Practice Problem #16 pg. 307: A researcher theorized that people can hear better when they have just eaten a large meal. Six individuals were randomly assigned to eat either a large meal or a small meal. After eating the meal, their hearing was tested. The hearing ability scores (high numbers indicate greater ability) are given in the following table. Using the .05 level, do the results support the researcher’s theory? (a) Use the steps of hypothesis testing, (b) sketch the distributions involved.. See Chart Below

A,B & C being the Big Meal Group and their corresponding scores, and D, E & F being the Small Meal Group and their corresponding scores

Subject Hearing

A 22

B 25

C 25
D 19
E 23
F 21

To determine whether the results support the researcher's theory that people can hear better when they have just eaten a large meal, we can follow the steps of hypothesis testing:

Step 1: State the hypotheses:
- Null hypothesis (H0): There is no difference in hearing ability between those who ate a large meal and those who ate a small meal.
- Alternative hypothesis (Ha): People who ate a large meal have better hearing ability than those who ate a small meal.

Step 2: Set the significance level:
The significance level, denoted as α, determines the threshold for rejecting the null hypothesis. In this case, α is given as .05, which means we will reject the null hypothesis if the probability of obtaining the observed data under the null hypothesis is less than 0.05.

Step 3: Calculate the test statistic:
Here, we can use an independent samples t-test to compare the means of two groups. We need to calculate the mean and standard deviation for each group, and then calculate the t-value.

Let's calculate the means and standard deviations for the two groups:

Big Meal Group:
Mean (μ1) = (22 + 25 + 25) / 3 = 24
Standard deviation (σ1) = √[(22-24)² + (25-24)² + (25-24)²] / √3 ≈ 1.73

Small Meal Group:
Mean (μ2) = (19 + 23 + 21) / 3 = 21
Standard deviation (σ2) = √[(19-21)² + (23-21)² + (21-21)²] / √3 ≈ 1.73

Now, we can calculate the t-value using the formula:
t = (μ1 - μ2) / √((σ1²/n1) + (σ2²/n2))
where n1 and n2 are the sample sizes (3 in this case).

t = (24 - 21) / √((1.73²/3) + (1.73²/3))
≈ 1.732 / √(0.99 + 0.99)
≈ 1.732 / √1.98
≈ 1.732 / 1.408
≈ 1.23

Step 4: Determine the critical value:
To determine the critical value, we need to refer to the t-distribution table with the degrees of freedom, which is (n1 + n2 - 2). In this case, the degrees of freedom are 3 + 3 - 2 = 4.

At α = 0.05 and 4 degrees of freedom, the critical value is approximately 2.776 (two-tailed test).

Step 5: Compare the test statistic with the critical value:
Since the absolute value of the calculated t-value (1.23) is less than the critical value (2.776), we fail to reject the null hypothesis.

Step 6: Make the decision:
Based on the hypothesis test, the results do not support the researcher's theory. There is not enough evidence to conclude that people can hear better after eating a large meal.

To sketch the distributions involved, you can create two separate histograms or boxplots for the hearing ability scores of the two groups (big meal vs. small meal). The x-axis would represent the hearing ability scores, and the y-axis would represent the frequency or count of each score in the group. The distributions can be compared visually to observe any differences.