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. 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, there is enough evidence to claim a difference in average scores across the different sections Since p ≥ 0.05, we do not have 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 section
Inverse Normal Distribution
The method used for finding the corresponding z-critical value in a normal distribution using the known probability is said to be an inverse normal distribution. The inverse normal distribution is a continuous probability distribution with a family of two parameters.
Mean, Median, Mode
It is a descriptive summary of a data set. It can be defined by using some of the measures. The central tendencies do not provide information regarding individual data from the dataset. However, they give a summary of the data set. The central tendency or measure of central tendency is a central or typical value for a probability distribution.
Z-Scores
A z-score is a unit of measurement used in statistics to describe the position of a raw score in terms of its distance from the mean, measured with reference to standard deviation from the mean. Z-scores are useful in statistics because they allow comparison between two scores that belong to different normal distributions.
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.
- 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, there is enough evidence to claim a difference in average scores across the different sections
- Since p ≥ 0.05, we do not have 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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