Question

A professor who teaches a large introductory statistics class (197 students) with eight discussion sections would like to test if student performance differs by discussion section, where each discussion section has a different teaching assistant. The summary table below shows the average score for each discussion section as well as the standard deviation of scores and the number of students in each section.

	Sec 1	Sec 2	Sec 3	Sec 4	Sec 5	Sec 6	Sec 7	Sec 8
ni	33	19	10	29	33	10	32	31
x̄i	92.94	91.11	91.8	92.45	89.3	88.3	90.12	93.45
si	4.21	5.58	3.43	5.92	9.32	7.27	6.93	4.57

The ANOVA output below can be used to test for differences between the average scores from the different discussion sections.

	Df	Sum Sq	Mean Sq	F value	Pr(>F)
section	7	525.01	75	1.87	0.0767
residuals	189	7584.11	40.13

Conduct a hypothesis test to determine if these data provide convincing evidence that the average score varies across some (or all) groups. Check conditions and describe any assumptions you must make to proceed with the test.
(a) Write the hypotheses for evaluating if there is a difference between average scores across sections in your own words.

• Ho: μ₁ = μ₂ = ... = μ₈
  Ha: At least one pair of means is the same
• Ho: μ₁ = μ₂ = ... = μ₈
  Ha: μ₁ ≠ μ₂ ≠ ... ≠ μ₈
• Ho: μ₁ = μ₂ = ... = μ₈
  Ha: At least one of the means is different

(b) Assume that the conditions required for this inference are satisfied.
(c) What is the test statistic associated with this ANOVA test?
(please round to two decimal places)
(d) What is the p-value associated with this ANOVA test?
(please round to four decimal places)

(e) Interpret the conclusion of the test in the context of the study:

• Since p ≥ 0.05, we do not have enough evidence to claim a difference in average scores across the different sections
• Since p ≥ 0.05, there is enough evidence to claim a difference in average scores across the different sections
• Since p is close to 0.05, we cannot make a decision
• Since p ≥ 0.05, there is no difference in average scores across the different sections

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Answer 1

we want to test for differences between the average scores from the different discussion sections.

Conduct a hypothesis test to determine if these data provide convincing evidence that the average score varies across some (or all) groups

(a) Write the hypotheses for evaluating if there is a difference between average scores across sections in your own words.

An alternative hypothesis will be at least one mean is different.

• Ho: μ1 = μ2 = ... = μ8
  Ha: At least one of the means is different

(b) Assume that the conditions required for this inference are satisfied.

(c) What is the test statistic associated with this ANOVA test?

From the table,

The test statistic value is F value in the table.

F value = 1.87

Answer:- 1.87

(d) What is the p-value associated with this ANOVA test?

From the table value corresponding to "Pr(>F) " is the P-value.

P-value = 0.0767

we reject Ho if the P-value < level of significance.

(e) Interpret the conclusion of the test in the context of the study:

p-value = 0.0767 > 0.05.we fail to reject Ho.

So we do not have sufficient evidence to support the claim of difference in means.

• Since p ≥ 0.05, we do not have enough evidence to claim a difference in average scores across the different sections