ap biology null hypothesis example

AP® Biology

The chi square test: ap® biology crash course.

• The Albert Team
• Last Updated On: March 7, 2024

The statistics section of the AP® Biology exam is without a doubt one of the most notoriously difficult sections. Biology students are comfortable with memorizing and understanding content, which is why this topic seems like the most difficult to master. In this article,  The Chi Square Test: AP® Biology Crash Course , we will teach you a system for how to perform the Chi Square test every time. We will begin by reviewing some topics that you must know about statistics before you can complete the Chi Square test. Next, we will simplify the equation by defining each of the Chi Square variables. We will then use a simple example as practice to make sure that we have learned every part of the equation. Finally, we will finish with reviewing a more difficult question that you could see on your AP® Biology exam .

Null and Alternative Hypotheses

As background information, first you need to understand that a scientist must create the null and alternative hypotheses prior to performing their experiment. If the dependent variable is not influenced by the independent variable , the null hypothesis will be accepted. If the dependent variable is influenced by the independent variable, the data should lead the scientist to reject the null hypothesis . The null and alternative hypotheses can be a difficult topic to describe. Let’s look at an example.

Consider an experiment about flipping a coin. The null hypothesis would be that you would observe the coin landing on heads fifty percent of the time and the coin landing on tails fifty percent of the time. The null hypothesis predicts that you will not see a change in your data due to the independent variable.

The alternative hypothesis for this experiment would be that you would not observe the coins landing on heads and tails an even number of times. You could choose to hypothesize you would see more heads, that you would see more tails, or that you would just see a different ratio than 1:1. Any of these hypotheses would be acceptable as alternative hypotheses.

Defining the Variables

Now we will go over the Chi-Square equation. One of the most difficult parts of learning statistics is the long and confusing equations. In order to master the Chi Square test, we will begin by defining the variables.

This is the Chi Square test equation. You must know how to use this equation for the AP® Bio exam. However, you will not need to memorize the equation; it will be provided to you on the AP® Biology Equations and Formulas sheet that you will receive at the beginning of your examination.

Now that you have seen the equation, let’s define each of the variables so that you can begin to understand it!

•   X 2  :The first variable, which looks like an x, is called chi squared. You can think of chi like x in algebra because it will be the variable that you will solve for during your statistical test. •   ∑ : This symbol is called sigma. Sigma is the symbol that is used to mean “sum” in statistics. In this case, this means that we will be adding everything that comes after the sigma together. •   O : This variable will be the observed data that you record during your experiment. This could be any quantitative data that is collected, such as: height, weight, number of times something occurs, etc. An example of this would be the recorded number of times that you get heads or tails in a coin-flipping experiment. •   E : This variable will be the expected data that you will determine before running your experiment. This will always be the data that you would expect to see if your independent variable does not impact your dependent variable. For example, in the case of coin flips, this would be 50 heads and 50 tails.

The equation should begin to make more sense now that the variables are defined.

Working out the Coin Flip

We have talked about the coin flip example and, now that we know the equation, we will solve the problem. Let’s pretend that we performed the coin flip experiment and got the following data:

Now we put these numbers into the equation:

Heads (55-50) 2 /50= .5

Tails (45-50) 2 /50= .5

Lastly, we add them together.

c 2 = .5+.5=1

Now that we have c 2 we must figure out what that means for our experiment! To do that, we must review one more concept.

Degrees of Freedom and Critical Values

Degrees of freedom is a term that statisticians use to determine what values a scientist must get for the data to be significantly different from the expected values. That may sound confusing, so let’s try and simplify it. In order for a scientist to say that the observed data is different from the expected data, there is a numerical threshold the scientist must reach, which is based on the number of outcomes and a chosen critical value.

Let’s return to our coin flipping example. When we are flipping the coin, there are two outcomes: heads and tails. To get degrees of freedom, we take the number of outcomes and subtract one; therefore, in this experiment, the degree of freedom is one. We then take that information and look at a table to determine our chi-square value:

We will look at the column for one degree of freedom. Typically, scientists use a .05 critical value. A .05 critical value represents that there is a 95% chance that the difference between the data you expected to get and the data you observed is due to something other than chance. In this example, our value will be 3.84.

Coin Flip Results

In our coin flip experiment, Chi Square was 1. When we look at the table, we see that Chi Square must have been greater than 3.84 for us to say that the expected data was significantly different from the observed data. We did not reach that threshold. So, for this example, we will say that we failed to reject the null hypothesis.

The best way to get better at these statistical questions is to practice. Next, we will go through a question using the Chi Square Test that you could see on your AP® Bio exam.

AP® Biology Exam Question

This question was adapted from the 2013 AP® Biology exam.

In an investigation of fruit-fly behavior, a covered choice chamber is used to test whether the spatial distribution of flies is affected by the presence of a substance placed at one end of the chamber. To test the flies’ preference for glucose, 60 flies are introduced into the middle of the choice chamber at the insertion point. A ripe banana is placed at one end of the chamber, and an unripe banana is placed at the other end. The positions of flies are observed and recorded after 1 minute and after 10 minutes. Perform a Chi Square test on the data for the ten minute time point. Specify the null hypothesis and accept or reject it.

Okay, we will begin by identifying the null hypothesis . The null hypothesis would be that the flies would be evenly distributed across the three chambers (ripe, middle, and unripe).

Next, we will perform the Chi-Square test just like we did in the heads or tails experiment. Because there are three conditions, it may be helpful to use this set up to organize yourself:

Ok, so we have a Chi Square of 48.9. Our degrees of freedom are 3(ripe, middle, unripe)-1=2. Let’s look at that table above for a confidence variable of .05. You should get a value of 5.99. Our Chi Square value of 48.9 is much larger than 5.99 so in this case we are able to reject the null hypothesis. This means that the flies are not randomly assorting themselves, and the banana is influencing their behavior.

The Chi Square test is something that takes practice. Once you learn the system of solving these problems, you will be able to solve any Chi Square problem using the exact same method every time! In this article, we have reviewed the Chi Square test using two examples. If you are still interested in reviewing the bio-statistics that will be on your AP® Biology Exam, please check out our article The Dihybrid Cross Problem: AP® Biology Crash Course . Let us know how studying is going and if you have any questions!

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Null hypothesis

Null hypothesis n., plural: null hypotheses [nʌl haɪˈpɒθɪsɪs] Definition: a hypothesis that is valid or presumed true until invalidated by a statistical test

Table of Contents

Null Hypothesis Definition

Null hypothesis is defined as “the commonly accepted fact (such as the sky is blue) and researcher aim to reject or nullify this fact”.

More formally, we can define a null hypothesis as “a statistical theory suggesting that no statistical relationship exists between given observed variables” .

In biology , the null hypothesis is used to nullify or reject a common belief. The researcher carries out the research which is aimed at rejecting the commonly accepted belief.

What Is a Null Hypothesis?

A hypothesis is defined as a theory or an assumption that is based on inadequate evidence. It needs and requires more experiments and testing for confirmation. There are two possibilities that by doing more experiments and testing, a hypothesis can be false or true. It means it can either prove wrong or true (Blackwelder, 1982).

For example, Susie assumes that mineral water helps in the better growth and nourishment of plants over distilled water. To prove this hypothesis, she performs this experiment for almost a month. She watered some plants with mineral water and some with distilled water.

In a hypothesis when there are no statistically significant relationships among the two variables, the hypothesis is said to be a null hypothesis. The investigator is trying to disprove such a hypothesis. In the above example of plants, the null hypothesis is:

There are no statistical relationships among the forms of water that are given to plants for growth and nourishment.

Usually, an investigator tries to prove the null hypothesis wrong and tries to explain a relation and association between the two variables.

An opposite and reverse of the null hypothesis are known as the alternate hypothesis . In the example of plants the alternate hypothesis is:

There are statistical relationships among the forms of water that are given to plants for growth and nourishment.

The example below shows the difference between null vs alternative hypotheses:

Alternate Hypothesis: The world is round Null Hypothesis: The world is not round.

Copernicus and many other scientists try to prove the null hypothesis wrong and false. By their experiments and testing, they make people believe that alternate hypotheses are correct and true. If they do not prove the null hypothesis experimentally wrong then people will not believe them and never consider the alternative hypothesis true and correct.

The alternative and null hypothesis for Susie’s assumption is:

• Null Hypothesis: If one plant is watered with distilled water and the other with mineral water, then there is no difference in the growth and nourishment of these two plants.
• Alternative Hypothesis:  If one plant is watered with distilled water and the other with mineral water, then the plant with mineral water shows better growth and nourishment.

The null hypothesis suggests that there is no significant or statistical relationship. The relation can either be in a single set of variables or among two sets of variables.

Most people consider the null hypothesis true and correct. Scientists work and perform different experiments and do a variety of research so that they can prove the null hypothesis wrong or nullify it. For this purpose, they design an alternate hypothesis that they think is correct or true. The null hypothesis symbol is H 0 (it is read as H null or H zero ).

Why is it named the “Null”?

The name null is given to this hypothesis to clarify and explain that the scientists are working to prove it false i.e. to nullify the hypothesis. Sometimes it confuses the readers; they might misunderstand it and think that statement has nothing. It is blank but, actually, it is not. It is more appropriate and suitable to call it a nullifiable hypothesis instead of the null hypothesis.

Why do we need to assess it? Why not just verify an alternate one?

In science, the scientific method is used. It involves a series of different steps. Scientists perform these steps so that a hypothesis can be proved false or true. Scientists do this to confirm that there will be any limitation or inadequacy in the new hypothesis. Experiments are done by considering both alternative and null hypotheses, which makes the research safe. It gives a negative as well as a bad impact on research if a null hypothesis is not included or a part of the study. It seems like you are not taking your research seriously and not concerned about it and just want to impose your results as correct and true if the null hypothesis is not a part of the study.

Development of the Null

In statistics, firstly it is necessary to design alternate and null hypotheses from the given problem. Splitting the problem into small steps makes the pathway towards the solution easier and less challenging. how to write a null hypothesis?

Writing a null hypothesis consists of two steps:

• Firstly, initiate by asking a question.
• Secondly, restate the question in such a way that it seems there are no relationships among the variables.

In other words, assume in such a way that the treatment does not have any effect.

The usual recovery duration after knee surgery is considered almost 8 weeks.

A researcher thinks that the recovery period may get elongated if patients go to a physiotherapist for rehabilitation twice per week, instead of thrice per week, i.e. recovery duration reduces if the patient goes three times for rehabilitation instead of two times.

Step 1: Look for the problem in the hypothesis. The hypothesis either be a word or can be a statement. In the above example the hypothesis is:

“The expected recovery period in knee rehabilitation is more than 8 weeks”

Step 2: Make a mathematical statement from the hypothesis. Averages can also be represented as μ, thus the null hypothesis formula will be.

In the above equation, the hypothesis is equivalent to H1, the average is denoted by μ and > that the average is greater than eight.

Step 3: Explain what will come up if the hypothesis does not come right i.e., the rehabilitation period may not proceed more than 08 weeks.

There are two options: either the recovery will be less than or equal to 8 weeks.

H 0 : μ ≤ 8

In the above equation, the null hypothesis is equivalent to H 0 , the average is denoted by μ and ≤ represents that the average is less than or equal to eight.

What will happen if the scientist does not have any knowledge about the outcome?

Problem: An investigator investigates the post-operative impact and influence of radical exercise on patients who have operative procedures of the knee. The chances are either the exercise will improve the recovery or will make it worse. The usual time for recovery is 8 weeks.

Step 1: Make a null hypothesis i.e. the exercise does not show any effect and the recovery time remains almost 8 weeks.

H 0 : μ = 8

In the above equation, the null hypothesis is equivalent to H 0 , the average is denoted by μ, and the equal sign (=) shows that the average is equal to eight.

Step 2: Make the alternate hypothesis which is the reverse of the null hypothesis. Particularly what will happen if treatment (exercise) makes an impact?

In the above equation, the alternate hypothesis is equivalent to H1, the average is denoted by μ and not equal sign (≠) represents that the average is not equal to eight.

Significance Tests

To get a reasonable and probable clarification of statistics (data), a significance test is performed. The null hypothesis does not have data. It is a piece of information or statement which contains numerical figures about the population. The data can be in different forms like in means or proportions. It can either be the difference of proportions and means or any odd ratio.

The following table will explain the symbols:

P-value is the chief statistical final result of the significance test of the null hypothesis.

• P-value = Pr(data or data more extreme | H 0 true)
• | = “given”
• Pr = probability
• H 0 = the null hypothesis

The first stage of Null Hypothesis Significance Testing (NHST) is to form an alternate and null hypothesis. By this, the research question can be briefly explained.

Null Hypothesis = no effect of treatment, no difference, no association Alternative Hypothesis = effective treatment, difference, association

When to reject the null hypothesis?

Researchers will reject the null hypothesis if it is proven wrong after experimentation. Researchers accept null hypothesis to be true and correct until it is proven wrong or false. On the other hand, the researchers try to strengthen the alternate hypothesis. The binomial test is performed on a sample and after that, a series of tests were performed (Frick, 1995).

Step 1: Evaluate and read the research question carefully and consciously and make a null hypothesis. Verify the sample that supports the binomial proportion. If there is no difference then find out the value of the binomial parameter.

Show the null hypothesis as:

H 0 :p= the value of p if H 0 is true

To find out how much it varies from the proposed data and the value of the null hypothesis, calculate the sample proportion.

Step 2: In test statistics, find the binomial test that comes under the null hypothesis. The test must be based on precise and thorough probabilities. Also make a list of pmf that apply, when the null hypothesis proves true and correct.

When H 0 is true, X~b(n, p)

N = size of the sample

P = assume value if H 0 proves true.

Step 3: Find out the value of P. P-value is the probability of data that is under observation.

Rise or increase in the P value = Pr(X ≥ x)

X = observed number of successes

P value = Pr(X ≤ x).

Step 4: Demonstrate the findings or outcomes in a descriptive detailed way.

• Sample proportion
• The direction of difference (either increases or decreases)

Perceived Problems With the Null Hypothesis

Variable or model selection and less information in some cases are the chief important issues that affect the testing of the null hypothesis. Statistical tests of the null hypothesis are reasonably not strong. There is randomization about significance. (Gill, 1999) The main issue with the testing of the null hypothesis is that they all are wrong or false on a ground basis.

There is another problem with the a-level . This is an ignored but also a well-known problem. The value of a-level is without a theoretical basis and thus there is randomization in conventional values, most commonly 0.q, 0.5, or 0.01. If a fixed value of a is used, it will result in the formation of two categories (significant and non-significant) The issue of a randomized rejection or non-rejection is also present when there is a practical matter which is the strong point of the evidence related to a scientific matter.

The P-value has the foremost importance in the testing of null hypothesis but as an inferential tool and for interpretation, it has a problem. The P-value is the probability of getting a test statistic at least as extreme as the observed one.

The main point about the definition is: Observed results are not based on a-value

Moreover, the evidence against the null hypothesis was overstated due to unobserved results. A-value has importance more than just being a statement. It is a precise statement about the evidence from the observed results or data. Similarly, researchers found that P-values are objectionable. They do not prefer null hypotheses in testing. It is also clear that the P-value is strictly dependent on the null hypothesis. It is computer-based statistics. In some precise experiments, the null hypothesis statistics and actual sampling distribution are closely related but this does not become possible in observational studies.

Some researchers pointed out that the P-value is depending on the sample size. If the true and exact difference is small, a null hypothesis even of a large sample may get rejected. This shows the difference between biological importance and statistical significance. (Killeen, 2005)

Another issue is the fix a-level, i.e., 0.1. On the basis, if a-level a null hypothesis of a large sample may get accepted or rejected. If the size of simple is infinity and the null hypothesis is proved true there are still chances of Type I error. That is the reason this approach or method is not considered consistent and reliable. There is also another problem that the exact information about the precision and size of the estimated effect cannot be known. The only solution is to state the size of the effect and its precision.

Null Hypothesis Examples

Here are some examples:

Example 1: Hypotheses with One Sample of One Categorical Variable

Among all the population of humans, almost 10% of people prefer to do their task with their left hand i.e. left-handed. Let suppose, a researcher in the Penn States says that the population of students at the College of Arts and Architecture is mostly left-handed as compared to the general population of humans in general public society. In this case, there is only a sample and there is a comparison among the known population values to the population proportion of sample value.

• Research Question: Do artists more expected to be left-handed as compared to the common population persons in society?
• Response Variable: Sorting the student into two categories. One category has left-handed persons and the other category have right-handed persons.
• Form Null Hypothesis: Arts and Architecture college students are no more predicted to be lefty as compared to the common population persons in society (Lefty students of Arts and Architecture college population is 10% or p= 0.10)

Example 2: Hypotheses with One Sample of One Measurement Variable

A generic brand of antihistamine Diphenhydramine making medicine in the form of a capsule, having a 50mg dose. The maker of the medicines is concerned that the machine has come out of calibration and is not making more capsules with the suitable and appropriate dose.

• Research Question: Does the statistical data recommended about the mean and average dosage of the population differ from 50mg?
• Response Variable: Chemical assay used to find the appropriate dosage of the active ingredient.
• Null Hypothesis: Usually, the 50mg dosage of capsules of this trade name (population average and means dosage =50 mg).

Example 3: Hypotheses with Two Samples of One Categorical Variable

Several people choose vegetarian meals on a daily basis. Typically, the researcher thought that females like vegetarian meals more than males.

• Research Question: Does the data recommend that females (women) prefer vegetarian meals more than males (men) regularly?
• Response Variable: Cataloguing the persons into vegetarian and non-vegetarian categories. Grouping Variable: Gender
• Null Hypothesis: Gender is not linked to those who like vegetarian meals. (Population percent of women who eat vegetarian meals regularly = population percent of men who eat vegetarian meals regularly or p women = p men).

Example 4: Hypotheses with Two Samples of One Measurement Variable

Nowadays obesity and being overweight is one of the major and dangerous health issues. Research is performed to confirm that a low carbohydrates diet leads to faster weight loss than a low-fat diet.

• Research Question: Does the given data recommend that usually, a low-carbohydrate diet helps in losing weight faster as compared to a low-fat diet?
• Response Variable: Weight loss (pounds)
• Explanatory Variable: Form of diet either low carbohydrate or low fat
• Null Hypothesis: There is no significant difference when comparing the mean loss of weight of people using a low carbohydrate diet to people using a diet having low fat. (population means loss of weight on a low carbohydrate diet = population means loss of weight on a diet containing low fat).

Example 5: Hypotheses about the relationship between Two Categorical Variables

A case-control study was performed. The study contains nonsmokers, stroke patients, and controls. The subjects are of the same occupation and age and the question was asked if someone at their home or close surrounding smokes?

• Research Question: Did second-hand smoke enhance the chances of stroke?
• Variables: There are 02 diverse categories of variables. (Controls and stroke patients) (whether the smoker lives in the same house). The chances of having a stroke will be increased if a person is living with a smoker.
• Null Hypothesis: There is no significant relationship between a passive smoker and stroke or brain attack. (odds ratio between stroke and the passive smoker is equal to 1).

Example 6: Hypotheses about the relationship between Two Measurement Variables

A financial expert observes that there is somehow a positive and effective relationship between the variation in stock rate price and the quantity of stock bought by non-management employees

• Response variable- Regular alteration in price
• Explanatory Variable- Stock bought by non-management employees
• Null Hypothesis: The association and relationship between the regular stock price alteration ($) and the daily stock-buying by non-management employees ($) = 0.

Example 7: Hypotheses about comparing the relationship between Two Measurement Variables in Two Samples

• Research Question: Is the relation between the bill paid in a restaurant and the tip given to the waiter, is linear? Is this relation different for dining and family restaurants?
• Explanatory Variable- total bill amount
• Response Variable- the amount of tip
• Null Hypothesis: The relationship and association between the total bill quantity at a family or dining restaurant and the tip, is the same.

Try to answer the quiz below to check what you have learned so far about the null hypothesis.

Choose the best answer. 

Send Your Results (Optional)

Time is Up!

• Blackwelder, W. C. (1982). “Proving the null hypothesis” in clinical trials. Controlled Clinical Trials , 3(4), 345–353.
• Frick, R. W. (1995). Accepting the null hypothesis. Memory & Cognition, 23(1), 132–138.
• Gill, J. (1999). The insignificance of null hypothesis significance testing. Political Research Quarterly , 52(3), 647–674.
• Killeen, P. R. (2005). An alternative to null-hypothesis significance tests. Psychological Science, 16(5), 345–353.

©BiologyOnline.com. Content provided and moderated by Biology Online Editors.

Last updated on June 16th, 2022

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AP Biology : Perform chi squared test

Study concepts, example questions & explanations for ap biology, all ap biology resources, example questions, example question #1 : perform chi squared test.

In a population of flowers, red is dominant to white. A true-breeding white flower is crossed with a heterozygous flower. Determine the expected ratios of this cross. Given observed values: 63 white flowers, 37 red flowers, determine:

1) The chi squared value

2) The degrees of freedom

3) The p-value

1) 6.76    

3) p>0.025

1) 14.44    

3) p>0.005

1) 3.35    

Example Question #2 : Perform Chi Squared Test

In a chi squared test, the greater the difference between the observed and expected frequencies of a trait, the ___ the p-value.

more positive

more negative

 A larger difference between observed and expected values will result in a larger chi square value. Using the critical values table, for a given degrees of freedom, as chi square value increases, p-value decreases.

Example Question #3 : Perform Chi Squared Test

If the p-value determined by a chi-square test is low, which is possible

The null hypothesis is rejected

The null hypothesis is accepted

The chi square value is also low

The alternative hypothesis is rejected

When the p-value is equal to or below the significance level (alpha), the null hypothesis is rejected. For the p-value to be low, the chi square value would need to be large (large difference between observed and expected values). The null hypothesis would not be accepted (under no circumstance is a null hypothesis “accepted”; onlay rejected or failed to reject). The alternative hypothesis would be accepted if the p-value is equal to or below the significance level (alpha).

Which of the following chi square values would likely result in rejection of the null hypothesis?

A larger difference between observed and expected values will result in a larger chi square value. Using the critical values table, for a given degrees of freedom, as chi square value increases, p-value decreases. A small p-value results in rejection of the null hypothesis.

Example Question #5 : Perform Chi Squared Test

Which is not a possible critical value of a chi square test?

If the chi square value is larger than the critical value at a given level of significance, what can be stated?

The null hypothesis fails to be rejected

If the chi square value is large, this indicates a large difference between the observed and expected values. This will subsequently result in a small p-value when using the critical value table. When the p-value is equal to or smaller than the significance level (alpha), the null hypothesis is rejected.

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7.5 Hardy-Weinberg Equilibrium

4 min read • january 11, 2023

Samantha Himegarner

Jed Quiaoit

The Hardy-Weinberg equilibrium is a theoretical model that describes how allele frequencies in a population will remain constant over time in the absence of certain influences, such as natural selection , genetic drift , mutation , and migration .

In general, The model assumes that the population is large, randomly mating, and not subject to any influences that would affect the frequency of alleles. (We'll dive deeper into these conditions later!) 🧉

Under these conditions, the frequency of alleles in the population will remain constant from generation to generation. Applications-wise, the Hardy-Weinberg equilibrium provides a baseline against which actual population data can be compared to assess whether evolution is occurring.

The Five Conditions

The Hardy-Weinberg equilibrium is a model in which allele frequencies will not change if these five conditions exist: 5️⃣

No mutation : new alleles are not added to the gene pool , so the existing alleles are passed on to the next generation.

No selection : no natural or artificial selection is acting on the population, so no allele is favored over another.

No gene flow : there is no movement of individuals between populations or migration that would introduce new alleles or remove existing alleles.

Infinite population size : the population is so large that random fluctuations in allele frequency ( genetic drift ) do not occur.

Random mating : mating is completely random and not influenced by any genetic or environmental factors.

Of course, each of these five things are impossible for a natural situation. A population will never be infinite, and mutation is not something that can be controlled! That’s why this simply acts as a model to calculate genotypic ratios under these assumptions (which, of course, are false).

Calculated Ratios

The calculated ratios under this control population can serve as a null hypothesis for evolution. The allele frequencies in a population are calculated from the genotypic ratios . Here’s how:

Although this equation may look like a lot, it actually isn’t too bad. If you think about it, we only have two variables! The ‘p’ in the equations can represent the frequency of allele 1 in the population, and the ‘q’ will represent the frequency of the second allele. These numbers will be decimals, as they will represent the percentage of the population that has each phenotype (which is why it adds up to 1!). 🎆

For the first equation, p2 represents the frequency of a homozygous dominant genotype, 2pq represents the frequency of a heterozygous genotype , and q2 represents the frequency of the homozygous recessive genotype.  Keeping all that in mind, the first thing to do in the process of calculating is to determine the frequency of the recessive genotype. This is the only genotype that can be determined from the phenotypic ratios (a dominant phenotype could be either homozygous dominant or heterozygous).

The frequency of the recessive genotype will be equal to q2. Therefore, q can be found by taking the square root of that number. Then, subtract this number from 1 to find p, or the frequency of the dominant phenotype in a population.

Once the variable p is found, it can be put into the first equation to find the genotypic ratios of a population.

Image courtesy of Giphy .

For example, if there is a population of 100 birds, 84 of them have white feathers and 16 of them have grey feathers. Here’s how we would solve:

Example Problem

A rare genetic disorder is caused by a recessive allele, "a", in a population. The frequency of the allele in the population is 0.02. Using the Hardy-Weinberg equilibrium , calculate the frequency of the dominant allele, "A", and the frequency of individuals who are homozygous recessive (aa) and heterozygous (Aa) for the disorder. Explain how you arrived at your calculations and what assumptions of the Hardy-Weinberg equilibrium are being made in this scenario. 🍞

To solve this question, you would first assign the frequency of the recessive allele, "a", as "q" and the frequency of the dominant allele, "A", as "p". Since "A" and "a" are codominant, the sum of p and q is equal to 1.

The equation for calculating Hardy-Weinberg equilibrium is p^2 + 2pq + q^2 = 1.

We know that q = 0.02, so we can substitute that into the equation, to calculate p:

p = 1 - q = 1 - 0.02 = 0.98

The frequency of individuals who are homozygous recessive (aa) is: q^2 = 0.02^2 = 0.0004, or 0.04%.

The frequency of individuals who are heterozygous (Aa) is: 2pq = 2(0.98)(0.02) = 0.0392, or 3.92%.

With this problem, we are making the assumptions of the Hardy-Weinberg equilibrium such as:

There is no mutation

No selection

No gene flow

Infinite population size

Random mating .

It is also good to remember that this equilibrium is theoretical and only holds true under these assumptions, real world populations almost never meet all these criteria.

Key Terms to Review ( 16 )

Allele Frequencies

Artificial Selection

Genetic Drift

Genotypic Ratios

Hardy-Weinberg Equilibrium

Heterozygous Genotype

Homozygous Dominant Genotype

Homozygous Recessive Genotype

Infinite Population Size

Natural Selection

Phenotypic Ratios

Random Mating

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Excel at Science

• Apr 29, 2020

AP Biology - How to Conduct Chi Square Tests

Chi-square tests are fair game for this year’s revised AP Biology exam and I’ve had multiple students asking about how to perform and use them. Because Chi-square tests are typically used one experimental data, this is likely to show up as part of Question #1 on the exam.

First, let’s clarify the purpose of a Chi-square test. It is a statistical test that determines whether there is a significant difference between different groups in an experiment (for instance, three groups of plants grown in different conditions). The null hypothesis, or default case, is that there is no difference between groups. The alternative hypothesis is that there is a difference between groups.

Of course, typically in an experiment, the goal is to show that there is indeed a difference between different treatment groups . For example, let’s say you put a tomato plant A near sunlight and another tomato plant B in the dark, all other factors held the same. The goal is to show whether there is a difference in growth between the plants after one month. As the experimenter, do you hope there is a difference in growth? Yes, of course you do. Then you can say that you have found this factor (sunlight) to be associated with plant growth.

Conduct a Chi-square Test

The Chi-square test computes the difference between experimental ( observed ) and expected values for the different groups involved. These calculations yield the Chi-square. That value is then compared to a critical value. We can find this value on the probability table provided on the exam using both the degrees of freedom (d.f., will be explained later) and the level of error (usually 0.05).

Below is an example of a probability table. If the experiment of interest has 3 groups and we aim for an error level of 0.05, what is the critical value?

Answer: It is 5.99, because the degrees of freedom is (3 - 1) = 2, and the error level is 0.05.

Drawing conclusions from the Chi-square test:

If the Chi-square value is greater than the critical value, we reject the null hypothesis and say the groups are significantly different.

If the Chi-square value is less than the critical value, we fail to reject the null hypothesis and say there is not a significant difference.

The diagram below summarizes the steps in a Chi-square test:

Practice Problems:

Now that we have walked through how to conduct Chi-square tests, it’s time to use them. It’s important to understand both how to do the tests and how to interpret the results of the test. Here are some good practice problems:

2013 #1 parts (c) and (d)

Chi Square Practice Worksheet

Note: this worksheet does not provide the probability table. You can easily find one on Google

Please comment below if you have any questions as you go through the problems. Happy studying!

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1.1 The Science of Biology

Learning objectives.

In this section, you will explore the following questions:

• What are the characteristics shared by the natural sciences?
• What are the steps of the scientific method?

Connection for AP ® courses

Biology is the science that studies living organisms and their interactions with one another and with their environment. The process of science attempts to describe and understand the nature of the universe by rational means. Science has many fields; those fields related to the physical world, including biology, are considered natural sciences. All of the natural sciences follow the laws of chemistry and physics. For example, when studying biology, you must remember living organisms obey the laws of thermodynamics while using free energy and matter from the environment to carry out life processes that are explored in later chapters, such as metabolism and reproduction.

Two types of logical reasoning are used in science: inductive reasoning and deductive reasoning. Inductive reasoning uses particular results to produce general scientific principles. Deductive reasoning uses logical thinking to predict results by applying scientific principles or practices. The scientific method is a step-by-step process that consists of: making observations, defining a problem, posing hypotheses, testing these hypotheses by designing and conducting investigations, and drawing conclusions from data and results. Scientists then communicate their results to the scientific community. Scientific theories are subject to revision as new information is collected.

The content presented in this section supports the Learning Objectives outlined in Big Idea 2 of the AP ® Biology Curriculum Framework. The Learning Objectives merge Essential Knowledge content with one or more of the seven Science Practices. These objectives provide a transparent foundation for the AP ® Biology course, along with inquiry-based laboratory experiences, instructional activities, and AP ® Exam questions.

Teacher Support

Illustrate uses of the scientific method in class. Divide students in groups of four or five and ask them to design experiments to test the existence of connections they have wondered about. Help them decide if they have a working hypothesis that can be tested and falsified. Give examples of hypotheses that are not falsifiable because they are based on subjective assessments. They are neither observable nor measurable. For example, birds like classical music is based on a subjective assessment. Ask if this hypothesis can be modified to become a testable hypothesis. Stress the need for controls and provide examples such as the use of placebos in pharmacology.

Biology is not a collection of facts to be memorized. Biological systems follow the law of physics and chemistry. Give as an example gas laws in chemistry and respiration physiology. Many students come with a 19th century view of natural sciences; each discipline is in its own sphere. Give as an example, bioinformatics which uses organism biology, chemistry, and physics to label DNA with light emitting reporter molecules (Next Generation sequencing). These molecules can then be scanned by light-sensing machinery, allowing huge amounts of information to be gathered on their DNA. Bring to their attention the fact that the analysis of these data is an application of mathematics and computer science.

For more information about next generation sequencing, check out this informative review .

What is biology? In simple terms, biology is the study of life. This is a very broad definition because the scope of biology is vast. Biologists may study anything from the microscopic or submicroscopic view of a cell to ecosystems and the whole living planet ( Figure 1.2 ). Listening to the daily news, you will quickly realize how many aspects of biology are discussed every day. For example, recent news topics include Escherichia coli ( Figure 1.3 ) outbreaks in spinach and Salmonella contamination in peanut butter. On a global scale, many researchers are committed to finding ways to protect the planet, solve environmental issues, and reduce the effects of climate change. All of these diverse endeavors are related to different facets of the discipline of biology.

The Process of Science

Biology is a science, but what exactly is science? What does the study of biology share with other scientific disciplines? Science (from the Latin scientia , meaning “knowledge”) can be defined as knowledge that covers general truths or the operation of general laws, especially when acquired and tested by the scientific method. It becomes clear from this definition that the application of the scientific method plays a major role in science. The scientific method is a method of research with defined steps that include experiments and careful observation.

The steps of the scientific method will be examined in detail later, but one of the most important aspects of this method is the testing of hypotheses by means of repeatable experiments. A hypothesis is a suggested explanation for an event, which can be tested. Although using the scientific method is inherent to science, it is inadequate in determining what science is. This is because it is relatively easy to apply the scientific method to disciplines such as physics and chemistry, but when it comes to disciplines like archaeology, psychology, and geology, the scientific method becomes less applicable as it becomes more difficult to repeat experiments.

These areas of study are still sciences, however. Consider archaeology—even though one cannot perform repeatable experiments, hypotheses may still be supported. For instance, an archaeologist can hypothesize that an ancient culture existed based on finding a piece of pottery. Further hypotheses could be made about various characteristics of this culture, and these hypotheses may be found to be correct or false through continued support or contradictions from other findings. A hypothesis may become a verified theory. A theory is a tested and confirmed explanation for observations or phenomena. Science may be better defined as fields of study that attempt to comprehend the nature of the universe.

Natural Sciences

What would you expect to see in a museum of natural sciences? Frogs? Plants? Dinosaur skeletons? Exhibits about how the brain functions? A planetarium? Gems and minerals? Or, maybe all of the above? Science includes such diverse fields as astronomy, biology, computer sciences, geology, logic, physics, chemistry, and mathematics ( Figure 1.4 ). However, those fields of science related to the physical world and its phenomena and processes are considered natural sciences . Thus, a museum of natural sciences might contain any of the items listed above.

There is no complete agreement when it comes to defining what the natural sciences include, however. For some experts, the natural sciences are astronomy, biology, chemistry, earth science, and physics. Other scholars choose to divide natural sciences into life sciences , which study living things and include biology, and physical sciences , which study nonliving matter and include astronomy, geology, physics, and chemistry. Some disciplines such as biophysics and biochemistry build on both life and physical sciences and are interdisciplinary. Natural sciences are sometimes referred to as “hard science” because they rely on the use of quantitative data; social sciences that study society and human behavior are more likely to use qualitative assessments to drive investigations and findings.

Not surprisingly, the natural science of biology has many branches or subdisciplines. Cell biologists study cell structure and function, while biologists who study anatomy investigate the structure of an entire organism. Those biologists studying physiology, however, focus on the internal functioning of an organism. Some areas of biology focus on only particular types of living things. For example, botanists explore plants, while zoologists specialize in animals.

Scientific Reasoning

One thing is common to all forms of science: an ultimate goal “to know.” Curiosity and inquiry are the driving forces for the development of science. Scientists seek to understand the world and the way it operates. To do this, they use two methods of logical thinking: inductive reasoning and deductive reasoning.

Inductive reasoning is a form of logical thinking that uses related observations to arrive at a general conclusion. This type of reasoning is common in descriptive science. A life scientist such as a biologist makes observations and records them. These data can be qualitative or quantitative, and the raw data can be supplemented with drawings, pictures, photos, or videos. From many observations, the scientist can infer conclusions (inductions) based on evidence. Inductive reasoning involves formulating generalizations inferred from careful observation and the analysis of a large amount of data. Brain studies provide an example. In this type of research, many live brains are observed while people are doing a specific activity, such as viewing images of food. The part of the brain that “lights up” during this activity is then predicted to be the part controlling the response to the selected stimulus, in this case, images of food. The “lighting up” of the various areas of the brain is caused by excess absorption of radioactive sugar derivatives by active areas of the brain. The resultant increase in radioactivity is observed by a scanner. Then, researchers can stimulate that part of the brain to see if similar responses result.

Deductive reasoning or deduction is the type of logic used in hypothesis-based science. In deductive reason, the pattern of thinking moves in the opposite direction as compared to inductive reasoning. Deductive reasoning is a form of logical thinking that uses a general principle or law to predict specific results. From those general principles, a scientist can deduce and predict the specific results that would be valid as long as the general principles are valid. Studies in climate change can illustrate this type of reasoning. For example, scientists may predict that if the climate becomes warmer in a particular region, then the distribution of plants and animals should change. These predictions have been made and tested, and many such changes have been found, such as the modification of arable areas for agriculture, with change based on temperature averages.

Both types of logical thinking are related to the two main pathways of scientific study: descriptive science and hypothesis-based science. Descriptive (or discovery) science , which is usually inductive, aims to observe, explore, and discover, while hypothesis-based science , which is usually deductive, begins with a specific question or problem and a potential answer or solution that can be tested. The boundary between these two forms of study is often blurred, and most scientific endeavors combine both approaches. The fuzzy boundary becomes apparent when thinking about how easily observation can lead to specific questions. For example, a gentleman in the 1940s observed that the burr seeds that stuck to his clothes and his dog’s fur had a tiny hook structure. On closer inspection, he discovered that the burrs’ gripping device was more reliable than a zipper. He eventually developed a company and produced the hook-and-loop fastener often used on lace-less sneakers and athletic braces. Descriptive science and hypothesis-based science are in continuous dialogue.

The Scientific Method

Biologists study the living world by posing questions about it and seeking science-based responses. This approach is common to other sciences as well and is often referred to as the scientific method. The scientific method was used even in ancient times, but it was first documented by England’s Sir Francis Bacon (1561–1626) ( Figure 1.5 ), who set up inductive methods for scientific inquiry. The scientific method is not exclusively used by biologists but can be applied to almost all fields of study as a logical, rational problem-solving method.

The scientific process typically starts with an observation (often a problem to be solved) that leads to a question. Let’s think about a simple problem that starts with an observation and apply the scientific method to solve the problem. One Monday morning, a student arrives at class and quickly discovers that the classroom is too warm. That is an observation that also describes a problem: the classroom is too warm. The student then asks a question: “Why is the classroom so warm?”

Proposing a Hypothesis

Recall that a hypothesis is a suggested explanation that can be tested. To solve a problem, several hypotheses may be proposed. For example, one hypothesis might be, “The classroom is warm because no one turned on the air conditioning.” But there could be other responses to the question, and therefore other hypotheses may be proposed. A second hypothesis might be, “The classroom is warm because there is a power failure, and so the air conditioning doesn’t work.”

Once a hypothesis has been selected, the student can make a prediction. A prediction is similar to a hypothesis but it typically has the format “If . . . then . . . .” For example, the prediction for the first hypothesis might be, “ If the student turns on the air conditioning, then the classroom will no longer be too warm.”

Testing a Hypothesis

A valid hypothesis must be testable. It should also be falsifiable , meaning that it can be disproven by experimental results. Importantly, science does not claim to “prove” anything because scientific understandings are always subject to modification with further information. This step—openness to disproving ideas—is what distinguishes sciences from non-sciences. The presence of the supernatural, for instance, is neither testable nor falsifiable. To test a hypothesis, a researcher will conduct one or more experiments designed to eliminate one or more of the hypotheses. Each experiment will have one or more variables and one or more controls. A variable is any part of the experiment that can vary or change during the experiment. The control group contains every feature of the experimental group except it is not given the manipulation that is hypothesized about. Therefore, if the results of the experimental group differ from the control group, the difference must be due to the hypothesized manipulation, rather than some outside factor. Look for the variables and controls in the examples that follow. To test the first hypothesis, the student would find out if the air conditioning is on. If the air conditioning is turned on but does not work, there should be another reason, and this hypothesis should be rejected. To test the second hypothesis, the student could check if the lights in the classroom are functional. If so, there is no power failure and this hypothesis should be rejected. Each hypothesis should be tested by carrying out appropriate experiments. Be aware that rejecting one hypothesis does not determine whether or not the other hypotheses can be accepted; it simply eliminates one hypothesis that is not valid ( see this figure ). Using the scientific method, the hypotheses that are inconsistent with experimental data are rejected.

While this “warm classroom” example is based on observational results, other hypotheses and experiments might have clearer controls. For instance, a student might attend class on Monday and realize she had difficulty concentrating on the lecture. One observation to explain this occurrence might be, “When I eat breakfast before class, I am better able to pay attention.” The student could then design an experiment with a control to test this hypothesis.

In hypothesis-based science, specific results are predicted from a general premise. This type of reasoning is called deductive reasoning: deduction proceeds from the general to the particular. But the reverse of the process is also possible: sometimes, scientists reach a general conclusion from a number of specific observations. This type of reasoning is called inductive reasoning, and it proceeds from the particular to the general. Inductive and deductive reasoning are often used in tandem to advance scientific knowledge ( see this figure ). In recent years a new approach of testing hypotheses has developed as a result of an exponential growth of data deposited in various databases. Using computer algorithms and statistical analyses of data in databases, a new field of so-called "data research" (also referred to as "in silico" research) provides new methods of data analyses and their interpretation. This will increase the demand for specialists in both biology and computer science, a promising career opportunity.

Science Practice Connection for AP® Courses

Think about it.

Almost all plants use water, carbon dioxide, and energy from the sun to make sugars. Think about what would happen to plants that don’t have sunlight as an energy source or sufficient water. What would happen to organisms that depend on those plants for their own survival?

Make a prediction about what would happen to the organisms living in a rain forest if 50% of its trees were destroyed. How would you test your prediction?

Use this example as a model to make predictions. Emphasize there is no rigid scientific method scheme. Active science is a combination of observations and measurement. Offer the example of ecology where the conventional scientific method is not always applicable because researchers cannot always set experiments in a laboratory and control all the variables.

Possible answers:

Destruction of the rain forest affects the trees, the animals which feed on the vegetation, take shelter on the trees, and large predators which feed on smaller animals. Furthermore, because the trees positively affect rain through massive evaporation and condensation of water vapor, drought follows deforestation.

Tell students a similar experiment on a grand scale may have happened in the past and introduce the next activity “What killed the dinosaurs?”

Some predictions can be made and later observations can support or disprove the prediction.

Ask, “what killed the dinosaurs?” Explain many scientists point to a massive asteroid crashing in the Yucatan peninsula in Mexico. One of the effects was the creation of smoke clouds and debris that blocked the Sun, stamped out many plants and, consequently, brought mass extinction. As is common in the scientific community, many other researchers offer divergent explanations.

Go to this site for a good example of the complexity of scientific method and scientific debate.

Visual Connection

In the example below, the scientific method is used to solve an everyday problem. Order the scientific method steps (numbered items) with the process of solving the everyday problem (lettered items). Based on the results of the experiment, is the hypothesis correct? If it is incorrect, propose some alternative hypotheses.

• The original hypothesis is correct. There is something wrong with the electrical outlet and therefore the toaster doesn’t work.
• The original hypothesis is incorrect. Alternative hypothesis includes that toaster wasn’t turned on.
• The original hypothesis is correct. The coffee maker and the toaster do not work when plugged into the outlet.
• The original hypothesis is incorrect. Alternative hypotheses includes that both coffee maker and toaster were broken.
• All flying birds and insects have wings. Birds and insects flap their wings as they move through the air. Therefore, wings enable flight.
• Insects generally survive mild winters better than harsh ones. Therefore, insect pests will become more problematic if global temperatures increase.
• Chromosomes, the carriers of DNA, are distributed evenly between the daughter cells during cell division. Therefore, each daughter cell will have the same chromosome set as the mother cell.
• Animals as diverse as humans, insects, and wolves all exhibit social behavior. Therefore, social behavior must have an evolutionary advantage.
• 1- Inductive, 2- Deductive, 3- Deductive, 4- Inductive
• 1- Deductive, 2- Inductive, 3- Deductive, 4- Inductive
• 1- Inductive, 2- Deductive, 3- Inductive, 4- Deductive
• 1- Inductive, 2-Inductive, 3- Inductive, 4- Deductive

The scientific method may seem too rigid and structured. It is important to keep in mind that, although scientists often follow this sequence, there is flexibility. Sometimes an experiment leads to conclusions that favor a change in approach; often, an experiment brings entirely new scientific questions to the puzzle. Many times, science does not operate in a linear fashion; instead, scientists continually draw inferences and make generalizations, finding patterns as their research proceeds. Scientific reasoning is more complex than the scientific method alone suggests. Notice, too, that the scientific method can be applied to solving problems that aren’t necessarily scientific in nature.

Two Types of Science: Basic Science and Applied Science

The scientific community has been debating for the last few decades about the value of different types of science. Is it valuable to pursue science for the sake of simply gaining knowledge, or does scientific knowledge only have worth if we can apply it to solving a specific problem or to bettering our lives? This question focuses on the differences between two types of science: basic science and applied science.

Basic science or “pure” science seeks to expand knowledge regardless of the short-term application of that knowledge. It is not focused on developing a product or a service of immediate public or commercial value. The immediate goal of basic science is knowledge for knowledge’s sake, though this does not mean that, in the end, it may not result in a practical application.

In contrast, applied science or “technology,” aims to use science to solve real-world problems, making it possible, for example, to improve a crop yield, find a cure for a particular disease, or save animals threatened by a natural disaster ( Figure 1.8 ). In applied science, the problem is usually defined for the researcher.

Some individuals may perceive applied science as “useful” and basic science as “useless.” A question these people might pose to a scientist advocating knowledge acquisition would be, “What for?” A careful look at the history of science, however, reveals that basic knowledge has resulted in many remarkable applications of great value. Many scientists think that a basic understanding of science is necessary before an application is developed; therefore, applied science relies on the results generated through basic science. Other scientists think that it is time to move on from basic science and instead to find solutions to actual problems. Both approaches are valid. It is true that there are problems that demand immediate attention; however, few solutions would be found without the help of the wide knowledge foundation generated through basic science.

One example of how basic and applied science can work together to solve practical problems occurred after the discovery of DNA structure led to an understanding of the molecular mechanisms governing DNA replication. Strands of DNA, unique in every human, are found in our cells, where they provide the instructions necessary for life. During DNA replication, DNA makes new copies of itself, shortly before a cell divides. Understanding the mechanisms of DNA replication enabled scientists to develop laboratory techniques that are now used to identify genetic diseases. Without basic science, it is unlikely that applied science could exist.

Another example of the link between basic and applied research is the Human Genome Project, a study in which each human chromosome was analyzed and mapped to determine the precise sequence of DNA subunits and the exact location of each gene. (The gene is the basic unit of heredity represented by a specific DNA segment that codes for a functional molecule.) Other less complex organisms have also been studied as part of this project in order to gain a better understanding of human chromosomes. The Human Genome Project ( Figure 1.9 ) relied on basic research carried out with simple organisms and, later, with the human genome. An important end goal eventually became using the data for applied research, seeking cures and early diagnoses for genetically related diseases.

While research efforts in both basic science and applied science are usually carefully planned, it is important to note that some discoveries are made by serendipity , that is, by means of a fortunate accident or a lucky surprise. Penicillin was discovered when biologist Alexander Fleming accidentally left a petri dish of Staphylococcus bacteria open. An unwanted mold grew on the dish, killing the bacteria. The mold turned out to be Penicillium , and a new antibiotic was discovered. Even in the highly organized world of science, luck—when combined with an observant, curious mind—can lead to unexpected breakthroughs.

Reporting Scientific Work

Whether scientific research is basic science or applied science, scientists must share their findings in order for other researchers to expand and build upon their discoveries. Collaboration with other scientists—when planning, conducting, and analyzing results—is important for scientific research. For this reason, important aspects of a scientist’s work are communicating with peers and disseminating results to peers. Scientists can share results by presenting them at a scientific meeting or conference, but this approach can reach only the select few who are present. Instead, most scientists present their results in peer-reviewed manuscripts that are published in scientific journals. Peer-reviewed manuscripts are scientific papers that are reviewed by a scientist’s colleagues, or peers. These colleagues are qualified individuals, often experts in the same research area, who judge whether or not the scientist’s work is suitable for publication. The process of peer review helps to ensure that the research described in a scientific paper or grant proposal is original, significant, logical, and thorough. Grant proposals, which are requests for research funding, are also subject to peer review. Scientists publish their work so other scientists can reproduce their experiments under similar or different conditions to expand on the findings.

A scientific paper is very different from creative writing. Although creativity is required to design experiments, there are fixed guidelines when it comes to presenting scientific results. First, scientific writing must be brief, concise, and accurate. A scientific paper needs to be succinct but detailed enough to allow peers to reproduce the experiments.

The scientific paper consists of several specific sections—introduction, materials and methods, results, and discussion. This structure is sometimes called the “IMRaD” format. There are usually acknowledgment and reference sections as well as an abstract (a concise summary) at the beginning of the paper. There might be additional sections depending on the type of paper and the journal where it will be published; for example, some review papers require an outline.

The introduction starts with brief, but broad, background information about what is known in the field. A good introduction also gives the rationale of the work; it justifies the work carried out and also briefly mentions the end of the paper, where the hypothesis or research question driving the research will be presented. The introduction refers to the published scientific work of others and therefore requires citations following the style of the journal. Using the work or ideas of others without proper citation is considered plagiarism .

The materials and methods section includes a complete and accurate description of the substances used, and the method and techniques used by the researchers to gather data. The description should be thorough enough to allow another researcher to repeat the experiment and obtain similar results, but it does not have to be verbose. This section will also include information on how measurements were made and what types of calculations and statistical analyses were used to examine raw data. Although the materials and methods section gives an accurate description of the experiments, it does not discuss them.

Some journals require a results section followed by a discussion section, but it is more common to combine both. If the journal does not allow the combination of both sections, the results section simply narrates the findings without any further interpretation. The results are presented by means of tables or graphs, but no duplicate information should be presented. In the discussion section, the researcher will interpret the results, describe how variables may be related, and attempt to explain the observations. It is indispensable to conduct an extensive literature search to put the results in the context of previously published scientific research. Therefore, proper citations are included in this section as well.

Finally, the conclusion section summarizes the importance of the experimental findings. While the scientific paper almost certainly answered one or more scientific questions that were stated, any good research should lead to more questions. Therefore, a well-done scientific paper leaves doors open for the researcher and others to continue and expand on the findings.

Review articles do not follow the IMRAD format because they do not present original scientific findings, or primary literature; instead, they summarize and comment on findings that were published as primary literature and typically include extensive reference sections.

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• Null and Alternative Hypotheses | Definitions & Examples

Null & Alternative Hypotheses | Definitions, Templates & Examples

Published on May 6, 2022 by Shaun Turney . Revised on June 22, 2023.

The null and alternative hypotheses are two competing claims that researchers weigh evidence for and against using a statistical test :

• Null hypothesis ( H 0 ): There’s no effect in the population .
• Alternative hypothesis ( H a or H 1 ) : There’s an effect in the population.

Table of contents

Answering your research question with hypotheses, what is a null hypothesis, what is an alternative hypothesis, similarities and differences between null and alternative hypotheses, how to write null and alternative hypotheses, other interesting articles, frequently asked questions.

The null and alternative hypotheses offer competing answers to your research question . When the research question asks “Does the independent variable affect the dependent variable?”:

• The null hypothesis ( H 0 ) answers “No, there’s no effect in the population.”
• The alternative hypothesis ( H a ) answers “Yes, there is an effect in the population.”

The null and alternative are always claims about the population. That’s because the goal of hypothesis testing is to make inferences about a population based on a sample . Often, we infer whether there’s an effect in the population by looking at differences between groups or relationships between variables in the sample. It’s critical for your research to write strong hypotheses .

You can use a statistical test to decide whether the evidence favors the null or alternative hypothesis. Each type of statistical test comes with a specific way of phrasing the null and alternative hypothesis. However, the hypotheses can also be phrased in a general way that applies to any test.

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The null hypothesis is the claim that there’s no effect in the population.

If the sample provides enough evidence against the claim that there’s no effect in the population ( p ≤ α), then we can reject the null hypothesis . Otherwise, we fail to reject the null hypothesis.

Although “fail to reject” may sound awkward, it’s the only wording that statisticians accept . Be careful not to say you “prove” or “accept” the null hypothesis.

Null hypotheses often include phrases such as “no effect,” “no difference,” or “no relationship.” When written in mathematical terms, they always include an equality (usually =, but sometimes ≥ or ≤).

You can never know with complete certainty whether there is an effect in the population. Some percentage of the time, your inference about the population will be incorrect. When you incorrectly reject the null hypothesis, it’s called a type I error . When you incorrectly fail to reject it, it’s a type II error.

Examples of null hypotheses

The table below gives examples of research questions and null hypotheses. There’s always more than one way to answer a research question, but these null hypotheses can help you get started.

*Note that some researchers prefer to always write the null hypothesis in terms of “no effect” and “=”. It would be fine to say that daily meditation has no effect on the incidence of depression and p 1 = p 2 .

The alternative hypothesis ( H a ) is the other answer to your research question . It claims that there’s an effect in the population.

Often, your alternative hypothesis is the same as your research hypothesis. In other words, it’s the claim that you expect or hope will be true.

The alternative hypothesis is the complement to the null hypothesis. Null and alternative hypotheses are exhaustive, meaning that together they cover every possible outcome. They are also mutually exclusive, meaning that only one can be true at a time.

Alternative hypotheses often include phrases such as “an effect,” “a difference,” or “a relationship.” When alternative hypotheses are written in mathematical terms, they always include an inequality (usually ≠, but sometimes < or >). As with null hypotheses, there are many acceptable ways to phrase an alternative hypothesis.

Examples of alternative hypotheses

The table below gives examples of research questions and alternative hypotheses to help you get started with formulating your own.

Null and alternative hypotheses are similar in some ways:

• They’re both answers to the research question.
• They both make claims about the population.
• They’re both evaluated by statistical tests.

However, there are important differences between the two types of hypotheses, summarized in the following table.

To help you write your hypotheses, you can use the template sentences below. If you know which statistical test you’re going to use, you can use the test-specific template sentences. Otherwise, you can use the general template sentences.

General template sentences

The only thing you need to know to use these general template sentences are your dependent and independent variables. To write your research question, null hypothesis, and alternative hypothesis, fill in the following sentences with your variables:

Does independent variable affect dependent variable ?

• Null hypothesis ( H 0 ): Independent variable does not affect dependent variable.
• Alternative hypothesis ( H a ): Independent variable affects dependent variable.

Test-specific template sentences

Once you know the statistical test you’ll be using, you can write your hypotheses in a more precise and mathematical way specific to the test you chose. The table below provides template sentences for common statistical tests.

Note: The template sentences above assume that you’re performing one-tailed tests . One-tailed tests are appropriate for most studies.

If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.

• Normal distribution
• Descriptive statistics
• Measures of central tendency
• Correlation coefficient

Methodology

• Cluster sampling
• Stratified sampling
• Types of interviews
• Cohort study
• Thematic analysis

Research bias

• Implicit bias
• Cognitive bias
• Survivorship bias
• Availability heuristic
• Nonresponse bias
• Regression to the mean

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is used by scientists to test specific predictions, called hypotheses , by calculating how likely it is that a pattern or relationship between variables could have arisen by chance.

Null and alternative hypotheses are used in statistical hypothesis testing . The null hypothesis of a test always predicts no effect or no relationship between variables, while the alternative hypothesis states your research prediction of an effect or relationship.

The null hypothesis is often abbreviated as H 0 . When the null hypothesis is written using mathematical symbols, it always includes an equality symbol (usually =, but sometimes ≥ or ≤).

The alternative hypothesis is often abbreviated as H a or H 1 . When the alternative hypothesis is written using mathematical symbols, it always includes an inequality symbol (usually ≠, but sometimes < or >).

A research hypothesis is your proposed answer to your research question. The research hypothesis usually includes an explanation (“ x affects y because …”).

A statistical hypothesis, on the other hand, is a mathematical statement about a population parameter. Statistical hypotheses always come in pairs: the null and alternative hypotheses . In a well-designed study , the statistical hypotheses correspond logically to the research hypothesis.

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Statistics Made Easy

How to Write a Null Hypothesis (5 Examples)

A hypothesis test uses sample data to determine whether or not some claim about a population parameter is true.

Whenever we perform a hypothesis test, we always write a null hypothesis and an alternative hypothesis, which take the following forms:

H 0 (Null Hypothesis): Population parameter =,  ≤, ≥ some value

H A  (Alternative Hypothesis): Population parameter <, >, ≠ some value

Note that the null hypothesis always contains the equal sign .

We interpret the hypotheses as follows:

Null hypothesis: The sample data provides no evidence to support some claim being made by an individual.

Alternative hypothesis: The sample data  does provide sufficient evidence to support the claim being made by an individual.

For example, suppose it’s assumed that the average height of a certain species of plant is 20 inches tall. However, one botanist claims the true average height is greater than 20 inches.

To test this claim, she may go out and collect a random sample of plants. She can then use this sample data to perform a hypothesis test using the following two hypotheses:

H 0 : μ ≤ 20 (the true mean height of plants is equal to or even less than 20 inches)

H A : μ > 20 (the true mean height of plants is greater than 20 inches)

If the sample data gathered by the botanist shows that the mean height of this species of plants is significantly greater than 20 inches, she can reject the null hypothesis and conclude that the mean height is greater than 20 inches.

Read through the following examples to gain a better understanding of how to write a null hypothesis in different situations.

Example 1: Weight of Turtles

A biologist wants to test whether or not the true mean weight of a certain species of turtles is 300 pounds. To test this, he goes out and measures the weight of a random sample of 40 turtles.

Here is how to write the null and alternative hypotheses for this scenario:

H 0 : μ = 300 (the true mean weight is equal to 300 pounds)

H A : μ ≠ 300 (the true mean weight is not equal to 300 pounds)

Example 2: Height of Males

It’s assumed that the mean height of males in a certain city is 68 inches. However, an independent researcher believes the true mean height is greater than 68 inches. To test this, he goes out and collects the height of 50 males in the city.

H 0 : μ ≤ 68 (the true mean height is equal to or even less than 68 inches)

H A : μ > 68 (the true mean height is greater than 68 inches)

Example 3: Graduation Rates

A university states that 80% of all students graduate on time. However, an independent researcher believes that less than 80% of all students graduate on time. To test this, she collects data on the proportion of students who graduated on time last year at the university.

H 0 : p ≥ 0.80 (the true proportion of students who graduate on time is 80% or higher)

H A : μ < 0.80 (the true proportion of students who graduate on time is less than 80%)

Example 4: Burger Weights

A food researcher wants to test whether or not the true mean weight of a burger at a certain restaurant is 7 ounces. To test this, he goes out and measures the weight of a random sample of 20 burgers from this restaurant.

H 0 : μ = 7 (the true mean weight is equal to 7 ounces)

H A : μ ≠ 7 (the true mean weight is not equal to 7 ounces)

Example 5: Citizen Support

A politician claims that less than 30% of citizens in a certain town support a certain law. To test this, he goes out and surveys 200 citizens on whether or not they support the law.

H 0 : p ≥ .30 (the true proportion of citizens who support the law is greater than or equal to 30%)

H A : μ < 0.30 (the true proportion of citizens who support the law is less than 30%)

Additional Resources

Introduction to Hypothesis Testing Introduction to Confidence Intervals An Explanation of P-Values and Statistical Significance

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AP®︎/College Statistics

Course: ap®︎/college statistics   >   unit 10.

• Idea behind hypothesis testing

Examples of null and alternative hypotheses

• Writing null and alternative hypotheses
• P-values and significance tests
• Comparing P-values to different significance levels
• Estimating a P-value from a simulation
• Estimating P-values from simulations
• Using P-values to make conclusions

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