5.46 Student performance across discussion sections: 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 final exam 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. A) Ho: μ1 = μ2 = ... = μ8 Ha: At least one of the means is different B) Ho: μ1 = μ2 = ... = μ8 Ha: μ1 ≠ μ2 ≠ ... ≠ μ8 C) Ho: μ1 = μ2 = ... = μ8 Ha: At least one pair of means is the same (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: A) Since p ≥ 0.05, there is no difference in average scores across the different sections B) Since p ≥ 0.05, we do not have enough evidence to claim a difference in average scores across the different sections C) Since p is close to 0.05, we cannot make a decision D) Since p ≥ 0.05, there is enough evidence to claim a difference in average scores across the different sections

5.46 Student performance across discussion sections: 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 final exam 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. A) Ho: μ1 = μ2 = ... = μ8 Ha: At least one of the means is different B) Ho: μ1 = μ2 = ... = μ8 Ha: μ1 ≠ μ2 ≠ ... ≠ μ8 C) Ho: μ1 = μ2 = ... = μ8 Ha: At least one pair of means is the same (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: A) Since p ≥ 0.05, there is no difference in average scores across the different sections B) Since p ≥ 0.05, we do not have enough evidence to claim a difference in average scores across the different sections C) Since p is close to 0.05, we cannot make a decision D) Since p ≥ 0.05, there is enough evidence to claim a difference in average scores across the different sections

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MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

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5.46 Student performance across discussion sections: 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 final exam 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
n_i	33	19	10	29	33	10	32	31
x̄_i	92.94	91.11	91.8	92.45	89.3	88.3	90.12	93.45
s_i	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.

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

B) Ho: μ₁ = μ₂ = ... = μ₈
Ha: μ₁ ≠ μ₂ ≠ ... ≠ μ₈

C) Ho: μ₁ = μ₂ = ... = μ₈
Ha: At least one pair of means is the same

(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:

A) Since p ≥ 0.05, there is no difference in average scores across the different sections

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

C) Since p is close to 0.05, we cannot make a decision

D) Since p ≥ 0.05, there is enough evidence to claim a difference in average scores across the different sections

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

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