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.
Big Meal Group Small Meal Group
Subject Hearing Subject Hearing
A 22 D 19
B 25 E 23
C 25 F 21
To determine whether the results support the researcher's theory, we can follow the steps of hypothesis testing. The steps include:
Step 1: State the null hypothesis (H0) and alternative hypothesis (Ha):
The null hypothesis states that there is no difference in hearing ability between people who have eaten a large meal and those who have eaten a small meal. The alternative hypothesis suggests that people who have eaten a large meal have better hearing ability.
H0: μ_bigMeal = μ_smallMeal
Ha: μ_bigMeal > μ_smallMeal
Step 2: Set the significance level (α):
The significance level, denoted as α, is the threshold at which we would reject the null hypothesis. In this case, the significance level is given as .05 (or 5%).
α = 0.05
Step 3: Determine the test statistic:
To compare the hearing ability between the two groups, we can use a t-test for independent samples since the groups are randomly assigned and independent. The test statistic, t, can be calculated using the following formula:
t = (X̄1 - X̄2) / sqrt((s1^2 / n1) + (s2^2 / n2))
Where:
- X̄1 and X̄2 are the means of the big meal and small meal groups, respectively.
- s1 and s2 are the standard deviations of the big meal and small meal groups, respectively.
- n1 and n2 are the sample sizes of the big meal and small meal groups, respectively.
Step 4: Determine the critical value(s):
Since we have a one-sided test (the alternative hypothesis suggests a directional difference), we need to compare the test statistic to the critical value from the t-distribution table with degrees of freedom equal to (n1 + n2 - 2).
Step 5: Calculate the p-value:
The p-value represents the probability of obtaining the observed test statistic (or a more extreme value) under the null hypothesis. To calculate the p-value, we can use the t-distribution and the calculated test statistic.
Step 6: Make a decision:
Based on the calculated p-value and the significance level, we can make a decision to either reject or fail to reject the null hypothesis.
Without the complete hearing ability scores for all individuals in both groups, we cannot proceed with the remaining steps to provide a definitive answer. Additionally, we would need the sample sizes (n1 and n2) and the standard deviations (s1 and s2) for each group.
Please provide the missing information, and we will be able to help you further with the hypothesis testing process and the decision.