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.
| 1 | 21 | 18 | 21 |
| 10 | 45 | 3 | 12 |
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:
| /E | |||
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 Examples
The null hypothesis (H 0 ) is the hypothesis that states there is no statistical difference between two sample sets. In other words, it assumes the independent variable does not have an effect on the dependent variable in a scientific experiment .
The null hypothesis is the most powerful type of hypothesis in the scientific method because it’s the easiest one to test with a high confidence level using statistics. If the null hypothesis is accepted, then it’s evidence any observed differences between two experiment groups are due to random chance. If the null hypothesis is rejected, then it’s strong evidence there is a true difference between test sets or that the independent variable affects the dependent variable.
- The null hypothesis is a nullifiable hypothesis. A researcher seeks to reject it because this result strongly indicates observed differences are real and not just due to chance.
- The null hypothesis may be accepted or rejected, but not proven. There is always a level of confidence in the outcome.
What Is the Null Hypothesis?
The null hypothesis is written as H 0 , which is read as H-zero, H-nought, or H-null. It is associated with another hypothesis, called the alternate or alternative hypothesis H A or H 1 . When the null hypothesis and alternate hypothesis are written mathematically, they cover all possible outcomes of an experiment.
An experimenter tests the null hypothesis with a statistical analysis called a significance test. The significance test determines the likelihood that the results of the test are not due to chance. Usually, a researcher uses a confidence level of 95% or 99% (p-value of 0.05 or 0.01). But, even if the confidence in the test is high, there is always a small chance the outcome is incorrect. This means you can’t prove a null hypothesis. It’s also a good reason why it’s important to repeat experiments.
Exact and Inexact Null Hypothesis
The most common type of null hypothesis assumes no difference between two samples or groups or no measurable effect of a treatment. This is the exact hypothesis . If you’re asked to state a null hypothesis for a science class, this is the one to write. It is the easiest type of hypothesis to test and is the only one accepted for certain types of analysis. Examples include:
There is no difference between two groups H 0 : μ 1 = μ 2 (where H 0 = the null hypothesis, μ 1 = the mean of population 1, and μ 2 = the mean of population 2)
Both groups have value of 100 (or any number or quality) H 0 : μ = 100
However, sometimes a researcher may test an inexact hypothesis . This type of hypothesis specifies ranges or intervals. Examples include:
Recovery time from a treatment is the same or worse than a placebo: H 0 : μ ≥ placebo time
There is a 5% or less difference between two groups: H 0 : 95 ≤ μ ≤ 105
An inexact hypothesis offers “directionality” about a phenomenon. For example, an exact hypothesis can indicate whether or not a treatment has an effect, while an inexact hypothesis can tell whether an effect is positive of negative. However, an inexact hypothesis may be harder to test and some scientists and statisticians disagree about whether it’s a true null hypothesis .
How to State the Null Hypothesis
To state the null hypothesis, first state what you expect the experiment to show. Then, rephrase the statement in a form that assumes there is no relationship between the variables or that a treatment has no effect.
Example: A researcher tests whether a new drug speeds recovery time from a certain disease. The average recovery time without treatment is 3 weeks.
- State the goal of the experiment: “I hope the average recovery time with the new drug will be less than 3 weeks.”
- Rephrase the hypothesis to assume the treatment has no effect: “If the drug doesn’t shorten recovery time, then the average time will be 3 weeks or longer.” Mathematically: H 0 : μ ≥ 3
This null hypothesis (inexact hypothesis) covers both the scenario in which the drug has no effect and the one in which the drugs makes the recovery time longer. The alternate hypothesis is that average recovery time will be less than three weeks:
H A : μ < 3
Of course, the researcher could test the no-effect hypothesis (exact null hypothesis): H 0 : μ = 3
The danger of testing this hypothesis is that rejecting it only implies the drug affected recovery time (not whether it made it better or worse). This is because the alternate hypothesis is:
H A : μ ≠ 3 (which includes μ <3 and μ >3)
Even though the no-effect null hypothesis yields less information, it’s used because it’s easier to test using statistics. Basically, testing whether something is unchanged/changed is easier than trying to quantify the nature of the change.
Remember, a researcher hopes to reject the null hypothesis because this supports the alternate hypothesis. Also, be sure the null and alternate hypothesis cover all outcomes. Finally, remember a simple true/false, equal/unequal, yes/no exact hypothesis is easier to test than a more complex inexact hypothesis.
| Does chewing willow bark relieve pain? | Pain relief is the same compared with a . (exact) Pain relief after chewing willow bark is the same or worse versus taking a placebo. (inexact) | Pain relief is different compared with a placebo. (exact) Pain relief is better compared to a placebo. (inexact) |
| Do cats care about the shape of their food? | Cats show no food preference based on shape. (exact) | Cat show a food preference based on shape. (exact) |
| Do teens use mobile devices more than adults? | Teens and adults use mobile devices the same amount. (exact) Teens use mobile devices less than or equal to adults. (inexact) | Teens and adults used mobile devices different amounts. (exact) Teens use mobile devices more than adults. (inexact) |
| Does the color of light influence plant growth? | The color of light has no effect on plant growth. (exact) | The color of light affects plant growth. (exact) |
- Adèr, H. J.; Mellenbergh, G. J. & Hand, D. J. (2007). Advising on Research Methods: A Consultant’s Companion . Huizen, The Netherlands: Johannes van Kessel Publishing. ISBN 978-90-79418-01-5 .
- Cox, D. R. (2006). Principles of Statistical Inference . Cambridge University Press. ISBN 978-0-521-68567-2 .
- Everitt, Brian (1998). The Cambridge Dictionary of Statistics . Cambridge, UK New York: Cambridge University Press. ISBN 978-0521593465.
- Weiss, Neil A. (1999). Introductory Statistics (5th ed.). ISBN 9780201598773.
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Genetics and Statistical Analysis
Once you have performed an experiment, how can you tell if your results are significant? For example, say that you are performing a genetic cross in which you know the genotypes of the parents. In this situation, you might hypothesize that the cross will result in a certain ratio of phenotypes in the offspring . But what if your observed results do not exactly match your expectations? How can you tell whether this deviation was due to chance? The key to answering these questions is the use of statistics , which allows you to determine whether your data are consistent with your hypothesis.
Forming and Testing a Hypothesis
The first thing any scientist does before performing an experiment is to form a hypothesis about the experiment's outcome. This often takes the form of a null hypothesis , which is a statistical hypothesis that states there will be no difference between observed and expected data. The null hypothesis is proposed by a scientist before completing an experiment, and it can be either supported by data or disproved in favor of an alternate hypothesis.
Let's consider some examples of the use of the null hypothesis in a genetics experiment. Remember that Mendelian inheritance deals with traits that show discontinuous variation, which means that the phenotypes fall into distinct categories. As a consequence, in a Mendelian genetic cross, the null hypothesis is usually an extrinsic hypothesis ; in other words, the expected proportions can be predicted and calculated before the experiment starts. Then an experiment can be designed to determine whether the data confirm or reject the hypothesis. On the other hand, in another experiment, you might hypothesize that two genes are linked. This is called an intrinsic hypothesis , which is a hypothesis in which the expected proportions are calculated after the experiment is done using some information from the experimental data (McDonald, 2008).
How Math Merged with Biology
But how did mathematics and genetics come to be linked through the use of hypotheses and statistical analysis? The key figure in this process was Karl Pearson, a turn-of-the-century mathematician who was fascinated with biology. When asked what his first memory was, Pearson responded by saying, "Well, I do not know how old I was, but I was sitting in a high chair and I was sucking my thumb. Someone told me to stop sucking it and said that if I did so, the thumb would wither away. I put my two thumbs together and looked at them a long time. ‘They look alike to me,' I said to myself, ‘I can't see that the thumb I suck is any smaller than the other. I wonder if she could be lying to me'" (Walker, 1958). As this anecdote illustrates, Pearson was perhaps born to be a scientist. He was a sharp observer and intent on interpreting his own data. During his career, Pearson developed statistical theories and applied them to the exploration of biological data. His innovations were not well received, however, and he faced an arduous struggle in convincing other scientists to accept the idea that mathematics should be applied to biology. For instance, during Pearson's time, the Royal Society, which is the United Kingdom's academy of science, would accept papers that concerned either mathematics or biology, but it refused to accept papers than concerned both subjects (Walker, 1958). In response, Pearson, along with Francis Galton and W. F. R. Weldon, founded a new journal called Biometrika in 1901 to promote the statistical analysis of data on heredity. Pearson's persistence paid off. Today, statistical tests are essential for examining biological data.
Pearson's Chi-Square Test for Goodness-of-Fit
One of Pearson's most significant achievements occurred in 1900, when he developed a statistical test called Pearson's chi-square (Χ 2 ) test, also known as the chi-square test for goodness-of-fit (Pearson, 1900). Pearson's chi-square test is used to examine the role of chance in producing deviations between observed and expected values. The test depends on an extrinsic hypothesis, because it requires theoretical expected values to be calculated. The test indicates the probability that chance alone produced the deviation between the expected and the observed values (Pierce, 2005). When the probability calculated from Pearson's chi-square test is high, it is assumed that chance alone produced the difference. Conversely, when the probability is low, it is assumed that a significant factor other than chance produced the deviation.
In 1912, J. Arthur Harris applied Pearson's chi-square test to examine Mendelian ratios (Harris, 1912). It is important to note that when Gregor Mendel studied inheritance, he did not use statistics, and neither did Bateson, Saunders, Punnett, and Morgan during their experiments that discovered genetic linkage . Thus, until Pearson's statistical tests were applied to biological data, scientists judged the goodness of fit between theoretical and observed experimental results simply by inspecting the data and drawing conclusions (Harris, 1912). Although this method can work perfectly if one's data exactly matches one's predictions, scientific experiments often have variability associated with them, and this makes statistical tests very useful.
The chi-square value is calculated using the following formula:
Using this formula, the difference between the observed and expected frequencies is calculated for each experimental outcome category. The difference is then squared and divided by the expected frequency . Finally, the chi-square values for each outcome are summed together, as represented by the summation sign (Σ).
Pearson's chi-square test works well with genetic data as long as there are enough expected values in each group. In the case of small samples (less than 10 in any category) that have 1 degree of freedom, the test is not reliable. (Degrees of freedom, or df, will be explained in full later in this article.) However, in such cases, the test can be corrected by using the Yates correction for continuity, which reduces the absolute value of each difference between observed and expected frequencies by 0.5 before squaring. Additionally, it is important to remember that the chi-square test can only be applied to numbers of progeny , not to proportions or percentages.
Now that you know the rules for using the test, it's time to consider an example of how to calculate Pearson's chi-square. Recall that when Mendel crossed his pea plants, he learned that tall (T) was dominant to short (t). You want to confirm that this is correct, so you start by formulating the following null hypothesis: In a cross between two heterozygote (Tt) plants, the offspring should occur in a 3:1 ratio of tall plants to short plants. Next, you cross the plants, and after the cross, you measure the characteristics of 400 offspring. You note that there are 305 tall pea plants and 95 short pea plants; these are your observed values. Meanwhile, you expect that there will be 300 tall plants and 100 short plants from the Mendelian ratio.
You are now ready to perform statistical analysis of your results, but first, you have to choose a critical value at which to reject your null hypothesis. You opt for a critical value probability of 0.01 (1%) that the deviation between the observed and expected values is due to chance. This means that if the probability is less than 0.01, then the deviation is significant and not due to chance, and you will reject your null hypothesis. However, if the deviation is greater than 0.01, then the deviation is not significant and you will not reject the null hypothesis.
So, should you reject your null hypothesis or not? Here's a summary of your observed and expected data:
Now, let's calculate Pearson's chi-square:
- For tall plants: Χ 2 = (305 - 300) 2 / 300 = 0.08
- For short plants: Χ 2 = (95 - 100) 2 / 100 = 0.25
- The sum of the two categories is 0.08 + 0.25 = 0.33
- Therefore, the overall Pearson's chi-square for the experiment is Χ 2 = 0.33
Next, you determine the probability that is associated with your calculated chi-square value. To do this, you compare your calculated chi-square value with theoretical values in a chi-square table that has the same number of degrees of freedom. Degrees of freedom represent the number of ways in which the observed outcome categories are free to vary. For Pearson's chi-square test, the degrees of freedom are equal to n - 1, where n represents the number of different expected phenotypes (Pierce, 2005). In your experiment, there are two expected outcome phenotypes (tall and short), so n = 2 categories, and the degrees of freedom equal 2 - 1 = 1. Thus, with your calculated chi-square value (0.33) and the associated degrees of freedom (1), you can determine the probability by using a chi-square table (Table 1).
Table 1: Chi-Square Table
&
(Table adapted from Jones, 2008)
Note that the chi-square table is organized with degrees of freedom (df) in the left column and probabilities (P) at the top. The chi-square values associated with the probabilities are in the center of the table. To determine the probability, first locate the row for the degrees of freedom for your experiment, then determine where the calculated chi-square value would be placed among the theoretical values in the corresponding row.
At the beginning of your experiment, you decided that if the probability was less than 0.01, you would reject your null hypothesis because the deviation would be significant and not due to chance. Now, looking at the row that corresponds to 1 degree of freedom, you see that your calculated chi-square value of 0.33 falls between 0.016, which is associated with a probability of 0.9, and 2.706, which is associated with a probability of 0.10. Therefore, there is between a 10% and 90% probability that the deviation you observed between your expected and the observed numbers of tall and short plants is due to chance. In other words, the probability associated with your chi-square value is much greater than the critical value of 0.01. This means that we will not reject our null hypothesis, and the deviation between the observed and expected results is not significant.
Level of Significance
Determining whether to accept or reject a hypothesis is decided by the experimenter, who is the person who chooses the "level of significance" or confidence. Scientists commonly use the 0.05, 0.01, or 0.001 probability levels as cut-off values. For instance, in the example experiment, you used the 0.01 probability. Thus, P ≥ 0.01 can be interpreted to mean that chance likely caused the deviation between the observed and the expected values (i.e. there is a greater than 1% probability that chance explains the data). If instead we had observed that P ≤ 0.01, this would mean that there is less than a 1% probability that our data can be explained by chance. There is a significant difference between our expected and observed results, so the deviation must be caused by something other than chance.
References and Recommended Reading
Harris, J. A. A simple test of the goodness of fit of Mendelian ratios. American Naturalist 46 , 741–745 (1912)
Jones, J. "Table: Chi-Square Probabilities." http://people.richland.edu/james/lecture/m170/tbl-chi.html (2008) (accessed July 7, 2008)
McDonald, J. H. Chi-square test for goodness-of-fit. From The Handbook of Biological Statistics . http://udel.edu/~mcdonald/statchigof.html (2008) (accessed June 9, 2008)
Pearson, K. On the criterion that a given system of deviations from the probable in the case of correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling. Philosophical Magazine 50 , 157–175 (1900)
Pierce, B. Genetics: A Conceptual Approach (New York, Freeman, 2005)
Walker, H. M. The contributions of Karl Pearson. Journal of the American Statistical Association 53 , 11–22 (1958)
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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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In a scientific experiment, the null hypothesis is the proposition that there is no effect or no relationship between phenomena or populations. If the null hypothesis is true, any observed difference in phenomena or populations would be due to sampling error (random chance) or experimental error. The null hypothesis is useful because it can be tested and found to be false, which then implies that there is a relationship between the observed data. It may be easier to think of it as a nullifiable hypothesis or one that the researcher seeks to nullify. The null hypothesis is also known as the H 0, or no-difference hypothesis.
The alternate hypothesis, H A or H 1 , proposes that observations are influenced by a non-random factor. In an experiment, the alternate hypothesis suggests that the experimental or independent variable has an effect on the dependent variable .
How to State a Null Hypothesis
There are two ways to state a null hypothesis. One is to state it as a declarative sentence, and the other is to present it as a mathematical statement.
For example, say a researcher suspects that exercise is correlated to weight loss, assuming diet remains unchanged. The average length of time to achieve a certain amount of weight loss is six weeks when a person works out five times a week. The researcher wants to test whether weight loss takes longer to occur if the number of workouts is reduced to three times a week.
The first step to writing the null hypothesis is to find the (alternate) hypothesis. In a word problem like this, you're looking for what you expect to be the outcome of the experiment. In this case, the hypothesis is "I expect weight loss to take longer than six weeks."
This can be written mathematically as: H 1 : μ > 6
In this example, μ is the average.
Now, the null hypothesis is what you expect if this hypothesis does not happen. In this case, if weight loss isn't achieved in greater than six weeks, then it must occur at a time equal to or less than six weeks. This can be written mathematically as:
H 0 : μ ≤ 6
The other way to state the null hypothesis is to make no assumption about the outcome of the experiment. In this case, the null hypothesis is simply that the treatment or change will have no effect on the outcome of the experiment. For this example, it would be that reducing the number of workouts would not affect the time needed to achieve weight loss:
H 0 : μ = 6
Null Hypothesis Examples
"Hyperactivity is unrelated to eating sugar " is an example of a null hypothesis. If the hypothesis is tested and found to be false, using statistics, then a connection between hyperactivity and sugar ingestion may be indicated. A significance test is the most common statistical test used to establish confidence in a null hypothesis.
Another example of a null hypothesis is "Plant growth rate is unaffected by the presence of cadmium in the soil ." A researcher could test the hypothesis by measuring the growth rate of plants grown in a medium lacking cadmium, compared with the growth rate of plants grown in mediums containing different amounts of cadmium. Disproving the null hypothesis would set the groundwork for further research into the effects of different concentrations of the element in soil.
Why Test a Null Hypothesis?
You may be wondering why you would want to test a hypothesis just to find it false. Why not just test an alternate hypothesis and find it true? The short answer is that it is part of the scientific method. In science, propositions are not explicitly "proven." Rather, science uses math to determine the probability that a statement is true or false. It turns out it's much easier to disprove a hypothesis than to positively prove one. Also, while the null hypothesis may be simply stated, there's a good chance the alternate hypothesis is incorrect.
For example, if your null hypothesis is that plant growth is unaffected by duration of sunlight, you could state the alternate hypothesis in several different ways. Some of these statements might be incorrect. You could say plants are harmed by more than 12 hours of sunlight or that plants need at least three hours of sunlight, etc. There are clear exceptions to those alternate hypotheses, so if you test the wrong plants, you could reach the wrong conclusion. The null hypothesis is a general statement that can be used to develop an alternate hypothesis, which may or may not be correct.
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AP Statistics : How to establish a null hypothesis
Study concepts, example questions & explanations for ap statistics, all ap statistics resources, example questions, example question #1 : how to establish a null hypothesis.
Jimmy thinks that Josh cannot shoot more than 50 points on average in a game. Josh disputes this claim and tells Jimmy that he is going to play 10 games and prove him wrong. What is the null hypothesis?
Josh cannot shoot more than 50 points.
Josh cannot shoot exactly 50 points.
Josh cannot shoot less than 50 points.
Josh cannot play 10 games.
The null hypothesis is what we intend to either reject or fail to reject using our sample data. In this case, the null hypothesis is that Josh cannot shoot more than 50 points on average, and Josh's performance in 10 games are the sample data we use to assess this hypothesis.
A student is beginning an analysis to determine whether there is a relationship between temperatures and traffic accidents. The student is trying to articulate a null hypothesis for the study. Which of the following is an acceptable null hypothesis?
There is no relationship between temperatures and frequency of traffic accidents
No variable can accurately predict whether traffic accidents will increase
Traffic accidents increase as temperatures decrease
There is a positive relationship between temperatures and traffic accidents
There is a negative relationship between temperatures and traffic accidents
The null hypothesis is the default hypothesis and predicts that there is no relationship between the variables in question. Each of the incorrect answer choices here either predicts a relationship between variables or makes a broad assertion that includes much more than the variables in question.
Example Question #3 : How To Establish A Null Hypothesis
Conditionally
Not enough information to make a decision.
The statistician has determined that she will only reject the null hypothesis if she has 95% confidence that there is a relationship between variables.
To have this level of confidence, the statistician must obtain a p value of 0.05 or lower.
Therefore, she should not reject the null hypothesis since 0.1 is greater that 0.05.
Example Question #4 : How To Establish A Null Hypothesis
The Environmental Protection Agency (EPA) wants to test the pollution level of the Colorado River. If the pollution level is too high, the water will be stopped from going into drinking water pipelines. The EPA randomly chooses different spots along the river to collect water samples from, and then tests the samples for their pollution levels. Which of the following decisions would result from the type I error?
Closing the drinking water pipelines for the river when the pollution levels are within the allowed limit.
Closing the drinking water pipelines when the pollution levels are higher than the allowed limit.
Keeping the drinking water pipelines open when the pollution levels are within the allowed limit.
Closing the drinking water pipelines because of the endangered frog population.
Keeping the drinking water pipelines open when the pollution levels are higher than the allowed limit.
The hypotheses tested here are:
The type I error occurs when the null hypothesis is rejected even though it is actually true. In this case, the type I error would be deciding that the mean pollution levels are higher than the allowed limit and closing the drinking water pipelines.
Example Question #5 : How To Establish A Null Hypothesis
A study would like to determine whether meditation helps students improve focus time. They know that the average focus time of an American 4 th grader is 23 minutes. They then recruit 50 meditators and calculate their average focus time. What is the appropriate null hypothesis for this study?
Example Question #15 : Significance Logic And Establishing Hypotheses
A researcher wants to determine whether there is a significant linear relationship between time spent meditating and time spent studying. What is the appropriate null hypothesis for this study?
This question is about a linear regression between time spent meditating and time spent studying. Therefore, the hypothesis is regarding Beta1, the slope of the line. We are testing a non-directional or bi-directional claim that the relationship is significant . Therefore, the null hypothesis is that the relationship is not significant, meaning the slope of the line is equal to zero.
A researcher wants to compare 3 different treatments to see if any of the treatments affects study time. The three treatments studied are control group, a group given vitamins, and a group given a placebo. They found that the average time spent studying with control students was 2 hours, with students given vitamins it was 3 hours, and with placebos students studied 5 hours. Which of the following is the correct null hypothesis?
Because we are comparing more than 2 groups, we must use an ANOVA for this problem. For an ANOVA problem, the null hypothesis is that all of the groups’ means are the same.
Example Question #21 : Significance Logic And Establishing Hypotheses
A researcher wants to investigate the claim that taking vitamins will help a student study longer. First, the researcher collects 32 students who do not take vitamins and determines their time spent studying. Then, the 32 students are given a vitamin for 1 week. After 1 week of taking vitamins, students are again tested to determine their time spent studying. Which of the following is the correct null hypothesis?
Because the same students are tested twice, this is a paired study, therefore we must use a hypothesis appropriate for a paired t-test. The hypothesis for a paired t-test regards the average of the differences between before and after treatment, called MuD. We are testing the claim that vitamins increase study time, which would mean that study time for vitamin users would be greater than that of the control. Therefore the null must include all other outcomes. The null hypothesis should state that the difference between before and after treatment is greater than or equal to zero.
Example Question #6 : How To Establish A Null Hypothesis
For her school science project, Susy wants to determine whether the ants in her neighborhood have smaller colonies than average. Research tells her that the average Harvester colony has around 4,000 ants. She counts the number of ants in 5 colonies in her neighborhood and determines the average colony size to be 3,700 ants. What is the appropriate null hypothesis for her science project?
Susy wants to know whether ants in her neighborhood have smaller colonies, so that will be her alternative hypothesis. Therefore her null hypothesis needs to cover all other outcomes, that the colony sizes are greater than or equal to average colony size of 4000 ants.
For his school science project, Timmy wants to determine whether the ants in his neighborhood have colonies that are sized differently than normal. His research shows that the average Harvester colony has around 4000 ants. He counts the number of ants in 5 colonies and determines that the average colony size is 3,700 ants. What is the appropriate null hypothesis for his science project?
Timmy does not have a directional hypothesis, he only wants to know whether local ant colonies are different from average. Therefore he thinks the colonies could be bigger or smaller than average. This means his alternative hypothesis is that the ant colonies are NOT equal to the average colony size of 4000 ants. His null hypothesis must include all other outcomes, which in this case is that local ant colonies are equal to the average size of 4000 ants.
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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.
Biology definition: A null hypothesis is an assumption or proposition where an observed difference between two samples of a statistical population is purely accidental and not due to systematic causes. It is the hypothesis to be investigated through statistical hypothesis testing so that when refuted indicates that the alternative hypothesis is true. . Thus, a null hypothesis is a hypothesis ...
AP® Biology 2022 Scoring Guidelines. Question 3: Scientific Investigation. 4 points. Fireflies emit light when the enzyme luciferase catalyzes a reaction in which its substrate, D-luciferin, reacts to form oxyluciferin and other products (Figure 1). In order to determine the optimal temperature for this enzyme, scientists added ATP to a ...
In this video, I begin discussing AP Biology Science Practice 3: Questions and Methods by explaining how questions and hypotheses are formed at the beginning...
AP® BIOLOGY 2017 SCORING GUIDELINES Question 1 (continued) Identification (3 points; 1 point per row) Null hypothesis . Increasing caffeine concentration has no effect (on the number of floral visits by bees). Control (Nectar/flowers with) no caffeine . Predicted results • The number of floral visits by bees is different at increasing caffeine
Lecture and Practice with Null Hypothesis, Alternative Hypothesis, and 95% Confidence Intervals for AP Bio. AP Biology.
In this video, students will learn to:-Write a null hypothesis that pertains to the investigation; -Determine the degrees of freedom (df) for an investigatio...
The null hypothesis is the claim against which we are looking for evidence in an investigation, specifically that the population proportions are what we would expect given random chance. For example, if we were rolling a standard six-sided die, our null hypothesis would be that the proportion of 1's, 2's, 3's, 4's, 5's, and 6's ...
Alternative hypothesis. Alternative hypothesis \(\left(H_{A}\right)\): If we conclude that the null hypothesis is false, or rather and more precisely, we find that we provisionally fail to reject the null hypothesis, then we provisionally accept the alternative hypothesis.The view then is that something other than random chance has influenced the sample observations.
A statement that is accepted if the sample data provide sufficient evidence that the null hypothesis is false. Chi-square test A statistical method of testing for an association between two categorical variables.
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.
An example of the null hypothesis is that light color has no effect on plant growth. The null hypothesis (H 0) is the hypothesis that states there is no statistical difference between two sample sets. In other words, it assumes the independent variable does not have an effect on the dependent variable in a scientific experiment.
To distinguish it from other hypotheses, the null hypothesis is written as H 0 (which is read as "H-nought," "H-null," or "H-zero"). A significance test is used to determine the likelihood that the results supporting the null hypothesis are not due to chance. A confidence level of 95% or 99% is common. Keep in mind, even if the confidence level is high, there is still a small chance the ...
the null hypothesis (that chance alone can explain the difference) or so far apart that the null hypothesis must be rejected. Accept the null hypothesis Reject the null hypothesis . Degrees of Freedom Probability 0.90 0.50 0.25 0.10 0.05 0.01 1 . 0.016 0.46 1.32 2.71 3.84 6.64 . 2 . 0.21 1.39 2.77 4.61 5.99 9.21 . 3
The null hypothesis is proposed by a scientist before completing an experiment, and it can be either supported by data or disproved in favor of an alternate hypothesis. Let's consider some ...
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.
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 ...
Null Hypothesis Examples. "Hyperactivity is unrelated to eating sugar " is an example of a null hypothesis. If the hypothesis is tested and found to be false, using statistics, then a connection between hyperactivity and sugar ingestion may be indicated. A significance test is the most common statistical test used to establish confidence in a ...
c. Do you reject the null hypothesis? Yes or No 3. When studying animal behavior, the distribution of organisms within a choice chamber can be studied to identify animal preferences. qq fruit flies are placed in a 2-choice choice chamber with large middle passage where flies may remain. Chamber A contains a 5 g sample of over ripe grapes; the
Paul Andersen shows you how to calculate the ch-squared value to test your null hypothesis. He explains the importance of the critical value and defines the...
If the calculated value is higher than the critical value in the table at the 0.05 level of significance, reject the null hypothesis and conclude that there IS a significant association between the variables. For example, with a DF=1, a value greater than 3.841 is required to be considered statistically significant (at p = 0.05).
Lesson 3: The idea of significance tests. 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.
Explanation: . The null hypothesis is what we intend to either reject or fail to reject using our sample data. In this case, the null hypothesis is that Josh cannot shoot more than 50 points on average, and Josh's performance in 10 games are the sample data we use to assess this hypothesis.