Question

A statistics teacher taught a large introductory statistics class, with 500 students having enrolled over many years. The mean score over all those students on the first midterm was u = 68 with standard deviation o = 20. One year, the teacher taught a much smaller class of only 25 students. The teacher wanted to know if teaching a smaller class affected scores in any way. We can consider the small class as an SRS of the students who took the large class over the years. The average midterm score was * = 78. The hypothesis the teacher tested was He : 4 = 68 vs. H. : #68. The P-value for this hypothesis was found to be: 0.0233 O 0.0248 O 0.0124. O 0.0062

Answer 1

A statistics instructor tested whether teaching a smaller class affected midterm scores by comparing a small class's average score to the historical average. Using hypothesis testing, the instructor found a statistically significant difference with a P-value less than 0.05.

Step-by-step explanation:

The question involves a statistics instructor who has taught large classes and is now analyzing the impact of teaching a smaller class on the students' midterm scores. Using a sample of 25 students, they found that the average score was 78, whereas the mean score over many years with larger classes was 68. To investigate whether this difference was statistically significant, the instructor performed a hypothesis test with the null hypothesis H0: μ = 68 and the alternative hypothesis H1: μ ≠ 68 (two-tailed test). A P-value was calculated, which is used to determine if the observed difference is likely due to chance or if there is evidence to suggest that the smaller class size had an effect on the scores.

Given that the P-value is less than the commonly used alpha level of 0.05, the result is statistically significant and suggests there may be an effect of class size on scores. Yet, without additional details such as effect size or confidence intervals, it is not possible to determine the practical significance of the finding.

Answer 2

The question is about conducting a hypothesis test in statistics to compare the mean score of a smaller class size to the mean of a larger class size over the years, and determining the significance of the result based on the P-value. A t-test would be used in this scenario due to the small sample size.

Step-by-step explanation:

The question is related to a statistics hypothesis testing scenario where a teacher wants to determine if the mean score of a small class is different from that of a larger class over many years. The null hypothesis (H0) being tested is that the mean score is 68 (μ = 68), with the alternative hypothesis (Ha) being that the mean score is not equal to 68 (μ ≠ 68). A P-value is provided as a measure of the strength of evidence against H0. The calculation of the P-value involves determining the probability of obtaining a sample mean as extreme as, or more extreme than, the one observed if the null hypothesis is true. The hypothesis test can be performed using a t-test for a small sample size when the population standard deviation is known, which seems to be the case here.

To decide whether the null hypothesis should be rejected, we compare the P-value to the chosen level of significance, typically α = 0.05. If the P-value is less than α, the null hypothesis is rejected, suggesting a statistically significant difference in means. In this case, the hypothetical answer to the P-value is not provided, but options are given (0.0233, 0.0248, 0.0124, and 0.0062). In such a test, if the given P-value were, for example, 0.0233, it would indicate that there is about a 2.33% chance of observing a sample mean of 78 or more extreme, assuming the true mean is 68. Since 0.0233 is less than 0.05, we would reject the null hypothesis at the 5% level of significance.