14. ANOVA
Introduction to ANOVA
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Four different high schools in local towns took random samples of 100 students in three grades, and collected data on the weekly time spent studying to see if students in each of these grades study on average for the same amount of time per week. The four schools ran ANOVA tests on their samples, and the F-Statistics were , , , and . Which F-Statistic is most likely to indicate the average study times across grades are not all the same?
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Step 1: Understand the purpose of the ANOVA test. ANOVA (Analysis of Variance) is used to determine if there are statistically significant differences between the means of three or more groups. In this case, it is used to compare the average study times across grades (10th, 11th, and 12th).
Step 2: Recall the role of the F-Statistic in ANOVA. The F-Statistic is a ratio of the variance between group means to the variance within the groups. A higher F-Statistic suggests that the differences between group means are more likely to be statistically significant.
Step 3: Compare the given F-Statistics. The F-Statistics provided are 2.35, 2.57, 2.81, and 3.93. The higher the F-Statistic, the stronger the evidence against the null hypothesis (which assumes that all group means are equal).
Step 4: Identify the F-Statistic most likely to indicate that the average study times are not all the same. Among the given values, 3.93 is the highest F-Statistic, which suggests the strongest evidence to reject the null hypothesis and conclude that the average study times across grades are not all the same.
Step 5: Note that the significance of the F-Statistic also depends on the critical value from the F-distribution table and the chosen significance level (e.g., α = 0.05). However, without additional information, the highest F-Statistic (3.93) is the most likely to indicate a significant difference.
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