the limitations of their sampling design when they discuss their conclusions.

In the following activities, you will have the opportunity to practice parts of the hypothesis test for a difference in two population means. On the next page, the activities focus on the entire process and also incorporate technology.

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Hypothesis Test: Difference Between Means

This lesson explains how to conduct a hypothesis test for the difference between two means. The test procedure, called the two-sample t-test , is appropriate when the following conditions are met:

The sampling method for each sample is simple random sampling .
The samples are independent .
Each population is at least 20 times larger than its respective sample .
The population distribution is normal.
The population data are symmetric , unimodal , without outliers , and the sample size is 15 or less.
The population data are slightly skewed , unimodal, without outliers, and the sample size is 16 to 40.
The sample size is greater than 40, without outliers.

This approach consists of four steps: (1) state the hypotheses, (2) formulate an analysis plan, (3) analyze sample data, and (4) interpret results.

State the Hypotheses

Every hypothesis test requires the analyst to state a null hypothesis and an alternative hypothesis . The hypotheses are stated in such a way that they are mutually exclusive. That is, if one is true, the other must be false; and vice versa.

The table below shows three sets of null and alternative hypotheses. Each makes a statement about the difference d between the mean of one population μ 1 and the mean of another population μ 2 . (In the table, the symbol ≠ means " not equal to ".)

The first set of hypotheses (Set 1) is an example of a two-tailed test , since an extreme value on either side of the sampling distribution would cause a researcher to reject the null hypothesis. The other two sets of hypotheses (Sets 2 and 3) are one-tailed tests , since an extreme value on only one side of the sampling distribution would cause a researcher to reject the null hypothesis.

When the null hypothesis states that there is no difference between the two population means (i.e., d = 0), the null and alternative hypothesis are often stated in the following form.

H o : μ 1 = μ 2

H a : μ 1 ≠ μ 2

Formulate an Analysis Plan

The analysis plan describes how to use sample data to accept or reject the null hypothesis. It should specify the following elements.

Significance level. Often, researchers choose significance levels equal to 0.01, 0.05, or 0.10; but any value between 0 and 1 can be used.
Test method. Use the two-sample t-test to determine whether the difference between means found in the sample is significantly different from the hypothesized difference between means.

Analyze Sample Data

Using sample data, find the standard error, degrees of freedom, test statistic, and the P-value associated with the test statistic.

SE = sqrt[ (s 1 2 /n 1 ) + (s 2 2 /n 2 ) ]

DF = (s 1 2 /n 1 + s 2 2 /n 2 ) 2 / { [ (s 1 2 / n 1 ) 2 / (n 1 - 1) ] + [ (s 2 2 / n 2 ) 2 / (n 2 - 1) ] }

t = [ ( x 1 - x 2 ) - d ] / SE

P-value. The P-value is the probability of observing a sample statistic as extreme as the test statistic. Since the test statistic is a t statistic, use the t Distribution Calculator to assess the probability associated with the t statistic, having the degrees of freedom computed above. (See sample problems at the end of this lesson for examples of how this is done.)

Interpret Results

If the sample findings are unlikely, given the null hypothesis, the researcher rejects the null hypothesis. Typically, this involves comparing the P-value to the significance level , and rejecting the null hypothesis when the P-value is less than the significance level.

Test Your Understanding

In this section, two sample problems illustrate how to conduct a hypothesis test of a difference between mean scores. The first problem involves a two-tailed test; the second problem, a one-tailed test.

Problem 1: Two-Tailed Test

Within a school district, students were randomly assigned to one of two Math teachers - Mrs. Smith and Mrs. Jones. After the assignment, Mrs. Smith had 30 students, and Mrs. Jones had 25 students.

At the end of the year, each class took the same standardized test. Mrs. Smith's students had an average test score of 78, with a standard deviation of 10; and Mrs. Jones' students had an average test score of 85, with a standard deviation of 15.

Test the hypothesis that Mrs. Smith and Mrs. Jones are equally effective teachers. Use a 0.10 level of significance. (Assume that student performance is approximately normal.)

Solution: The solution to this problem takes four steps: (1) state the hypotheses, (2) formulate an analysis plan, (3) analyze sample data, and (4) interpret results. We work through those steps below:

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: μ 1 - μ 2 = 0

Alternative hypothesis: μ 1 - μ 2 ≠ 0

Formulate an analysis plan . For this analysis, the significance level is 0.10. Using sample data, we will conduct a two-sample t-test of the null hypothesis.

SE = sqrt[(s 1 2 /n 1 ) + (s 2 2 /n 2 )]

SE = sqrt[(10 2 /30) + (15 2 /25] = sqrt(3.33 + 9)

SE = sqrt(12.33) = 3.51

DF = (10 2 /30 + 15 2 /25) 2 / { [ (10 2 / 30) 2 / (29) ] + [ (15 2 / 25) 2 / (24) ] }

DF = (3.33 + 9) 2 / { [ (3.33) 2 / (29) ] + [ (9) 2 / (24) ] } = 152.03 / (0.382 + 3.375) = 152.03/3.757 = 40.47

t = [ ( x 1 - x 2 ) - d ] / SE = [ (78 - 85) - 0 ] / 3.51 = -7/3.51 = -1.99

where s 1 is the standard deviation of sample 1, s 2 is the standard deviation of sample 2, n 1 is the size of sample 1, n 2 is the size of sample 2, x 1 is the mean of sample 1, x 2 is the mean of sample 2, d is the hypothesized difference between the population means, and SE is the standard error.

Since we have a two-tailed test , the P-value is the probability that a t statistic having 40 degrees of freedom is more extreme than -1.99; that is, less than -1.99 or greater than 1.99.

We use the t Distribution Calculator to find P(t < -1.99) is about 0.027.

If you enter 1.99 as the sample mean in the t Distribution Calculator, you will find the that the P(t ≤ 1.99) is about 0.973. Therefore, P(t > 1.99) is 1 minus 0.973 or 0.027. Thus, the P-value = 0.027 + 0.027 = 0.054.
Interpret results . Since the P-value (0.054) is less than the significance level (0.10), we cannot accept the null hypothesis.

Note: If you use this approach on an exam, you may also want to mention why this approach is appropriate. Specifically, the approach is appropriate because the sampling method was simple random sampling, the samples were independent, the sample size was much smaller than the population size, and the samples were drawn from a normal population.

Problem 2: One-Tailed Test

The Acme Company has developed a new battery. The engineer in charge claims that the new battery will operate continuously for at least 7 minutes longer than the old battery.

To test the claim, the company selects a simple random sample of 100 new batteries and 100 old batteries. The old batteries run continuously for 190 minutes with a standard deviation of 20 minutes; the new batteries, 200 minutes with a standard deviation of 40 minutes.

Test the engineer's claim that the new batteries run at least 7 minutes longer than the old. Use a 0.05 level of significance. (Assume that there are no outliers in either sample.)

Null hypothesis: μ 1 - μ 2 <= 7

Alternative hypothesis: μ 1 - μ 2 > 7

where μ 1 is battery life for the new battery, and μ 2 is battery life for the old battery.

Formulate an analysis plan . For this analysis, the significance level is 0.05. Using sample data, we will conduct a two-sample t-test of the null hypothesis.

SE = sqrt[(40 2 /100) + (20 2 /100]

SE = sqrt(16 + 4) = 4.472

DF = (40 2 /100 + 20 2 /100) 2 / { [ (40 2 / 100) 2 / (99) ] + [ (20 2 / 100) 2 / (99) ] }

DF = (20) 2 / { [ (16) 2 / (99) ] + [ (2) 2 / (99) ] } = 400 / (2.586 + 0.162) = 145.56

t = [ ( x 1 - x 2 ) - d ] / SE = [(200 - 190) - 7] / 4.472 = 3/4.472 = 0.67

Here is the logic of the analysis: Given the alternative hypothesis (μ 1 - μ 2 > 7), we want to know whether the observed difference in sample means is big enough (i.e., sufficiently greater than 7) to cause us to reject the null hypothesis.

Interpret results . Suppose we replicated this study many times with different samples. If the true difference in population means were actually 7, we would expect the observed difference in sample means to be 10 or less in 75% of our samples. And we would expect to find an observed difference to be more than 10 in 25% of our samples Therefore, the P-value in this analysis is 0.25.

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9.2: Comparing Two Independent Population Means (Hypothesis test)

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The two independent samples are simple random samples from two distinct populations.
if the sample sizes are small, the distributions are important (should be normal)
if the sample sizes are large, the distributions are not important (need not be normal)

The test comparing two independent population means with unknown and possibly unequal population standard deviations is called the Aspin-Welch $t$-test. The degrees of freedom formula was developed by Aspin-Welch.

The comparison of two population means is very common. A difference between the two samples depends on both the means and the standard deviations. Very different means can occur by chance if there is great variation among the individual samples. In order to account for the variation, we take the difference of the sample means, $\bar{X}_{1} - \bar{X}_{2}$, and divide by the standard error in order to standardize the difference. The result is a t-score test statistic.

Because we do not know the population standard deviations, we estimate them using the two sample standard deviations from our independent samples. For the hypothesis test, we calculate the estimated standard deviation, or standard error , of the difference in sample means , $\bar{X}_{1} - \bar{X}_{2}$.

The standard error is:

\[\sqrt{\dfrac{(s_{1})^{2}}{n_{1}} + \dfrac{(s_{2})^{2}}{n_{2}}}\]

The test statistic ( t -score) is calculated as follows:

\[\dfrac{(\bar{x}-\bar{x}) - (\mu_{1} - \mu_{2})}{\sqrt{\dfrac{(s_{1})^{2}}{n_{1}} + \dfrac{(s_{2})^{2}}{n_{2}}}}\]

$s_{1}$ and $s_{2}$, the sample standard deviations, are estimates of $\sigma_{1}$ and $\sigma_{1}$, respectively.
$\sigma_{1}$ and $\sigma_{2}$ are the unknown population standard deviations.
$\bar{x}_{1}$ and $\bar{x}_{2}$ are the sample means. $\mu_{1}$ and $\mu_{2}$ are the population means.

The number of degrees of freedom ($df$) requires a somewhat complicated calculation. However, a computer or calculator calculates it easily. The $df$ are not always a whole number. The test statistic calculated previously is approximated by the Student's t -distribution with $df$ as follows:

Degrees of freedom

\[df = \dfrac{\left(\dfrac{(s_{1})^{2}}{n_{1}} + \dfrac{(s_{2})^{2}}{n_{2}}\right)^{2}}{\left(\dfrac{1}{n_{1}-1}\right)\left(\dfrac{(s_{1})^{2}}{n_{1}}\right)^{2} + \left(\dfrac{1}{n_{2}-1}\right)\left(\dfrac{(s_{2})^{2}}{n_{2}}\right)^{2}}\]

We can also use a conservative estimation of degree of freedom by taking DF to be the smallest of $n_{1}-1$ and $n_{2}-1$

When both sample sizes $n_{1}$ and $n_{2}$ are five or larger, the Student's t approximation is very good. Notice that the sample variances $(s_{1})^{2}$ and $(s_{2})^{2}$ are not pooled. (If the question comes up, do not pool the variances.)

It is not necessary to compute the degrees of freedom by hand. A calculator or computer easily computes it.

Example $\PageIndex{1}$: Independent groups

The average amount of time boys and girls aged seven to 11 spend playing sports each day is believed to be the same. A study is done and data are collected, resulting in the data in Table $\PageIndex{1}$. Each populations has a normal distribution.

Is there a difference in the mean amount of time boys and girls aged seven to 11 play sports each day? Test at the 5% level of significance.

The population standard deviations are not known. Let g be the subscript for girls and b be the subscript for boys. Then, $\mu_{g}$ is the population mean for girls and $\mu_{b}$ is the population mean for boys. This is a test of two independent groups, two population means.

Random variable: $\bar{X}_{g} - \bar{X}_{b} =$ difference in the sample mean amount of time girls and boys play sports each day.

$H_{0}: \mu_{g} = \mu_{b}$
$H_{0}: \mu_{g} - \mu_{b} = 0$
$H_{a}: \mu_{g} \neq \mu_{b}$
$H_{a}: \mu_{g} - \mu_{b} \neq 0$

The words "the same" tell you $H_{0}$ has an "=". Since there are no other words to indicate $H_{a}$, assume it says "is different." This is a two-tailed test.

Distribution for the test: Use $t_{df}$ where $df$ is calculated using the $df$ formula for independent groups, two population means. Using a calculator, $df$ is approximately 18.8462. Do not pool the variances.

Calculate the p -value using a Student's t -distribution: $p\text{-value} = 0.0054$

$This is a normal distribution curve representing the difference in the average amount of time girls and boys play sports all day. The mean is equal to zero, and the values -1.2, 0, and 1.2 are labeled on the horizontal axis. Two vertical lines extend from -1.2 and 1.2 to the curve. The region to the left of x = -1.2 and the region to the right of x = 1.2 are shaded to represent the p-value. The area of each region is 0.0028.$

\[s_{g} = 0.866\]

\[s_{b} = 1\]

\[\bar{x}_{g} - \bar{x}_{b} = 2 - 3.2 = -1.2\]

Half the $p\text{-value}$ is below –1.2 and half is above 1.2.

Make a decision: Since $\alpha > p\text{-value}$, reject $H_{0}$. This means you reject $\mu_{g} = \mu_{b}$. The means are different.

Press STAT . Arrow over to TESTS and press 4:2-SampTTest . Arrow over to Stats and press ENTER . Arrow down and enter 2 for the first sample mean, $\sqrt{0.866}$ for Sx1, 9 for n1, 3.2 for the second sample mean, 1 for Sx2, and 16 for n2. Arrow down to μ1: and arrow to does not equal μ2. Press ENTER . Arrow down to Pooled: and No . Press ENTER . Arrow down to Calculate and press ENTER . The $p\text{-value}$ is $p = 0.0054$, the dfs are approximately 18.8462, and the test statistic is -3.14. Do the procedure again but instead of Calculate do Draw.

Conclusion: At the 5% level of significance, the sample data show there is sufficient evidence to conclude that the mean number of hours that girls and boys aged seven to 11 play sports per day is different (mean number of hours boys aged seven to 11 play sports per day is greater than the mean number of hours played by girls OR the mean number of hours girls aged seven to 11 play sports per day is greater than the mean number of hours played by boys).

Exercise $\PageIndex{1}$

Two samples are shown in Table. Both have normal distributions. The means for the two populations are thought to be the same. Is there a difference in the means? Test at the 5% level of significance.

The $p\text{-value}$ is $0.4125$, which is much higher than 0.05, so we decline to reject the null hypothesis. There is not sufficient evidence to conclude that the means of the two populations are not the same.

When the sum of the sample sizes is larger than $30 (n_{1} + n_{2} > 30)$ you can use the normal distribution to approximate the Student's $t$.

Example $\PageIndex{2}$

A study is done by a community group in two neighboring colleges to determine which one graduates students with more math classes. College A samples 11 graduates. Their average is four math classes with a standard deviation of 1.5 math classes. College B samples nine graduates. Their average is 3.5 math classes with a standard deviation of one math class. The community group believes that a student who graduates from college A has taken more math classes, on the average. Both populations have a normal distribution. Test at a 1% significance level. Answer the following questions.

Is this a test of two means or two proportions?
Are the populations standard deviations known or unknown?
Which distribution do you use to perform the test?
What is the random variable?
What are the null and alternate hypotheses? Write the null and alternate hypotheses in words and in symbols.
Is this test right-, left-, or two-tailed?
What is the $p\text{-value}$?
Do you reject or not reject the null hypothesis?
Student's t
$\bar{X}_{A} - \bar{X}_{B}$
$H_{0}: \mu_{A} \leq \mu_{B}$ and $H_{a}: \mu_{A} > \mu_{B}$

$alt$

h. Do not reject.
i. At the 1% level of significance, from the sample data, there is not sufficient evidence to conclude that a student who graduates from college A has taken more math classes, on the average, than a student who graduates from college B.

Exercise $\PageIndex{2}$

A study is done to determine if Company A retains its workers longer than Company B. Company A samples 15 workers, and their average time with the company is five years with a standard deviation of 1.2. Company B samples 20 workers, and their average time with the company is 4.5 years with a standard deviation of 0.8. The populations are normally distributed.

Are the population standard deviations known?
Conduct an appropriate hypothesis test. At the 5% significance level, what is your conclusion?
They are unknown.
The $p\text{-value} = 0.0878$. At the 5% level of significance, there is insufficient evidence to conclude that the workers of Company A stay longer with the company.

Example $\PageIndex{3}$

A professor at a large community college wanted to determine whether there is a difference in the means of final exam scores between students who took his statistics course online and the students who took his face-to-face statistics class. He believed that the mean of the final exam scores for the online class would be lower than that of the face-to-face class. Was the professor correct? The randomly selected 30 final exam scores from each group are listed in Table $\PageIndex{3}$ and Table $\PageIndex{4}$.

Is the mean of the Final Exam scores of the online class lower than the mean of the Final Exam scores of the face-to-face class? Test at a 5% significance level. Answer the following questions:

Are the population standard deviations known or unknown?
What are the null and alternative hypotheses? Write the null and alternative hypotheses in words and in symbols.
Is this test right, left, or two tailed?
At the ___ level of significance, from the sample data, there ______ (is/is not) sufficient evidence to conclude that ______.

(See the conclusion in Example, and write yours in a similar fashion)

Be careful not to mix up the information for Group 1 and Group 2!

Student's $t$
$\bar{X}_{1} - \bar{X}_{2}$
$H_{0}: \mu_{1} = \mu_{2}$ Null hypothesis: the means of the final exam scores are equal for the online and face-to-face statistics classes.
$H_{a}: \mu_{1} < \mu_{2}$ Alternative hypothesis: the mean of the final exam scores of the online class is less than the mean of the final exam scores of the face-to-face class.
left-tailed

$This is a normal distribution curve with mean equal to zero. A vertical line near the tail of the curve to the left of zero extends from the axis to the curve. The region under the curve to the left of the line is shaded representing p-value = 0.0011.$

Figure $\PageIndex{3}$.

Reject the null hypothesis

At the 5% level of significance, from the sample data, there is (is/is not) sufficient evidence to conclude that the mean of the final exam scores for the online class is less than the mean of final exam scores of the face-to-face class.

First put the data for each group into two lists (such as L1 and L2). Press STAT. Arrow over to TESTS and press 4:2SampTTest. Make sure Data is highlighted and press ENTER. Arrow down and enter L1 for the first list and L2 for the second list. Arrow down to $\mu_{1}$: and arrow to $\neq \mu_{1}$ (does not equal). Press ENTER. Arrow down to Pooled: No. Press ENTER. Arrow down to Calculate and press ENTER.

Cohen's Standards for Small, Medium, and Large Effect Sizes

Cohen's $d$ is a measure of effect size based on the differences between two means. Cohen’s $d$, named for United States statistician Jacob Cohen, measures the relative strength of the differences between the means of two populations based on sample data. The calculated value of effect size is then compared to Cohen’s standards of small, medium, and large effect sizes.

Cohen's $d$ is the measure of the difference between two means divided by the pooled standard deviation: $d = \dfrac{\bar{x}_{2}-\bar{x}_{2}}{s_{\text{pooled}}}$ where $s_{pooled} = \sqrt{\dfrac{(n_{1}-1)s^{2}_{1} + (n_{2}-1)s^{2}_{2}}{n_{1}+n_{2}-2}}$

Example $\PageIndex{4}$

Calculate Cohen’s d for Example. Is the size of the effect small, medium, or large? Explain what the size of the effect means for this problem.

$\mu_{1} = 4 s_{1} = 1.5 n_{1} = 11$

$\mu_{2} = 3.5 s_{2} = 1 n_{2} = 9$

$d = 0.384$

The effect is small because 0.384 is between Cohen’s value of 0.2 for small effect size and 0.5 for medium effect size. The size of the differences of the means for the two colleges is small indicating that there is not a significant difference between them.

Example $\PageIndex{5}$

Calculate Cohen’s $d$ for Example. Is the size of the effect small, medium or large? Explain what the size of the effect means for this problem.

$d = 0.834$; Large, because 0.834 is greater than Cohen’s 0.8 for a large effect size. The size of the differences between the means of the Final Exam scores of online students and students in a face-to-face class is large indicating a significant difference.

Example 10.2.6

Weighted alpha is a measure of risk-adjusted performance of stocks over a period of a year. A high positive weighted alpha signifies a stock whose price has risen while a small positive weighted alpha indicates an unchanged stock price during the time period. Weighted alpha is used to identify companies with strong upward or downward trends. The weighted alpha for the top 30 stocks of banks in the northeast and in the west as identified by Nasdaq on May 24, 2013 are listed in Table and Table, respectively.

Is there a difference in the weighted alpha of the top 30 stocks of banks in the northeast and in the west? Test at a 5% significance level. Answer the following questions:

Calculate Cohen’s d and interpret it.
Student’s-t
$H_{0}: \mu_{1} = \mu_{2}$ Null hypothesis: the means of the weighted alphas are equal.
$H_{a}: \mu_{1} \neq \mu_{2}$ Alternative hypothesis : the means of the weighted alphas are not equal.
$p\text{-value} = 0.8787$
Do not reject the null hypothesis

$This is a normal distribution curve with mean equal to zero. Both the right and left tails of the curve are shaded. Each tail represents 1/2(p-value) = 0.4394.$

Figure $\PageIndex{4}$.

$d = 0.040$, Very small, because 0.040 is less than Cohen’s value of 0.2 for small effect size. The size of the difference of the means of the weighted alphas for the two regions of banks is small indicating that there is not a significant difference between their trends in stocks.
Data from Graduating Engineer + Computer Careers. Available online at www.graduatingengineer.com
Data from Microsoft Bookshelf .
Data from the United States Senate website, available online at www.Senate.gov (accessed June 17, 2013).
“List of current United States Senators by Age.” Wikipedia. Available online at en.Wikipedia.org/wiki/List_of...enators_by_age (accessed June 17, 2013).
“Sectoring by Industry Groups.” Nasdaq. Available online at www.nasdaq.com/markets/barcha...&base=industry (accessed June 17, 2013).
“Strip Clubs: Where Prostitution and Trafficking Happen.” Prostitution Research and Education, 2013. Available online at www.prostitutionresearch.com/ProsViolPosttrauStress.html (accessed June 17, 2013).
“World Series History.” Baseball-Almanac, 2013. Available online at http://www.baseball-almanac.com/ws/wsmenu.shtml (accessed June 17, 2013).

Two population means from independent samples where the population standard deviations are not known

Random Variable: $\bar{X}_{1} - \bar{X}_{2} =$ the difference of the sampling means
Distribution: Student's t -distribution with degrees of freedom (variances not pooled)

Formula Review

Standard error: \[SE = \sqrt{\dfrac{(s_{1}^{2})}{n_{1}} + \dfrac{(s_{2}^{2})}{n_{2}}}\]

Test statistic ( t -score): \[t = \dfrac{(\bar{x}_{1}-\bar{x}_{2}) - (\mu_{1}-\mu_{2})}{\sqrt{\dfrac{(s_{1})^{2}}{n_{1}} + \dfrac{(s_{2})^{2}}{n_{2}}}}\]

Degrees of freedom:

\[df = \dfrac{\left(\dfrac{(s_{1})^{2}}{n_{1}} + \dfrac{(s_{2})^{2}}{n_{2}}\right)^{2}}{\left(\dfrac{1}{n_{1} - 1}\right)\left(\dfrac{(s_{1})^{2}}{n_{1}}\right)^{2}} + \left(\dfrac{1}{n_{2} - 1}\right)\left(\dfrac{(s_{2})^{2}}{n_{2}}\right)^{2}\]

$s_{1}$ and $s_{2}$ are the sample standard deviations, and n 1 and n 2 are the sample sizes.
$x_{1}$ and $x_{2}$ are the sample means.

OR use the DF to be the smallest of $n_{1}-1$ and $n_{2}-1$

Cohen’s $d$ is the measure of effect size:

\[d = \dfrac{\bar{x}_{1} - \bar{x}_{2}}{s_{\text{pooled}}}\]

\[s_{\text{pooled}} = \sqrt{\dfrac{(n_{1} - 1)s^{2}_{1} + (n_{2} - 1)s^{2}_{2}}{n_{1} + n_{2} - 2}}\]

The domain of the random variable (RV) is not necessarily a numerical set; the domain may be expressed in words; for example, if $X =$ hair color, then the domain is {black, blond, gray, green, orange}.
We can tell what specific value x of the random variable $X$ takes only after performing the experiment.

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Statistics Quizzes

Requirements : Two normally distributed but independent populations, σ is unknown

Hypothesis test

An experiment is conducted to determine whether intensive tutoring (covering a great deal of material in a fixed amount of time) is more effective than paced tutoring (covering less material in the same amount of time). Two randomly chosen groups are tutored separately and then administered proficiency tests. Use a significance level of α < 0.05.

Let μ 1 represent the population mean for the intensive tutoring group and μ 2 represent the population mean for the paced tutoring group.

null hypothesis : H 0 : μ 1 = μ 2

or H 0 : μ 1 – μ 2 = 0

alternative hypothesis : H a : μ 1 > μ 2

or: H a : μ 1 – μ 2 > 0

The degrees of freedom parameter is the smaller of (12 – 1) and (10 – 1), or 9. Because this is a one‐tailed test, the alpha level (0.05) is not divided by two. The next step is to look up t .05,9 in the t‐ table (Table 3 in "Statistics Tables"), which gives a critical value of 1.833. The computed t of 1.166 does not exceed the tabled value, so the null hypothesis cannot be rejected. This test has not provided statistically significant evidence that intensive tutoring is superior to paced tutoring.

Estimate a 90 percent confidence interval for the difference between the number of raisins per box in two brands of breakfast cereal.

The interval is (–2.81, 19.81).

You can be 90 percent confident that Brand A cereal has between 2.81 fewer and 19.81 more raisins per box than Brand B. The fact that the interval contains 0 means that if you had performed a test of the hypothesis that the two population means are different (using the same significance level), you would not have been able to reject the null hypothesis of no difference.

If the two population distributions can be assumed to have the same variance—and, therefore, the same standard deviation— s 1 and s 2 can be pooled together, each weighted by the number of cases in each sample. Although using pooled variance in a t‐ test is generally more likely to yield significant results than using separate variances, it is often hard to know whether the variances of the two populations are equal. For this reason, the pooled variance method should be used with caution. The formula for the pooled estimator of σ 2 is

where s 1 and s 2 are the standard deviations of the two samples and n 1 and n 2 are the sizes of the two samples.

The formula for comparing the means of two populations using pooled variance is

df = n 1 + n 2 – 2

Does right‐ or left‐handedness affect how fast people type? Random samples of students from a typing class are given a typing speed test (words per minute), and the results are compared. Significance level for the test: 0.10. Because you are looking for a difference between the groups in either direction (right‐handed faster than left, or vice versa), this is a two‐tailed test.

or: H 0 : μ 1 – μ 2 = 0

alternative hypothesis : H a : μ 1 ≠ μ 2

or: H a : μ 1 – μ 2 ≠ 0

First, calculate the pooled variance:

Next, calculate the t‐ value:

The degrees‐of ‐ freedom parameter is 16 + 9 – 2, or 23. This test is a two‐tailed one, so you divide the alpha level (0.10) by two. Next, you look up t .05,23 in the t‐ table (Table 3 in "Statistics Tables"), which gives a critical value

of 1.714. This value is larger than the absolute value of the computed t of –1.598, so the null hypothesis of equal population means cannot be rejected. There is no evidence that right‐ or left ‐ handedness has any effect on typing speed.

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Inference for Comparing 2 Population Means (HT for 2 Means, independent samples)

More of the good stuff! We will need to know how to label the null and alternative hypothesis, calculate the test statistic, and then reach our conclusion using the critical value method or the p-value method.

The Test Statistic for a Test of 2 Means from Independent Samples:

[latex]t = \displaystyle \frac{(\bar{x_1} - \bar{x_2}) - (\mu_1 - \mu_2)}{\sqrt{\displaystyle \frac{s_1^2}{n_1} + \displaystyle \frac{s_2^2}{n_2}}}[/latex]

What the different symbols mean:

[latex]n_1[/latex] is the sample size for the first group

[latex]n_2[/latex] is the sample size for the second group

[latex]df[/latex], the degrees of freedom, is the smaller of [latex]n_1 - 1[/latex] and [latex]n_2 - 1[/latex]

[latex]\mu_1[/latex] is the population mean from the first group

[latex]\mu_2[/latex] is the population mean from the second group

[latex]\bar{x_1}[/latex] is the sample mean for the first group

[latex]\bar{x_2}[/latex] is the sample mean for the second group

[latex]s_1[/latex] is the sample standard deviation for the first group

[latex]s_2[/latex] is the sample standard deviation for the second group

[latex]\alpha[/latex] is the significance level , usually given within the problem, or if not given, we assume it to be 5% or 0.05

Assumptions when conducting a Test for 2 Means from Independent Samples:

We do not know the population standard deviations, and we do not assume they are equal
The two samples or groups are independent
Both samples are simple random samples
Both populations are Normally distributed OR both samples are large ([latex]n_1 > 30[/latex] and [latex]n_2 > 30[/latex])

Steps to conduct the Test for 2 Means from Independent Samples:

Identify all the symbols listed above (all the stuff that will go into the formulas). This includes [latex]n_1[/latex] and [latex]n_2[/latex], [latex]df[/latex], [latex]\mu_1[/latex] and [latex]\mu_2[/latex], [latex]\bar{x_1}[/latex] and [latex]\bar{x_2}[/latex], [latex]s_1[/latex] and [latex]s_2[/latex], and [latex]\alpha[/latex]
Identify the null and alternative hypotheses
Calculate the test statistic, [latex]t = \displaystyle \frac{(\bar{x_1} - \bar{x_2}) - (\mu_1 - \mu_2)}{\sqrt{\displaystyle \frac{s_1^2}{n_1} + \displaystyle \frac{s_2^2}{n_2}}}[/latex]
Find the critical value(s) OR the p-value OR both
Apply the Decision Rule
Write up a conclusion for the test

Example 1: Study on the effectiveness of stents for stroke patients [1]

In this study , researchers randomly assigned stroke patients to two groups: one received the current standard care (control) and the other received a stent surgery in addition to the standard care (stent treatment). If the stents work, the treatment group should have a lower average disability score . Do the results give convincing statistical evidence that the stent treatment reduces the average disability from stroke?

Since we are being asked for convincing statistical evidence, a hypothesis test should be conducted. In this case, we are dealing with averages from two samples or groups (the patients with stent treatment and patients receiving the standard care), so we will conduct a Test of 2 Means.

[latex]n_1 = 98[/latex] is the sample size for the first group
[latex]n_2 = 93[/latex] is the sample size for the second group
[latex]df[/latex], the degrees of freedom, is the smaller of [latex]98 - 1 = 97[/latex] and [latex]93 - 1 = 92[/latex], so [latex]df = 92[/latex]
[latex]\bar{x_1} = 2.26[/latex] is the sample mean for the first group
[latex]\bar{x_2} = 3.23[/latex] is the sample mean for the second group
[latex]s_1 = 1.78[/latex] is the sample standard deviation for the first group
[latex]s_2 = 1.78[/latex] is the sample standard deviation for the second group
[latex]\alpha = 0.05[/latex] (we were not told a specific value in the problem, so we are assuming it is 5%)
One additional assumption we extend from the null hypothesis is that [latex]\mu_1 - \mu_2 = 0[/latex]; this means that in our formula, those variables cancel out
[latex]H_{0}: \mu_1 = \mu_2[/latex]
[latex]H_{A}: \mu_1 < \mu_2[/latex]
[latex]t = \displaystyle \frac{(\bar{x_1} - \bar{x_2}) - (\mu_1 - \mu_2)}{\sqrt{\displaystyle \frac{s_1^2}{n_1} + \displaystyle \frac{s_2^2}{n_2}}} = \displaystyle \frac{(2.26 - 3.23) - 0)}{\sqrt{\displaystyle \frac{1.78^2}{98} + \displaystyle \frac{1.78^2}{93}}} = -3.76[/latex]
StatDisk : We can conduct this test using StatDisk. The nice thing about StatDisk is that it will also compute the test statistic. From the main menu above we click on Analysis, Hypothesis Testing, and then Mean Two Independent Samples. From there enter the 0.05 significance, along with the specific values as outlined in the picture below in Step 2. Notice the alternative hypothesis is the [latex]<[/latex] option. Enter the sample size, mean, and standard deviation for each group, and make sure that unequal variances is selected. Now we click on Evaluate. If you check the values, the test statistic is reported in the Step 3 display, as well as the P-Value of 0.00011.
Applying the Decision Rule: We now compare this to our significance level, which is 0.05. If the p-value is smaller or equal to the alpha level, we have enough evidence for our claim, otherwise we do not. Here, [latex]p-value = 0.00011[/latex], which is definitely smaller than [latex]\alpha = 0.05[/latex], so we have enough evidence for the alternative hypothesis…but what does this mean?
Conclusion: Because our p-value of [latex]0.00011[/latex] is less than our [latex]\alpha[/latex] level of [latex]0.05[/latex], we reject [latex]H_{0}[/latex]. We have convincing statistical evidence that the stent treatment reduces the average disability from stroke.

Example 2: Home Run Distances

In 1998, Sammy Sosa and Mark McGwire (2 players in Major League Baseball) were on pace to set a new home run record. At the end of the season McGwire ended up with 70 home runs, and Sosa ended up with 66. The home run distances were recorded and compared (sometimes a player’s home run distance is used to measure their “power”). Do the results give convincing statistical evidence that the home run distances are different from each other? Who would you say “hit the ball farther” in this comparison?

Since we are being asked for convincing statistical evidence, a hypothesis test should be conducted. In this case, we are dealing with averages from two samples or groups (the home run distances), so we will conduct a Test of 2 Means.

[latex]n_1 = 70[/latex] is the sample size for the first group
[latex]n_2 = 66[/latex] is the sample size for the second group
[latex]df[/latex], the degrees of freedom, is the smaller of [latex]70 - 1 = 69[/latex] and [latex]66 - 1 = 65[/latex], so [latex]df = 65[/latex]
[latex]\bar{x_1} = 418.5[/latex] is the sample mean for the first group
[latex]\bar{x_2} = 404.8[/latex] is the sample mean for the second group
[latex]s_1 = 45.5[/latex] is the sample standard deviation for the first group
[latex]s_2 = 35.7[/latex] is the sample standard deviation for the second group
[latex]H_{A}: \mu_1 \neq \mu_2[/latex]
[latex]t = \displaystyle \frac{(\bar{x_1} - \bar{x_2}) - (\mu_1 - \mu_2)}{\sqrt{\displaystyle \frac{s_1^2}{n_1} + \displaystyle \frac{s_2^2}{n_2}}} = \displaystyle \frac{(418.5 - 404.8) - 0)}{\sqrt{\displaystyle \frac{45.5^2}{70} + \displaystyle \frac{35.7^2}{65}}} = 1.95[/latex]
StatDisk : We can conduct this test using StatDisk. The nice thing about StatDisk is that it will also compute the test statistic. From the main menu above we click on Analysis, Hypothesis Testing, and then Mean Two Independent Samples. From there enter the 0.05 significance, along with the specific values as outlined in the picture below in Step 2. Notice the alternative hypothesis is the [latex]\neq[/latex] option. Enter the sample size, mean, and standard deviation for each group, and make sure that unequal variances is selected. Now we click on Evaluate. If you check the values, the test statistic is reported in the Step 3 display, as well as the P-Value of 0.05221.
Applying the Decision Rule: We now compare this to our significance level, which is 0.05. If the p-value is smaller or equal to the alpha level, we have enough evidence for our claim, otherwise we do not. Here, [latex]p-value = 0.05221[/latex], which is larger than [latex]\alpha = 0.05[/latex], so we do not have enough evidence for the alternative hypothesis…but what does this mean?
Conclusion: Because our p-value of [latex]0.05221[/latex] is larger than our [latex]\alpha[/latex] level of [latex]0.05[/latex], we fail to reject [latex]H_{0}[/latex]. We do not have convincing statistical evidence that the home run distances are different.
Follow-up commentary: But what does this mean? There actually was a difference, right? If we take McGwire’s average and subtract Sosa’s average we get a difference of 13.7. What this result indicates is that the difference is not statistically significant; it could be due more to random chance than something meaningful. Other factors, such as sample size, could also be a determining factor (with a larger sample size, the difference may have been more meaningful).
Adapted from the Skew The Script curriculum ( skewthescript.org ), licensed under CC BY-NC-Sa 4.0 ↵

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Module 10: Inference for Means

Hypothesis Test for a Difference in Two Population Means (1 of 2)

Learning outcomes.

Under appropriate conditions, conduct a hypothesis test about a difference between two population means. State a conclusion in context.

Using the Hypothesis Test for a Difference in Two Population Means

Step 1: Determine the hypotheses.

H 0 : μ 1 – μ 2 = 0, which is the same as H 0 : μ 1 = μ 2

The alternative hypothesis, H a , can be any one of the following.

H a : μ 1 – μ 2 < 0, which is the same as H a : μ 1 < μ 2
H a : μ 1 – μ 2 > 0, which is the same as H a : μ 1 > μ 2
H a : μ 1 – μ 2 ≠ 0, which is the same as H a : μ 1 ≠ μ 2

Step 2: Collect the data.

As usual, how we collect the data determines whether we can use it in the inference procedure. We have our usual two requirements for data collection.

Samples must be random to remove or minimize bias.
Samples must be representative of the populations in question.

We use this hypothesis test when the data meets the following conditions.

The two random samples are independent .
The variable is normally distributed in both populations . If this variable is not known, samples of more than 30 will have a difference in sample means that can be modeled adequately by the t-distribution. As we discussed in “Hypothesis Test for a Population Mean,” t-procedures are robust even when the variable is not normally distributed in the population. If checking normality in the populations is impossible, then we look at the distribution in the samples. If a histogram or dotplot of the data does not show extreme skew or outliers, we take it as a sign that the variable is not heavily skewed in the populations, and we use the inference procedure. (Note: This is the same condition we used for the one-sample t-test in “Hypothesis Test for a Population Mean.”)

Step 3: Assess the evidence.

If the conditions are met, then we calculate the t-test statistic. The t-test statistic has a familiar form.

[latex]T=\frac{Observeddifferenceinsamplemeans-Hypothesizeddiferenceinpopulationmeans}{ standarderror}[/latex]

[latex]T=\frac{(\bar{x}_{1}-\bar{x}_{2})-(\mu_{1}-\mu_{2})}{\sqrt{\frac{s_{1}^{2}}{n_{1}}}+\frac{s_{2}^{2}}{n_{2}}}[/latex]

Since the null hypothesis assumes there is no difference in the population means, the expression (μ 1 – μ 2 ) is always zero.

Step 4: State a conclusion.

To state a conclusion, we follow what we have done with other hypothesis tests. We compare our P-value to a stated level of significance.

If the P-value ≤ α, we reject the null hypothesis in favor of the alternative hypothesis.
If the P-value > α, we fail to reject the null hypothesis. We do not have enough evidence to support the alternative hypothesis.

As always, we state our conclusion in context, usually by referring to the alternative hypothesis.

“Context and Calories”

Step 1: Stating the hypotheses.

The null hypothesis is always H 0 : μ 1 – μ 2 = 0, which is the same as H 0 : μ 1 = μ 2 .

The alternative hypothesis H a : μ 1 – μ 2 > 0, which is the same as H a : μ 1 > μ 2 .

Step 2: Collect Data.

As usual, there are two major things to keep in mind when considering the collection of data.

Samples need to be representative of the population in question.
Samples need to be random in order to remove or minimize bias.

Representative Samples?

Random Samples?

Do the data meet the conditions for use of a t-test?

The researchers reported the following sample statistics.

In a sample of 45 women dining with other women, the average number of calories ordered was 850, and the standard deviation was 252.
In a sample of 27 women dining with men, the average number of calories ordered was 719, and the standard deviation was 322.

As noted previously, the researchers reported the following sample statistics.

To compute the t-test statistic, make sure sample 1 corresponds to population 1. Here our population 1 is “women eating with other women.” So x 1 = 850, s 1 = 252, n 1 =45, and so on.

[latex]T=\frac{\bar{x}_{1}-\bar{x}_{2}}{\sqrt{\frac{s_{1}^{2}}{n_{1}}}+\frac{s_{2}^{2}}{n_{2}}}= \frac{850-719}{\sqrt{\frac{252^{2}}{45}+\frac{322^{2}}{27}}}\approx \frac{131}{72.47}\approx 1.81[/latex]

Generic Conclusion

The hypotheses for this test are H 0 : μ 1 – μ 2 = 0 and H a : μ 1 – μ 2 > 0. Since the P-value is less than the significance level (0.0385 < 0.05), we reject H 0 and accept H a .

Conclusion in context

Comment about Conclusions

Both men and women appear to choose larger portions when they eat with women, and both men and women choose smaller portions when they eat in the company of men, according to new research published in the Journal of Applied Social Psychology . The study, conducted among a sample of 127 college students, suggests that both men and women are influenced by unconscious scripts about how to behave in each other’s company. And these scripts change the way men and women eat when they eat together and when they eat apart.

National Health and Nutrition Survey

Concepts in Statistics. Provided by : Open Learning Initiative. Located at : http://oli.cmu.edu . License : CC BY: Attribution

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5.2 - writing hypotheses.

The first step in conducting a hypothesis test is to write the hypothesis statements that are going to be tested. For each test you will have a null hypothesis ($H_0$) and an alternative hypothesis ($H_a$).

When writing hypotheses there are three things that we need to know: (1) the parameter that we are testing (2) the direction of the test (non-directional, right-tailed or left-tailed), and (3) the value of the hypothesized parameter.

At this point we can write hypotheses for a single mean ($\mu$), paired means($\mu_d$), a single proportion ($p$), the difference between two independent means ($\mu_1-\mu_2$), the difference between two proportions ($p_1-p_2$), a simple linear regression slope ($\beta$), and a correlation ($\rho$).
The research question will give us the information necessary to determine if the test is two-tailed (e.g., "different from," "not equal to"), right-tailed (e.g., "greater than," "more than"), or left-tailed (e.g., "less than," "fewer than").
The research question will also give us the hypothesized parameter value. This is the number that goes in the hypothesis statements (i.e., $\mu_0$ and $p_0$). For the difference between two groups, regression, and correlation, this value is typically 0.

Hypotheses are always written in terms of population parameters (e.g., $p$ and $\mu$). The tables below display all of the possible hypotheses for the parameters that we have learned thus far. Note that the null hypothesis always includes the equality (i.e., =).

10.5 Hypothesis Testing for Two Means and Two Proportions

Hypothesis testing for two means and two proportions.

Student Learning Outcomes

The student will select the appropriate distributions to use in each case.
The student will conduct hypothesis tests and interpret the results.
The business section from two consecutive days’ newspapers
Three small packages of multicolored chocolates
Five small packages of peanut butter candies

Increasing Stocks Survey Look at yesterday’s newspaper business section. Conduct a hypothesis test to determine if the proportion of New York Stock Exchange (NYSE) stocks that increased is greater than the proportion of NASDAQ stocks that increased. As randomly as possible, choose 40 NYSE stocks and 32 NASDAQ stocks and complete the following statements.

H 0 : _________
H a : _________
In words, define the random variable.
The distribution to use for the test is _____________.
Calculate the test statistic using your data.
Calculate the p value.
Do you reject or not reject the null hypothesis? Why?
Write a clear conclusion using a complete sentence.

Decreasing Stocks Survey Randomly pick eight stocks from the newspaper. Using two consecutive days’ business sections, test whether the stocks went down, on average, for the second day.

H 0 : ________
H a : ________
Calculate the p value:

Candy Survey Buy three small packages of multicolored chocolates and five small packages of peanut butter candies (same net weight as the multicolored chocolates). Test whether the mean number of candy pieces per package is the same for the two brands.

What distribution should be used for this test?

Shoe Survey Test whether women have, on average, more pairs of shoes than men. Include all forms of sneakers, shoes, sandals, and boots. Use your class as the sample.

The distribution to use for the test is ________________.

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Two Population Calculator

Related: hypothesis testing calculator, confidence interval, hypothesis testing.

When computing confidence intervals for two population means, we are interested in the difference between the population means ($ \mu_1 - \mu_2 $). A confidence interval is made up of two parts, the point estimate and the margin of error. The point estimate of the difference between two population means is simply the difference between two sample means ($ \bar{x}_1 - \bar{x}_2 $). The standard error of $ \bar{x}_1 - \bar{x}_2 $, which is used in computing the margin of error, is given by the formula below.

The formula for the margin of error depends on whether the population standard deviations ($\sigma_1$ and $\sigma_2$) are known or unknown. If the population standard deviations are known, then they are used in the formula. If they are unknown, then the sample standard deviations ($s_1$ and $s_2$)are used in their place. To change from $\sigma$ known to $\sigma$ unknown, click on $\boxed{σ}$ and select $\boxed{s}$ in the Two Population Calculator.

While the formulas for the margin of error in the two population case are similar to those in the one population case, the formula for the degrees of freedom is quite a bit more complicated. Although this formula does seem intimidating at first sight, there is a shortcut to get the answer faster. Notice that the terms $\frac{s_1^2}{n_1}$ and $\frac{s_2^2}{n_2}$ each repeat twice. The terms are actually computed previously when finding the margin of error so they don't need to be calculated again.

If the two population variances are assumed to be equal, an alternative formula for computing the degrees of freedom is used. It's simply df = n1 + n2 - 2. This is a simple extension of the formula for the one population case. In the one population case the degrees of freedom is given by df = n - 1. If we add up the degrees of freedom for the two samples we would get df = (n1 - 1) + (n2 - 1) = n1 + n2 - 2. This formula gives a pretty good approximation of the more complicated formula above.

Just like in hypothesis tests about a single population mean, there are lower-tail, upper-tail and two tailed tests. However, the null and alternative are slightly different. First of all, instead of having mu on the left side of the equality, we have $\mu_1 - \mu_2$. On the right side of the equality, we don't have $\mu_0$, the hypothesized value of the population mean. Instead we have $D_0$, the hypothesized difference between the population means. To switch from a lower tail test to an upper tail or two-tailed test, click on $\boxed{\geq}$ and select $\boxed{\leq}$ or $\boxed{=}$, respectively.

Again, hypothesis testing for a single population mean is very similar to hypothesis testing for two population means. For a single population mean, the test statistics is the difference between mu and mu0 dividied by the standard error. For two population means, the test statistic is the difference between $\bar{x}_1 - \bar{x}_2$ and $D_0$ divided by the standard error. The procedure after computing the test statistic is identical to the one population case. That is, you proceed with the p-value approach or critical value approach in the same exact way.

The calculator above computes confidence intervals and hypothesis tests for the difference between two population means. The simpler version of this is confidence intervals and hypothesis tests for a single population mean. For confidence intervals about a single population mean, visit the Confidence Interval Calculator . For hypothesis tests about a single population mean, visit the Hypothesis Testing Calculator .

Difference in Means Hypothesis Test Calculator

Use the calculator below to analyze the results of a difference in sample means hypothesis test. Enter your sample means, sample standard deviations, sample sizes, hypothesized difference in means, test type, and significance level to calculate your results.

You will find a description of how to conduct a two sample t-test below the calculator.

Define the Two Sample t-test

The difference between the sample means under the null distribution, conducting a hypothesis test for the difference in means.

When two populations are related, you can compare them by analyzing the difference between their means.

A hypothesis test for the difference in samples means can help you make inferences about the relationships between two population means.

Testing for a Difference in Means

For the results of a hypothesis test to be valid, you should follow these steps:

Check Your Conditions

State your hypothesis, determine your analysis plan, analyze your sample, interpret your results.

To use the testing procedure described below, you should check the following conditions:

Independence of Samples - Your samples should be collected independently of one another.
Simple Random Sampling - You should collect your samples with simple random sampling. This type of sampling requires that every occurrence of a value in a population has an equal chance of being selected when taking a sample.
Normality of Sample Distributions - The sampling distributions for both samples should follow the Normal or a nearly Normal distribution. A sampling distribution will be nearly Normal when the samples are collected independently and when the population distribution is nearly Normal. Generally, the larger the sample size, the more normally distributed the sampling distribution. Additionally, outlier data points can make a distribution less Normal, so if your data contains many outliers, exercise caution when verifying this condition.

You must state a null hypothesis and an alternative hypothesis to conduct an hypothesis test of the difference in means.

The null hypothesis is a skeptical claim that you would like to test.

The alternative hypothesis represents the alternative claim to the null hypothesis.

Your null hypothesis and alternative hypothesis should be stated in one of three mutually exclusive ways listed in the table below.

D is the hypothesized difference between the populations' means that you would like to test.

Before conducting a hypothesis test, you must determine a reasonable significance level, α, or the probability of rejecting the null hypothesis assuming it is true. The lower your significance level, the more confident you can be of the conclusion of your hypothesis test. Common significance levels are 10%, 5%, and 1%.

To evaluate your hypothesis test at the significance level that you set, consider if you are conducting a one or two tail test:

Two-tail tests divide the rejection region, or critical region, evenly above and below the null distribution, i.e. to the tails of the null sampling distribution. For example, in a two-tail test with a 5% significance level, your rejection region would be the upper and lower 2.5% of the null distribution. An alternative hypothesis of μ 1 - μ 2 ≠ D requires a two tail test.
One-tail tests place the rejection region entirely on one side of the distribution i.e. to the right or left tail of the null distribution. For example, in a one-tail test evaluating if the actual difference in means, D, is above the null distribution with a 5% significance level, your rejection region would be the upper 5% of the null distribution. μ 1 - μ 2 > D and μ 1 - μ 2 < D alternative hypotheses require one-tail tests.

The graphical results section of the calculator above shades rejection regions blue.

After checking your conditions, stating your hypothesis, determining your significance level, and collecting your sample, you are ready to analyze your hypothesis.

Sample means follow the Normal distribution with the following parameters:

The Difference in the Population Means, D - The true difference in the population means is unknown, but we use the hypothesized difference in the means, D, from the null hypothesis in the calculations.
The Standard Error, SE - The standard error of the difference in the sample means can be computed as follows: SE = (s 1 2 /n 1 + s 2 2 /n 2 ) (1/2) with s 1 being the standard deviation of sample one, n 1 being the sample size of sample one, s 2 being the standard deviation of sample one, and n 2 being the sample size of sample two. The standard error defines how differences in sample means are expected to vary around the null difference in means sampling distribution given the sample sizes and under the assumption that the null hypothesis is true.
The Degrees of Freedom, DF - The degrees of freedom calculation can be estimated as the smaller of n 1 - 1 or n 2 - 1. For more accurate results, use the following formula for the degrees of freedom (DF): DF = (s 1 2 /n 1 + s 2 2 /n 2 ) 2 / ((s 1 2 /n 1 ) 2 / (n 1 - 1) + (s 2 2 /n 2 ) 2 / (n 2 - 1))

In a difference in means hypothesis test, we calculate the probability that we would observe the difference in sample means (x̄ 1 - x̄ 2 ), assuming the null hypothesis is true, also known as the p-value . If the p-value is less than the significance level, then we can reject the null hypothesis.

You can determine a precise p-value using the calculator above, but we can find an estimate of the p-value manually by calculating the t-score, or t-statistic, as follows: t = (x̄ 1 - x̄ 2 - D) / SE

The t-score is a test statistic that tells you how far our observation is from the null hypothesis's difference in means under the null distribution. Using any t-score table, you can look up the probability of observing the results under the null distribution. You will need to look up the t-score for the type of test you are conducting, i.e. one or two tail. A hypothesis test for the difference in means is sometimes known as a two sample mean t-test because of the use of a t-score in analyzing results.

The conclusion of a hypothesis test for the difference in means is always either:

Reject the null hypothesis
Do not reject the null hypothesis

If you reject the null hypothesis, you cannot say that your sample difference in means is the true difference between the means. If you do not reject the null hypothesis, you cannot say that the hypothesized difference in means is true.

A hypothesis test is simply a way to look at evidence and conclude if it provides sufficient evidence to reject the null hypothesis.

Example: Hypothesis Test for the Difference in Two Means

Let’s say you are a manager at a company that designs batteries for smartphones. One of your engineers believes that she has developed a battery that will last more than two hours longer than your standard battery.

Before you can consider if you should replace your standard battery with the new one, you need to test the engineer’s claim. So, you decided to run a difference in means hypothesis test to see if her claim that the new battery will last two hours longer than the standard one is reasonable.

You direct your team to run a study. They will take a sample of 100 of the new batteries and compare their performance to 1,000 of the old standard batteries.

Check the conditions - Your test consists of independent samples . Your team collects your samples using simple random sampling , and you have reason to believe that all your batteries' performances are always close to normally distributed . So, the conditions are met to conduct a two sample t-test.
State Your Hypothesis - Your null hypothesis is that the charge of the new battery lasts at most two hours longer than your standard battery (i.e. μ 1 - μ 2 ≤ 2). Your alternative hypothesis is that the new battery lasts more than two hours longer than the standard battery (i.e. μ 1 - μ 2 > 2).
Determine Your Analysis Plan - You believe that a 1% significance level is reasonable. As your test is a one-tail test, you will evaluate if the difference in mean charge between the samples would occur at the upper 1% of the null distribution.
Analyze Your Sample - After collecting your samples (which you do after steps 1-3), you find the new battery sample had a mean charge of 10.4 hours, x̄ 1 , with a 0.8 hour standard deviation, s 1 . Your standard battery sample had a mean charge of 8.2 hours, x̄ 2 , with a standard deviation of 0.2 hours, s 2 . Using the calculator above, you find that a difference in sample means of 2.2 hours [2 = 10.4 – 8.2] would results in a t-score of 2.49 under the null distribution, which translates to a p-value of 0.72%.
Interpret Your Results - Since your p-value of 0.72% is less than the significance level of 1%, you have sufficient evidence to reject the null hypothesis.

In this example, you found that you can reject your null hypothesis that the new battery design does not result in more than 2 hours of extra battery life. The test does not guarantee that your engineer’s new battery lasts two hours longer than your standard battery, but it does give you strong reason to believe her claim.

Open access
Published: 18 March 2024

Linking undergraduates’ future work self and employability: a moderated mediation model

Yaju Ma 1 ,
Lingyan Hou 2 ,
Wenjing Cai 3 ,
Xiaopei Gao 2 &
Lin Jiang 3

BMC Psychology volume 12 , Article number: 160 ( 2024 ) Cite this article

83 Accesses

Metrics details

The career intentions of students play a crucial role in shaping the growth of the hospitality and tourism industry. Previous research underlines the significance of future work self in predicting outcomes related to one’s career. However, there is limited knowledge regarding the precise ways, timing, and conditions under which the future work self of undergraduate students can enhance their employability.

This paper aims to address the existing research gap by employing career construction theory and self-determination theory to propose a moderated mediation model—i.e., career exploration serves as a mediator and job market knowledge functions as a moderator in the relationship between future work self and employability. We conducted two independent studies (i.e., an experimental study and a time-lagged field study) to test the proposed model. Specifically, in Study 1 we employed an experimental research design to recruit 61 students majoring in tourism management to participate. They were randomly assigned to two scenarios (future work self: high vs. low), and we manipulated different levels of future work self by means of scenario descriptions. In Study 2, we used the time-lagged research design to collect data via submitting questionnaires among 253 Chinese undergraduates who majored in hospitality and tourism at a university in the middle area of China.

The results indicate a positive correlation between undergraduates’ future work self and their employability. Furthermore, this relationship is mediated by a mediator of career exploration. It is important to note that this mediating relationship is also contingent upon the moderator variable of undergraduates’ job market knowledge when considering the impact of career exploration on employability.

The findings contribute to enriching the current understanding of the positive effects of future work self on undergraduates’ desirable outcomes in employability.

Peer Review reports

Introduction

Human resources, as a significant determinant of the growth of the hospitality industry, highlights the importance of undergraduates specializing in the hospitality and tourism industry [ 1 ]. A recent review correspondingly indicated that students’ career intentions determine the need for hospitality and tourism management programs from the perspective of the hospitality and tourism industry [ 2 ]. In light of the uncertainties surrounding employment opportunities and the substantial decrease in job openings during the transition from school to work due to the COVID-19 pandemic [ 3 ], universities are prioritizing the development of undergraduates’ employability to help them gain employment and be successful in their chosen occupations after graduation [ 4 ], especially for educating undergraduates’ majored in hospitality and tourism relevant areas. Given the heightened risk of COVID-19 transmission within the hospitality sector [ 5 ], these undergraduates’ pessimistic perception about the current and future workforce of the tourism industry significantly influences their career attitudes and behaviors toward their future jobs [ 6 , 7 ]. In addition, some research findings have suggested that newcomers, especially graduates, encounter higher occurrences of job mismatching and underemployment [ 8 ]. Consequently, with the goal of increasing undergraduates’ capabilities to transition from school to work, scholars have empirically demonstrated that developing students’ personal characteristics (e.g., proactivity, career adaptability, and knowledge, skills, and attitudes) increases their employability in the future job market [ 9 ].

Scholars argue the necessity for universities to not only focus on improving students’ employability skills but also on fostering their career motivation [ 10 ]. For students who have not yet entered the workplace, their future work selves—i.e., their thoughts and hopes about their future jobs—are the driving force behind their proactive career preparations and early job search behaviors [ 11 ]. Future work self is an important motivational resource for proactive career behavior, individuals with a high level of future work self tend to engage in proactive career behaviors (e.g., career planning) toward a better employment status in the future [ 12 ]. Existing research suggests that motivation is a significant and substantial predictor of student employability [ 13 , 14 ]. However, empirical research on the relationship between students’ future work self, an important motivational resource, and perceived employability is still lacking, and the mechanism of whether and how future work self act on students’ perceived employability remain unclear. Thus, this study aims to address the following research question: whether, how and when undergraduates’ future work self contributes to their perceived employability .

Career construction theory proposes that the concept of future work self serves as a source of motivation, encouraging individuals to invest more effort in career-related behaviors by developing goals and strategies for their future work [ 11 , 15 , 16 ]. Meanwhile, self-determination theory emphasizes that intrinsic motivation (i.e., future work self) is positively associated with key attitudes and behaviors [ 17 ]. Specifically, a strong future work self, which represents a significant intrinsic motivation, enables individuals to adopt an exploratory approach in navigating the uncertainty surrounding their future work. This is achieved by developing and accomplishing self-derived goals and strategies that contribute to positive work and career outcomes [ 18 ]. Therefore, we expect that undergraduates with a salient future work self would have a self-starting motive to explore their career toward boosting their success in the future job market (i.e., employability). Furthermore, as career construction theory’s suggestion that environmental factors can influence an individual’s career development [ 19 ], those who are aware of relevant contextual cues can actively process career-related information and advance their careers [ 20 ]. Accordingly, students who possess comprehensive knowledge about the job market are more likely to be concerned about their future career trajectory and engage in more activities related to career exploration, all in the pursuit of increasing their employability in the future job market, compared to those lacking such knowledge [ 21 ]. Taking all of this into consideration, we propose our hypothesized model in Fig. 1 .

Conceptual model

With this research, we make two main contributions to the literature. First, by linking the positive association between the future work self and undergraduates’ perceived employability, we enrich the current academic understandings of career construction theory [ 19 ] and self-determination theory [ 22 ]. That is, we empirically link the motivational benefits of the future work self to undergraduates’ employability in the future job market. Meanwhile, with the guidance of two distinct theoretical perspectives [ 23 , 24 , 25 ], we propose and test career exploration, a central process in students’ career development [ 26 ], as a key mediator in explaining the relationship between the future work self and employability. In this vein, the results enhance the present comprehension of the influence that the future work self has on employability. Second, by examining job market knowledge as a boundary condition for the indirect relationship between future work self and employability in undergraduates’ career exploration, we included the potential moderator of the acquisition of job-related knowledge in the study, which helped to elucidate the role of future work self in relation to career exploration from the perspective of individual dependency characteristics.

Literature Review and Hypothesis Development

Future work self and perceived employability.

In the age of VUCA, the flexible employment relationship and the blurring of organizational boundaries are making individuals’ careers discontinuous and “boundaryless” [ 27 ]. As career paths become more uncertain, individuals need to engage in increasingly proactive career behaviors to enhance their employability [ 28 ] and access jobs and careers that match their values and needs [ 29 ]. To better prepare for the transition from school to work [ 30 ] and to develop employability in a changing organizational environment, it is critical for students to proactively shape their career future and actively manage their careers [ 31 ].

With reference to the research of Rothwell, Herbert, & Rothwell (2008), we defined the employability as the ability perceived by university students to maintain existing jobs and obtain desired jobs. It is categorized into internal and external dimensions [ 32 ]. Specifically, internal employability refers to the self-evaluation and career value perception felt by employees in the organization, while external employability refers to the willingness and ability of employees to transfer to other organizations, reflecting the value of employees in the external labor market [ 33 ]. Scholars have also suggested that the perception of employability is influenced by self-concept [ 34 ]. In other words, students’ personal traits play a significant role in predicting their future employability perception [ 35 ]. When students become job seekers, they begin to focus more on their future career direction than before and are concerned about their future employability [ 36 ]. We consider that students’ perceived employability while constructing their careers is closely related to their future work selves.

The future work self is a conceptual representation of an individual’s aspirations and hopes for their future self in the work domain [ 24 ]. Compared to the general concept of the “possible self”, the future work self is future-oriented, work-related, and includes the two attributes of salience and elaboration [ 37 ]. Specifically, future work self-salience refers to the degree to which the person’s future work self is clear and imaginable. Future work self-elaboration can be extrapolated from the complex and detailed descriptions of representations of the future self [ 24 ]. According to career construction theory, individuals should consider their past memory, current experience, and future aspirations to make their career behavioral choices [ 19 ]. The future work self potentially expands undergraduates’ aspirations and develops their thinking about future career possibilities [ 16 ], which enables them to proactively prepare for enhancing their employability [ 38 ]. According to self-determination theory, the future work self serves as a motivational career resource that can motivate students to engage in current goal-setting and goal-striving behaviors to achieve a desired future [ 22 , 39 ]. Students who have a higher level of salience and elaboration about their future work self can not only clearly depict the image of their future work [ 40 ] but also take the initiative to learn job-related knowledge and skills needed in career development [ 12 ] to purposefully enhance a series of comprehensive abilities and strengthen their employability [ 41 ]. It follows that the future work self plays a motivating role in increasing students’ perceived employability. Accordingly, we propose the hypothesis as followed:

Hypothesis 1. Future work self is positively related to an undergraduate’s perceived employability.

The mediating role of career exploration

Career exploration is the most essential stage in the career development of students [ 42 ]. Sufficient and proactive exploration contributes to better self-awareness [ 43 ] and greater career-related outcomes. Career exploration involves the exploration of the self and the employment environment, which focuses on carrying out career options, developing abilities, accumulating experiences, and reaching goals [ 26 ]. Students who actively explore their internal and external surroundings can consciously relate their motivations [ 44 ], interests, and abilities to acceptable occupational roles and engage in more goal-oriented behaviors than those who do not [ 45 ]. Through explorations of the self and the environment, students gain a full understanding of their internal characteristics and occupational traits [ 20 ], which helps them seize job opportunities and sustain their competitiveness in the labor market [ 46 ].

Drawing upon career construction theory, we examine the role of the future work self in inspiring career exploration and, subsequently, driving perceived employability among students. Career construction theory posits that career development is an action-oriented process in which individuals establish careers and design their own lives [ 47 ]. Individuals who are willing or flexible to make changes are more likely to engage in career-related activities [ 19 ]. Meanwhile, the career construction model of adaption divides the adaptive construct process into four links: adaptive readiness, adaptability resources, adapting responses and adaptation results [ 19 ]. Career exploration is an important expression of adaptive response, which can help students better cope with career development tasks and changes in the job market environment [ 11 , 23 ]. Specifically, future work self helps students envision desirable futures and highlights discrepancies between current and ideal states. Recognizing these differences enables individuals to visualize the potential challenges they may encounter in pursuing their future career goals and to proactively explore opportunities in their career development process to prepare for these challenges [ 48 ]. Additionally, individuals actively use career resources to adapt to the demands of the dynamic work environment while constructing their careers [ 19 ]. Based on this framework, the future work self enables students to explore and rediscover themselves through the process of identity construction and to actively work toward a future that is consistent with their goals [ 48 ]. As it constitutes the positive possible selves and is a motivational resource in the context of work [ 11 ], studies have suggested that the future work self is positively linked to proactive career behaviors [ 16 , 49 ], such as career planning and skill development. Moreover, when shaping their future work self, students are more likely to seek relevant information and suggestions on environmental clues [ 15 ], which not only provides a clearer image of their occupational self-concept [ 50 ] but also forms “personalized” career planning for constructing themselves [ 51 ]. Therefore, it can be inferred that there is a positive correlation between the future work self and career exploration.

In today’s complex, dynamically shifting labor market, it is crucial to hold a positive perception of employability. A proper assessment of employability can help students proactively choose the right career path that suits their career planning [ 52 ] and cope with work-related challenges and unexpected job transitions [ 53 ]. Scholars have shown that career preparatory behaviors (e.g., exploration) can lead to the development of career-related ideas and attitudes [ 54 ]. According to self-determination theory, when faced with an event that has a significant impact on their career, individuals are motivated to explore new ideas, adjust their behavior and engage with ongoing change to cope with the changing environment and achieve positive career-related outcomes [ 25 ]. When students experience a period of career role transformation and the transition from education to social work, they need to engage in more career exploration activities to actively seek career-relevant experiences [ 55 ], construct their possible selves, and clarify their career path [ 56 ]. Through career exploration, students re-examine themselves, strengthen their skills and formulate strategies to achieve goals [ 57 ], which in turn enhance their employability. This suggests that career exploration is positively associated with perceived employability.

From the standpoint of career construction theory, the concept of the future work self is seen as a valuable source of motivation that empowers individuals to invest greater effort in career-related actions and achieve favorable career results by continually developing and exploring future work objectives and strategies [ 11 , 15 , 16 , 58 ]. Concurrently, according to the self-determination theory, individuals are inclined to actively explore and shape their present roles, leading to positive career outcomes, when they experience strong intrinsic motivation, such as that provided by the future work self [ 18 , 26 ]. Thus, the future work self motivates students to consider their future aspirations, promotes meaningful career exploration behaviors, and thus enhances perceived employability. This means that the future work self is positively associated with career exploration, which, in turn, is positively related to perceived employability. Therefore, we propose the hypothesis as followed:

Hypothesis 2. Career exploration mediates the relationship between future work self and perceived employability.

The moderating role of job market knowledge

From the perspective of career exploration, students exhibit variations in their career motivations. Research indicates that students’ engagement in career preparatory activities is influenced by their personal resources [ 59 ]. These resources can both trigger and constrain career preparatory behaviors [ 59 ], thereby impacting their career development and overall well-being [ 60 ]. As a personally relevant resource, job market knowledge plays an important role in judging the employment situation, making career decisions, and promoting career success [ 61 ]. Job market knowledge refers to the degree to which students are familiar with current labor market developments and future trends [ 62 ]. Research has demonstrated that students who acquire more job market knowledge in their education can be self-motivated to perform specific actions related to career development [ 63 ]. Thus, we posit that the positive impact of career exploration on the perception of employability can be reinforced when students possess a high level of job market knowledge.

According to career construct theory, the environment in which a career develops provides the driving force and guidance for how individuals construct their careers [ 19 ]. Being attentive to contextual cues allows individuals to actively process career-related information and make progress in their careers [ 20 ]. Accordingly, students who have well-equipped job market knowledge are more concerned about their future career direction, engage in more career exploration activities, and become more proactive in developing their careers than those who do not. This well-equipped understanding of the job market serves as a valuable resource, aiding students in better understanding themselves and the external environment [ 64 ], and as a result, enhancing their employability prospects. In contrast, when students possess less job market knowledge, they are blindly optimistic about the labor market and are more reluctant to break out of their comfort zone to carry out career strategies [ 65 ] and enhance their career competencies. As such, due to a lack of awareness of the job market, they exhibit fewer career exploration behaviors and are reserved in boosting their employability [ 66 ]. From the above analysis, we infer that the relationship between career exploration and perceived employability is enhanced when students have a higher level of job market knowledge. Accordingly, we propose the hypothesis as followed:

Hypothesis 3. Job market knowledge moderates the positive relationship between career exploration and perceived employability, such that when an undergraduate’s job market knowledge is higher, this relationship becomes stronger.

The moderated mediation model

Based on the aforementioned hypothesis, we argue that job market knowledge moderates the indirect effects of the future work self on perceived employability through career exploration. Specifically, the influence of the future work self on perceived employability, via career exploration, is amplified when a student possesses a greater level of job market knowledge. According to career construction theory [ 19 ], students with sufficient job market knowledge have a clearer orientation of their future selves in the context of work [ 11 ]. Furthermore, they are more likely to engage in extensive career exploration in order to continuously comprehend their interests [ 67 ], motivations and career aspirations [ 57 ] compared to those who lack such knowledge. Consequently, their perceived employability is enhanced through the creation of a wider spectrum of future possibilities. Conversely, when students possess less job market knowledge, i.e., when they have less knowledge about the future labor market and the current employment situation, they have a vague self-image related to their future jobs [ 38 ] and engage in less career exploration behavior, thus limiting the development of their perceived employability. Drawing on these findings, we propose the hypothesis as followed:

Hypothesi s 4 . Job market knowledge positively moderates the indirect relationship between future work self and perceived employability through career exploration, such that the relationship becomes stronger when an undergraduate’s job market knowledge is higher.

Sampling and procedure

We recruited a total of 65 students majoring in tourism management from a junior college located in central China to participate in a scenario experiment. After excluding four participants who failed the attention check question, we derived data from a valid sample of 61 individuals. Among these participants ( N = 61), 21 were males (34.4%) and 40 were females (65.6%). Their average age was 19.31 years ( SD = 1.36). Participants were randomly assigned to two scenarios (future work self: high vs. low). We manipulated different levels of future work self by means of scenario descriptions. After reading the experimental material on future work self, participants were asked to complete the future work self scale based on the scenario material read above. Immediately following this, participants were asked to report information on other variables (career exploration and employability) and provide demographic information based on their true feelings in the scenario. At the end of the experiment, participants were rewarded with a bonus pack.

Manipulation and measures

We developed experimental materials for future work self based on the research by Strauss and Parker [ 68 ]. The specific content of the experimental material was in the appendix.

Future work self. After reading the experimental materials, participants were asked to complete the 5-item scale developed by Strauss et al. (2012) which was widely used to measure future work self in previous studies [ 11 ]. A representative item was “I am very clear about who and what I want to become in my future work (1 = strongly disagree, 7 = strongly agree). ” The Cronbach’s α was 0.95.

Career exploration. We used the 12-item scale by Stumpf, Colarelli, and Hartman (1983) to access career exploration [ 69 ]. A 7-point scale was used (1 = strongly disagree to 7 = strongly agree) to show the extent to which participants agreed with each item (e.g., “I prepared mentally for my work”). The Cronbach’s α was 0.66.

Perceived employability. We used a 16-item scale developed by Rothwell, Herbert, and Rothwell (2008) to measure student’s employability [ 32 ]. A sample item was “The knowledge and skills I possess are what employers are looking for (1 = strongly disagree, 7 = strongly agree). ” The Cronbach’s α was 0.85.

Manipulation check

The results of the ANOVA indicated that participant’s perceived level of future work self was significantly higher in the high level of future work self condition ( M = 6.49, SD = 0.47) than in the low level of future work self condition ( M = 2.50, SD = 0.45), and the difference between the two conditions was significant (F(1, 59) = 1143.33, p < 0.001, ${\eta }_{p}^{2}$ = 0.95). Thus, we successfully manipulated the future work self.

Hypotheses testing

Descriptive statistics such as mean, standard deviation and correlation coefficients of the variables were given in Table 1 .

First, we conducted a one-way ANOVA with future work self as the independent variable and employability as the dependent variable. The results showed that different levels of future work self had significantly different effects on students’ perceived employability (F(1, 59) = 14.42, p < 0.001, ${\eta }_{p}^{2}$ = 0.20). Specifically, the high level of future work self condition ( M = 3.59, SD = 0.61) led to the higher level of perceived employability compared to the low level of future work self condition ( M = 2.94, SD = 0.91). Therefore, hypothesis 1 was verified.

Second, we used PROCESS to conduct mediation effect test. The results showed that future work self was significantly and positively correlated with career exploration ( b = 0.43, p < 0.01). Career exploration was significantly and positively associated with employability ( b = 0.93, p < 0.001). Bootstrapping results from a sample of 5,000 showed that the indirect effect of future work self on perceived employability via career exploration was 0.40. And the bootstrapped confidence interval [95% CI: (0.12,0.68)] did not include zero. Thus, the mediating effect was significant, supporting hypothesis 2.

Discussion of study 1

A scenario-based experimental approach was employed in Study 1 to test the model. The experimental results showed that the main and mediating effects of the theoretical hypotheses model proposed in this study were valid. The experimental study tested the causal relationships between the independent and mediating variables, and between the dependent and outcome variables, further enhancing the validity of the findings of the study on the mechanism of the influence of future work self on students’ perceived employability. To further test the impact of the moderating variables, Study 2, a questionnaire study, was conducted.

We used the time-lagged research design to collect data via submitting questionnaires among Chinese undergraduates at a university in the middle area of China. One of the authors, as a teaching assistant of a career development course at this university, extended an invitation to the undergraduate students to complete the questionnaires in the classroom. Specifically, the author introduced the topic of this study to 495 students who majored in hospitality and tourism, and asked them to participate in this study. After receiving conformation to participate from 288 undergraduates, the author submitted the online questionnaire to them. Specifically, the questionnaire was uploaded to Wenjuanxing which is an online questionnaire system widely used in academic study in China. The author subsequently shared the questionnaire link, generated by Wenjuanxing, with the students on WeChat, the most prominent Chinese social media platform, and invited them to participate in filling out the questionnaire on WeChat. The time-lagged research design was employed in the current study with an eight-week time interval. At time 1 (T1), these undergraduates were asked to report their future work self, career exploration, job market knowledge, and their demographic information. Eight weeks later, at time 2 (T2), they were asked to rate their perception about their employability. After matching their two sets of responses, a valid sample of 253 undergraduates were used in the study. Of those reporting, participants included 153 males (60.5%) and 100 females (39.5%), and their average age was 21.68 years old ( SD = 3.20). The sample consisted of 70.8% undergraduate students ( N = 179), 18.5% master students ( N = 47) and 10.7% doctoral students ( N = 27).

All the measurement scales are mature English scales, and the translation-back translation method is used to ensure the Chinese versions can accurately express the original concepts [ 70 ]. Before the formal distribution of the questionnaires, we invited undergraduates to take a pre-survey and revised certain items that were inaccurately stated, inappropriate and difficult to understand based on undergraduates’ feedbacks and experts’ advice.

Future work self and career exploration scales used in Study 2 were all consistent with those used in Study 1. The Cronbach’s α for the two scales were 0.91 and 0.92. We employed the three-item scale developed by Hodzic, Ripoll, Lira, and Zenasni (2015) to measure undergraduate’s employability [ 71 ]. The scale combined with the Likert-7 point scoring method has been widely used to evaluate employability ( 1 = strongly disagree to 7 = strongly agree ). A representative item was “in the current job market situation, I think it is possible to find an interesting job.” The Cronbach’s α of this scale was 0.90. Job market knowledge was measured using a three-item scale compiled by Hirschi, Nagy, Baumeler, Johnston, and Spurk (2018) [ 72 ]. Participants were asked to rate a 5-point Likert scale of 1 = strongly disagree to 5 = strongly agree. The representative item was “I have a good knowledge of the job market.” The Cronbach’s α for job market knowledge was 0.96.

We controlled respondents’ age, gender, and education level. According to previous studies, an individual’s employability increases with age [ 73 ] and the level of education [ 74 ]. We also noticed that women appear to be more confident in their employment opportunities when they are unemployed [ 75 ]. Therefore, we statistically controlled these variables for their potential influences. In addition, since researchers have suggested that such environmental-oriented factor as supports from family and schools may exert influences on students’ career-related behaviors and attitudes during school-to-work transition (e.g., perceived employability, and career explorations) [ 76 ], we in the current study controlled career support from school by using the 6-item scale from Sturges et al. (2002) [ 77 ]. A representative item was “I have been given training to help develop my career.” The Cronbach’s α of this scale was 0.77.

Analytical strategy

Firstly, we conducted reliability and validity tests on the data using SPSS 26.0 and AMOS 26.0. Next, we use hierarchical regression to test for mediating and moderating effects with SPSS 26.0 to support the hypotheses. Finally, to further elucidate the indirect effect and the validation of the result, we used the PROCESS procedure by Hayes developed in SPSS [ 78 ] to generate a confidence interval (CI) using a bootstrap program with 5000 sample size.

Confirmatory factor analysis

Before testing hypotheses, we adopted AMOS 26.0 to conduct confirmatory factor analysis on four variables: future work self, career exploration, job market knowledge, and perceived employability to examine the discriminant validity among the variables. As shown in Table 2 , the four-factor model was significantly better than the other competing models and demonstrated a good fit (χ 2 / df = 2.79, CFI = 0.93, RMSEA = 0.08, IFI = 0.93, TLI = 0.91), which indicated that the variables of the measurement model have good discriminant validity.

Collinearity evaluation is also carried out to find out whether there is collinearity in the model. To test collinearity, VIF calculation is needed for each construct. If the VIF score is higher than 5, then the model has collinearity in the educational psychology domain [ 79 ]. The results of the collinearity assessments showed that all VIF scores were less than 4.4, meaning that no pathological collinearity issue existed in the model.

In addition, since the data was collected form only one source (i.e., students), we conducted two methods to identify the potential for common method bias (CMB). We first followed the explanatory factor analysis from Harman (1976) [ 80 ], and the results showed that one factor accounted for 31.25%, which is below the accepted threshold of 40%. Meanwhile, we conducted the test of the one-factor measurement model [ 81 ], generating a poor fit to the data. Taken together, CMB is not a serious problem in our study.

Descriptive statistics

Table 3 presents the results of descriptive statistics such as the mean, standard deviation, and correlation coefficient of the variables. In line with our expectations, the results of Pearson correlation analysis show that future work self is significantly related to career exploration ( r = 0.61, p < 0.01) and is positively related to perceived employability ( r = 0.48, p < 0.01). Moreover, career exploration presents a positive relationship with participants’ perceived employability ( r = 0.51, p < 0.01). These results give initial support for the hypotheses.

In the current paper, the hypotheses were verified by means of the hierarchical regression approach and Bootstrap method. We used SPSS 26.0 and the SPSS macro program PROCESS for data analysis. Table 4 reports our results and the specific analysis results are as follows.

As shown in Model 4 in Table 4 , when participants’ gender, age, and education were controlled for, the positive effect of future work self on undergraduate’s perceived employability is significant ( b = 0.35, p < 0.001). Thus, the results support hypothesis 1.

After considering all the control variables, the results shown in Table 4 Model 2 indicate that future work self is significantly and positively associated with career exploration ( b = 0.421, p < 0.001) As shown in Table 4 Model 5, career exploration positively affects undergraduates’ perceived employability after controlling for future work self ( b = 0.40, p < 0.001). Further, we adopted the Bootstrap method to probe the indirect effects [ 82 ] and set 5000 bootstrapped samples. The results show that the indirect effect of future work self on undergraduate’s perceived employability via career exploration was 0.15 with a 95% confidence interval of [0.08,0.24], and the upper and lower intervals do not contain zero which suggests mediation is indicated. Therefore, hypothesis 2 is supported.

Hypothesis 3 proposes that job market knowledge moderates the relationship between career exploration and perceived employability. As shown in Table 4 Model 7, the interaction term of “career exploration” × “job market knowledge” is significantly and positively related to the perceived employability ( b = 0.14, p < 0.01). To further test the moderating effect of job market knowledge, interaction effects are plotted at high (+ 1 SD) and low (-1 SD) levels of job market knowledge. As showed in Fig. 2 , a simple slope test reveals that career exploration shows a significant tendency to enhance perceived employability at high levels ( b = 0.61, t = 6.40, p < 0.001) and low levels ( b = 0.35, t = 3.88, p < 0.001) of job market knowledge. Thus, hypothesis 3 is supported.

Interaction between career exploration and job market knowledge on perceived employability

Further, we employed the Bootstrap method to test for moderated mediation effect and set 5000 repeated sampling times to obtain an indirect effect and 95% confidence intervals for future work self on perceived employability when job market knowledge is one standard deviation higher or lower than the mean. As can be seen from Table 5 , for undergraduates with less job market knowledge, the indirect effect is 0.13, and the bootstrapped confidence interval (95% CI: [0.01, 0.23]) excludes zero. For undergraduates with medium job market knowledge, the indirect effect was 0.19, with a 95% confidence interval [0.07,0.27], excluding zero. For undergraduates with high job market knowledge, the indirect effect is 0.26, and a 95% confidence interval is [0.12, 0.32] excluding zero. The indirect effect of the difference between two conditions (high and low conditions of job market knowledge) is 0.16 with a 95% confidence interval of [0.01, 0.30]. The interval excludes 0 and the difference is significant. In conclusion, job market knowledge significantly moderates the indirect effect of future work self on undergraduates’ perceived employability. Hypothesis 4 is verified.

Discussion of study 2

Study 2 used a questionnaire method to test the overall model and the data results supported the hypotheses of this study. The findings revealed that the positive relationship between future work self and employability was mediated by career exploration. In addition, job market knowledge positively moderated the indirect relationship between future work self and perceived employability through career exploration, such that the relationship became stronger when an undergraduate’s job market knowledge is higher.

Discussions

Overview of findings.

Aiming at examining how and when undergraduates’ future work self contributes to their perceived employability by utilizing career construction theory and self-determination theory, the current research conducts two independent studies (i.e., an experimental study and a time-lagged field study) to investigate the role of career exploration as a mediator and job market knowledge as a moderator. The results indicate that undergraduates’ future work self is positively related to their perceived employability through increasing their career exploration. In addition, when undergraduates’ job market knowledge is high, their career exploration is more likely to boost their employability, and their future work self is also more likely to improve their employability via enhancing their career exploration.

Theoretical implications

By employing career construction theory and self-determination theory, we provide theoretical implications. First, the results demonstrate that individuals’ positive self-concepts have significant and beneficial effects on the development of undergraduates’ perceived employability [ 83 ]. Specifically, the typical self-concept—i.e., the future work self—highlighting future orientations represents individuals’ strong motivations, perceptions, and behaviors [ 84 ], which effectively guides individuals’ processing of self-relevant information toward a better outcome [ 85 ]. In the domain of career development, since graduates’ future work selves serve as guides or references for developing their career-related abilities, knowledge and skills in the future workplace [ 28 ], undergraduates with a high level of future work self have identity-based motivation toward career planning, skill development, and networking [ 16 ]. That is, their current career-related behavior is consistent with their characteristics and aimed toward the attainment of their desired future, such as being employable [ 86 ]. These findings are aligned with career construction theory [ 19 ], suggesting that positive self-concepts tend to expand undergraduates’ aspirations and develop their thinking about future career possibilities [ 16 ]. It, thus, significantly allows them to redefine their future self and proactively promote their employability [ 38 ]. At the same time, these results are consistent with self-determination theory [ 22 ], which posits that positive intrinsic motivation (e.g., future work self) is an effective predictor of positive job and career outcomes (e.g., employability) [ 17 ].

Meanwhile, by integrating insights from career construction theory and self-determination theory, we propose and find a mediated relationship between the future work self, career exploration, and employability. That is, we contribute to unfolding the black box of behavioral processes by demonstrating that graduates’ future work self could trigger career explorative behaviors toward enhancing employability. Consistent with career construction theory and self-determination theory, which posit that career exploration is a key mediator in explaining the relationship between students’ career motivation and positive career outcomes [ 23 , 24 , 25 ], our findings indicate that individual self-factors with proactive motivations generate internal goals that boost career development behaviors [ 11 , 16 ], which are conducive to positive career outcomes, such as employability [ 18 , 46 ].

Our results also demonstrate that graduates’ job market knowledge positively moderates the relationship between the future work self and employability via career exploration. Specifically, an undergraduate who has a higher level of both future work self and job market knowledge is more likely to engage in career explorative behaviors, which in turn increases his or her employability in the future job market. These findings extend previous studies on treating career-related knowledge and skills as personally relevant resources by demonstrating that obtaining job market knowledge can strengthen individuals’ career behaviors and outcomes (e.g., judging the employment situation, making career decisions, and promoting career success) [ 61 ]. Existing research drawing on career construction theory has suggested that some environmental factors can be used to actively process career-related information and advance individuals’ career development [ 20 ]; that is, well-examined boundary conditions are context-oriented [ 87 ]. In the current study, we go one step further by showing that this motivational process (i.e., the future work self boosts employability by increasing career exploration) is further strengthened in the presence of boundary conditions such as individuals’ knowledge about their future jobs from the perspective of personal-dependent characteristics. Thus, we enrich the current understanding that undergraduates should have a comprehensive thought on their jobs and better understand what competencies, knowledge and skills are necessary to successfully search for a job to sustain their future employability.

Practical implications

According to the findings in the current studies, there are some practical implications that can be provided. First, universities should emphasize on developing undergraduates’ future work self which is amenable to intervention and change [ 16 , 88 ]. Specifically, counseling interventions and strategies can be designed for use with undergraduates facing career transitions, such as offering courses regarding to planning careers in the future and job searching strategies.

On the one hand, according to the mediator of career exploration, we encourage students to develop such proactive behaviors as exploring their future career. For example, after making a list of some possibilities of future jobs, students should engage more in activities of proposed career options (e.g., getting involved in the workplace for valuable insight into a career workday). On the other hand, it is suggested that educators and counselors guide students to identify the discrepancies between their current states and future resource requirements, which in turn stimulate students to take steps to cope with these challenges towards a more promising future. Meanwhile, educators and counselors could provide more external opportunities (e.g., building professional network) to evaluate students’ career interests, enhance students’ career abilities for their future their career choices. In this vein, students could discover the jobs that are available to them after their graduation from universities.

Finally, some job lessons should be integrated into current course designs [ 21 ]. For example, teachers in universities should be trained to employ active learning methods in class towards supporting students to develop a realistic perception of the job market in their community as well as to their own interests and strength. In this vein, undergraduates can be well prepared to make realistic and personal decisions regarding their educational and professional future.

Limitations and future research

Some limitations can be noted in the current research. First, we collected data from one source (i.e., undergraduates). Although we have tested that CMV is not a potential problem in the study, we encouraged future research to invite others to rate undergraduates’ employability (e.g., teachers), which would increase the objectiveness of the results. A related limitation is about broadening the samples. Specifically, since an increasing number of Chinese undergraduates are pursuing a master’s degree, it is highly recommended to replicate our results among postgraduates. Thus, the generalization of the results reported in the current study awaits further empirical examination.

In addition, to further improve the overall robustness and rigor of the current research, it is highly recommended to rate students’ employability by other’s rating. We in this research, theoretically, aim to investigate students’ personal perceptions of their employability in the future job market; thus, we invited them to rate their perceived employability. Empirically, our examinations also precluded the possibility of CMB. However, others’ rating would provide more valid results on students’ employability. For example, following Roessler, Brolin, & Johnson (1990) [ 89 ], researchers in the future could invited employers to assess students’ employability in the real workplace, which may reveal the extent to which students are employable.

The final limitation regards to the research design. Specifically, although we conducted two independent studies in the current research by employing the experimental design (i.e., Study 1) and the time-lagged design (i.e., Study 2), it limits our ability to determine the direction of causality among the variables to the most extent. For instance, the findings may be influenced by opposite or bidirectional relationships due to the potential for undergraduates who have explored their career to enhance their development of future work self. This is because individuals are able to thoroughly examine their internal attributes, which facilitates the formation of a clear self-image in relation to work [ 39 , 90 ]. As research indicated the reciprocal relationship between future work self and career exploration [ 48 ], whether Chinese students’ future work self and their perceived employability are reciprocally related over time. Thus, we suggest that scholars in the future should conduct a more rigorous research design (e.g., the time-lagged research design) to further validate our research findings in terms of reciprocal relationship.

Availability of data and materials

The data resulting from this study is stored and protected according to the Data Management rules of the School of Business and Economics of the Vrije Universiteit Amsterdam, and so are not publicly available. Data are however available from the authors upon reasonable request and with permission of the School of Business and Economics of the Vrije Universiteit Amsterdam.

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Ma, Y., Hou, L., Cai, W. et al. Linking undergraduates’ future work self and employability: a moderated mediation model. BMC Psychol 12 , 160 (2024). https://doi.org/10.1186/s40359-024-01530-1

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10: Hypothesis Testing with Two Samples

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You have learned to conduct hypothesis tests on single means and single proportions. You will expand upon that in this chapter. You will compare two means or two proportions to each other. The general procedure is still the same, just expanded. To compare two means or two proportions, you work with two groups. The groups are classified either as independent or matched pairs. Independent groups consist of two samples that are independent, that is, sample values selected from one population are not related in any way to sample values selected from the other population. Matched pairs consist of two samples that are dependent. The parameter tested using matched pairs is the population mean. The parameters tested using independent groups are either population means or population proportions.

10.1: Prelude to Hypothesis Testing with Two Samples This chapter deals with the following hypothesis tests: Independent groups (samples are independent) Test of two population means. Test of two population proportions. Matched or paired samples (samples are dependent) Test of the two population proportions by testing one population mean of differences.
10.2: Two Population Means with Unknown Standard Deviations The comparison of two population means is very common. A difference between the two samples depends on both the means and the standard deviations. Very different means can occur by chance if there is great variation among the individual samples.
10.3: Two Population Means with Known Standard Deviations Even though this situation is not likely (knowing the population standard deviations is not likely), the following example illustrates hypothesis testing for independent means, known population standard deviations.
10.4: Comparing Two Independent Population Proportions Comparing two proportions, like comparing two means, is common. If two estimated proportions are different, it may be due to a difference in the populations or it may be due to chance. A hypothesis test can help determine if a difference in the estimated proportions reflects a difference in the population proportions.
10.5: Matched or Paired Samples When using a hypothesis test for matched or paired samples, the following characteristics should be present: Simple random sampling is used. Sample sizes are often small. Two measurements (samples) are drawn from the same pair of individuals or objects. Differences are calculated from the matched or paired samples. The differences form the sample that is used for the hypothesis test. Either the matched pairs have differences that come from a population that is normal or the number of difference
10.6: Hypothesis Testing for Two Means and Two Proportions (Worksheet) A statistics Worksheet: The student will select the appropriate distributions to use in each case. The student will conduct hypothesis tests and interpret the results.
10.E: Hypothesis Testing with Two Samples (Exercises) These are homework exercises to accompany the Textmap created for "Introductory Statistics" by OpenStax.

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Comparing Two Means
Hypothesis Testing
Best Example of How to Write a Hypothesis 2024
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Hypothesis Test for a Difference Between 2 Means, Statistics Given
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What is Hypothesis #hypothesis
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hypothesis testing difference between two means part 1
Difference between two means
Hypothesis Testing
Notes 11.1 Hypothesis Tests for Difference Between Two Means: Independent Samples

COMMENTS

10.29: Hypothesis Test for a Difference in Two Population Means (1 of 2
The hypotheses for a difference in two population means are similar to those for a difference in two population proportions. The null hypothesis, H 0, is again a statement of "no effect" or "no difference.". H 0: μ 1 - μ 2 = 0, which is the same as H 0: μ 1 = μ 2. The alternative hypothesis, H a, can be any one of the following.
Hypothesis Test for a Difference in Two Population Means (1 of 2
Step 1: Determine the hypotheses. The hypotheses for a difference in two population means are similar to those for a difference in two population proportions. The null hypothesis, H 0, is again a statement of "no effect" or "no difference.". H 0: μ 1 - μ 2 = 0, which is the same as H 0: μ 1 = μ 2. The alternative hypothesis, H a ...
Hypothesis Test: Difference Between Means
The first step is to state the null hypothesis and an alternative hypothesis. Null hypothesis: μ 1 - μ 2 = 0. Alternative hypothesis: μ 1 - μ 2 ≠ 0. Note that these hypotheses constitute a two-tailed test. The null hypothesis will be rejected if the difference between sample means is too big or if it is too small.
12.3: Difference between Two Means
Figure 12.3.1 shows that the probability value for a two-tailed test is 0.0164. The two-tailed test is used when the null hypothesis can be rejected regardless of the direction of the effect. As shown in Figure 12.3.1, it is the probability of a t < − 2.533 or a t > 2.533. Figure 12.3.1: The two-tailed probability.
10.5: Difference of Two Means
Instead a point estimate of the difference in average 10 mile times for men and women, μw −μm μ w − μ m, can be found using the two sample means: x¯w −x¯m = 102.13 − 87.65 = 14.48 (10.5.1) (10.5.1) x ¯ w − x ¯ m = 102.13 − 87.65 = 14.48. Figure 10.5.1 10.5. 1: A histogram of time for the sample Cherry Blossom Race data.
7.3
We are 99% confident that the difference between the two population mean times is between -2.012 and -0.167. ... The same process for the hypothesis test for one mean can be applied. The test for the mean difference may be referred to as the paired t-test or the test for paired means.
9.2: Comparing Two Independent Population Means (Hypothesis test)
This is a test of two independent groups, two population means. Random variable: ˉXg − ˉXb = difference in the sample mean amount of time girls and boys play sports each day. H0: μg = μb. H0: μg − μb = 0. Ha: μg ≠ μb. Ha: μg − μb ≠ 0.
Writing hypotheses to test the difference of means
Writing hypotheses to test the difference of means. An exercise scientist wanted to test the effectiveness of a new program designed to increase the flexibility of senior citizens. They recruited participants and rated their flexibility according to a standard scale before starting the program. The participants all went through the program and ...
Hypothesis test for difference of means (video)
And our alternative hypothesis, I'll write over here. It's just that it actually does do something. And let's say that it actually has an improvement. So that would mean that we have more weight loss. So if we have the mean of Group One, the population mean of Group One minus the population mean of Group Two should be greater then zero.
Hypothesis Testing
This statistics video explains how to perform hypothesis testing with two sample means using the t-test with the student's t-distribution and the z-test with...
Mean Difference / Difference in Means (MD)
The standardized mean difference (SMD) is a way to measure effect size; it standardizes test results so that they can be compared. For example, a SMD of 0.60 based on outcomes A from one study is equal in comparison to a SMD of 0.60 calculated on the same outcome A in a separate study (SMDs are typically rounded off to two decimal places). The ...
Two Sample t test for Comparing Two Means
Requirements: Two normally distributed but independent populations, σ is unknown. Hypothesis test. Formula: . where and are the means of the two samples, Δ is the hypothesized difference between the population means (0 if testing for equal means), s 1 and s 2 are the standard deviations of the two samples, and n 1 and n 2 are the sizes of the two samples. The number of degrees of freedom for ...
Hypothesis Testing: 2 Means (Independent Samples)
Since we are being asked for convincing statistical evidence, a hypothesis test should be conducted. In this case, we are dealing with averages from two samples or groups (the home run distances), so we will conduct a Test of 2 Means. n1 = 70 n 1 = 70 is the sample size for the first group. n2 = 66 n 2 = 66 is the sample size for the second group.
Hypothesis Test for a Difference in Two Population Means (1 of 2)
The hypotheses for a difference in two population means are similar to those for a difference in two population proportions. The null hypothesis, H 0, is again a statement of "no effect" or "no difference." H 0: μ 1 - μ 2 = 0, which is the same as H 0: μ 1 = μ 2; The alternative hypothesis, H a, can be any one of the following.
Two-sample t test for difference of means
If you switched A and B in the subtraction, you would just get a negative result (similar to how 5 - 3 = 2, but 3 - 5 = -2). Then when you used a t-table or the tcdf() function, you would just have to find the area of the high end of the distribution instead of the area of the low end (or vise versa).
5.2
Alternative Hypothesis. The statement that there is some difference in the population (s), denoted as H a or H 1. When writing hypotheses there are three things that we need to know: (1) the parameter that we are testing (2) the direction of the test (non-directional, right-tailed or left-tailed), and (3) the value of the hypothesized parameter.
10.5 Hypothesis Testing for Two Means and Two Proportions
Conduct a hypothesis test to determine if the proportion of New York Stock Exchange (NYSE) stocks that increased is greater than the proportion of NASDAQ stocks that increased. As randomly as possible, choose 40 NYSE stocks and 32 NASDAQ stocks and complete the following statements. In words, define the random variable.
Two Population Calculator with Steps
The calculator above computes confidence intervals and hypothesis tests for the difference between two population means. The simpler version of this is confidence intervals and hypothesis tests for a single population mean. For confidence intervals about a single population mean, visit the Confidence Interval Calculator.
10.2: Two Population Means with Unknown Standard Deviations
The test comparing two independent population means with unknown and possibly unequal population standard deviations is called the Aspin-Welch t t -test. The degrees of freedom formula was developed by Aspin-Welch. The comparison of two population means is very common.
Difference in Means Hypothesis Test Calculator
Calculate the results of your two sample t-test. Use the calculator below to analyze the results of a difference in sample means hypothesis test. Enter your sample means, sample standard deviations, sample sizes, hypothesized difference in means, test type, and significance level to calculate your results. You will find a description of how to ...
5.3: Difference of Two Means
When considering the difference of two means, there are two common cases: the two samples are paired or they are independent. (There are instances where the data are neither paired nor independent.) The paired case was treated in Section 5.1, where the one-sample methods were applied to the differences from the paired observations.
Linking undergraduates' future work self and employability: a moderated
With this research, we make two main contributions to the literature. First, by linking the positive association between the future work self and undergraduates' perceived employability, we enrich the current academic understandings of career construction theory [] and self-determination theory [].That is, we empirically link the motivational benefits of the future work self to ...
7.7: Comparing Two Independent Population Means
A graph may help to understand this concept. Figure 7.7.2 7.7. 2. Pictured are two distributions of data, X1 X 1 and X2 X 2, with unknown means and standard deviations. The second panel shows the sampling distribution of the newly created random variable ( X¯¯¯¯1 −X¯¯¯¯2 X ¯ 1 − X ¯ 2 ). This distribution is the theoretical ...
10: Hypothesis Testing with Two Samples
This chapter deals with the following hypothesis tests: Independent groups (samples are independent) Test of two population means. Test of two population proportions. Matched or paired samples (samples are dependent) Test of the two population proportions by testing one population mean of differences. 10.2: Two Population Means with Unknown ...

Module 10: Inference for Means

Using the Hypothesis Test for a Difference in Two Population Means

Step 1: Determine the hypotheses.

Step 2: Collect the data.

“Context and Calories”

Comment about Conclusions

National Health and Nutrition Survey

Hypothesis Test: Difference Between Means

State the Hypotheses

Formulate an Analysis Plan

Analyze Sample Data

Interpret Results

Test Your Understanding

9.2: Comparing Two Independent Population Means (Hypothesis test)

Degrees of freedom

Example \(\PageIndex{1}\): Independent groups

Exercise \(\PageIndex{1}\)

Example \(\PageIndex{2}\)

Exercise \(\PageIndex{2}\)

Example \(\PageIndex{3}\)

Cohen's Standards for Small, Medium, and Large Effect Sizes

Example \(\PageIndex{4}\)

Example \(\PageIndex{5}\)

Example 10.2.6

Formula Review

Inference for Comparing 2 Population Means (HT for 2 Means, independent samples)

The Test Statistic for a Test of 2 Means from Independent Samples:

What the different symbols mean:

Assumptions when conducting a Test for 2 Means from Independent Samples:

Steps to conduct the Test for 2 Means from Independent Samples:

Example 1: Study on the effectiveness of stents for stroke patients [1]

Example 2: Home Run Distances

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Linking undergraduates’ future work self and employability: a moderated mediation model

Introduction

Literature Review and Hypothesis Development

The mediating role of career exploration

The moderating role of job market knowledge

The moderated mediation model

Sampling and procedure

Manipulation and measures

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Hypotheses testing

Discussion of study 1

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