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Statistical Analysis

Introduction

In this laboratory, we will practice measuring length and weight, using the metric system, performing a statistical analysis, and using scientific method. A case study in which Sara makes an observation concerning the shape of pine trees is presented. Students will create a hypothesis, design a test of the hypothesis, collect data by taking measurements, perform a statistical analysis and then draw conclusions about the hypothesis.

Statistical Analysis

  Suppose that we flipped a coin 10 times and obtained 8 heads and two tails. We would like to know if one side of the coin is heavier because we expected 5 heads and 5 tails. We must use statistical analysis to determine what the probability is of getting a 8 heads if the expected number is 5. This is like asking "does 8 heads and 2 tails differ significantly from the expected 5 heads and 5 tails?"

Statistical tests test the hypothesis that there is no difference. In this case, we test the hypothesis that 8 heads and 2 tails is not different than 5 heads and 5 tails. The hypothesis of no difference is called the null hypothesis (abbreviated H₀). The alternative hypothesis (H_A) in this case is that 8 heads did not happen by chance. Perhaps the coin really is heavier on one side.

It is theoretically possible to flip a coin 1000 times and get all heads. It is very unlikely to get 1000 heads in a row but it is still possible. Suppose that you get 427 heads and 573 tails. You expect 500 and 500. If the probability of obtaining 427-573 is 30% (or 0.3), then we will conclude that it  could happen and that there is no reason to conclude that the coin is heavier on one side. On the other hand, if the probability is 0.000001 (one out of one million), we would obviously conclude that it is very unlikely; the coin must be heavier on one side.

A statistical test is done to determine the probability that the hypothesis of no difference (null hypothesis) is true. Normally, if the probability is less than 0.05 (1 in 20), then we conclude that it is different, that is, 8 and 2 is unlikely to have happened by chance. We will be wrong 5% of the time.

Normally, when doing statistical tests, we reject the null hypothesis if p < .05. If a 5% error rate is not acceptable, we could reject the null hypothesis only if p < .01 or .001 or .00000001. The advantage of this is that if we reject the null hypothesis and conclude that the two are really different, we are more likely to be correct. The disadvantage is that we are more likely to accept the null hypothesis when it is really incorrect. Increased sample size decreases the likelihood of this kind of error.

Example

Suppose that a researcher wishes to test if a certain kind of growth hormone will produce faster growth in mice. She injects 10 mice with the hormone and uses another 10 as a control. Three weeks later, she weighs the mice and discovers that the mean weight of mice that have received the injections is 12.1 g and the mean weight of control mice is 9.3 g. These values indicate that the mice receiving the hormone are heavier. Is her value of 12.1 significantly different than 9.3? Is it possible that the hormone has no effect, that the weight difference between the two groups is due to chance? This is like flipping a coin 10 times. You expect 5 heads and 5 tails but you might get 6 heads or 7 heads or 8 heads. Similarly, if the hormone does not work, you expect the mean for the two groups to be similar but it may not be exactly the same.

Statistical tests are used to test the hypothesis that there is no difference. They tell the probability that the null hypothesis is true. In this example, we must test the hypothesis that the mean weight of hormone-treated mice is not different than the mean weight of control mice. The mean can be abbreviated using the Greek letter . The null hypothesis is: the means of the two groups do not differ.

H₀:

The alternative hypothesis is: mice receiving the hormone are heavier.

H_A:

We have no reason to suspect that the mean of the control might be greater. If we were not sure which group to expect to be heavier, the alternative hypothesis would be:

H_A:

A statistical test can be used to test the null hypothesis. The test will give the probability that the null hypothesis is true. If the probability of the null hypothesis being true is <0.05, then we will reject the null hypothesis and accept the alternative hypothesis. There are several tests available for this hypothesis. A commonly used test for data that are normally distributed (have a bell-shaped curve) is a t-test.

Group 1 - Hormone -  Weight (grams)	Group 2 – No Hormone -  Weight (grams)
12.5	12
13	8.5
12	10
12	8
13	8
14	13.5
13	9
10.5	8.5
9.5	6.5
11	9
Mean = 12.1	Mean = 9.3

Sara's Hypothesis: Newborn mice injected with the hormone will be heavier after 3 weeks of growth than mice without the hormone.

For statistical purposes:

The null hypothesis (H₀) for a t-test is: mean (group1) = mean (group 2).

The alternative hypothesis (H_A) is: mean (group1) > mean (group 2).

A t-test reveals that p (the probability that the null hypothesis is correct) = 0.0011. Because p < 0.05, we reject the null hypothesis and accept the alternative hypothesis. Acceptance of the alternative hypothesis leads us to also accept Sara's original hypothesis.

Number of Tails

There are two ways to perform this test. The following alternative hypothesis would lead us to perform a one tailed test:

H_A: The mean weight of mice injected with the hormone will be greater than the mean weight of the control mice.

The following alternative hypothesis would lead us to perform a two tailed test:

H_A: The mean weight of mice injected with the hormone will be different than the mean weight of the control mice.

Case Study

Sara is studying human physiology and she wishes to know if the following affect breathing.

sex

age

fitness level

smoking

She also wishes to know if these four items affect heart rate.

Hypotheses

There are 8 items listed above. Create a hypothesis that addresses one of these items. 

Your hypothesis should be a statement, not a question because a statement can be tested but a question cannot. The following is not a hypothesis because it cannot be tested; it is neither true nor false: Do males have a faster heart rate than females? The data collected will not be able to support or reject this question.

Create a hypothesis that can be tested during this class period and write the hypothesis in the space provided on your answer sheet. Create your hypothesis before you collect data.

After you create your hypothesis, inform your instructor to verify that you created a hypothesis that can be tested today. After your hypothesis is approved, collect the data as described below. 

Data Collection

You will test your hypothesis using data from the entire class.

Sit still for 2 minutes and then measure your breathing rate (breaths per second) and heart rate (beats per minute). Give this information to your instructor or record it with the class data on the board. You will also record your age, whether or not you smoke, and whether or not you exercise regularly (3 times per week for at least 30 minutes).

Copy the class data to the data table in the answer sheet.

Data Analysis

Choose two variables in the class data (see below) and calculate a mean and a standard deviation for each using the Excel spreadsheet provided. Record the two means and standard deviations of each in the answer sheet. Some examples of variables are:

heart rate of males and heart rate of females

heart rate of the oldest half of the class and heart rate of the youngest half of the class

heart rate of people who do regular exercise and heart rate of people who do not

heart rate of smokers and heart rate of nonsmokers

breathing rate of males and breathing rate of females

breathing rate of the oldest half of the class and breathing rate of the youngest half of the class

breathing rate of people who do regular exercise and breathing rate of people who do not

breathing rate of smokers and breathing rate of nonsmokers

Scatter Plot

It may be interesting to determine if breathing rate is related to heart rate. Use Create A Graph to create a scatter plot of breathing rate and heart rate.

Is there a relationship between breathing rate and heart rate? How do you know?  Answer these questions on the answer sheet and Attach a scatter plot.

T-test

Before performing a statistical analysis, it is necessary to state a null hypothesis (H₀) and an alternative hypothesis (H_A) for your data. Record your null and alternative hypotheses in the data table. Some possible null and alternative hypotheses are below. In the hypotheses below, = the mean of one variable such as heart rate of males; = the mean of the other variable such as heart rate of females. Select one hypothesis from this list and record it as the null hypothesis (H₀) on the answer sheet. Select one and record it as the alternative hypothesis (H_A).

Choose from these possible hypotheses:

An Excel spreadsheet for performing a t-test has been provided. Use this spreadsheet to perform a t-test on your data.

Record the value of p on your answer sheet.

Do you accept or reject the null hypothesis (H₀)? Do you accept the alternative (H_A)? Explain why on the answer sheet.

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