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High school students can be categorized into two groups by the amount of activities they are involved in. Let group 1 consist of all high school students who are very involved in sports and other activities and group 2 consist of all high school students who aren't. The distributions of GPAs in both groups are approximately normal. The mean and standard deviation for group 1 are 2.9 and 0.4, respectively. The mean and standard deviation for group 2 are 2.7 and 0.5, respectively. Independent random samples of 50 high school students are to be selected from both groups (for a total of 100 students).
-What is the probability that the average GPA in the sample of students who are not so involved is higher than the average GPA in the sample of students who are very involved?
A)

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Use the following information for questions:
 High school students can be categorized into two groups by the amount of activities they are involved in. Let group 1 consist of all high school students who are very involved in sports and other activities and group 2 consist of all high school students who aren't. The distributions of GPAs in both groups are approximately normal. The mean and standard deviation for group 1 are 2.9 and 0.4, respectively. The mean and standard deviation for group 2 are 2.7 and 0.5, respectively. Independent random samples of 50 high school students are to be selected from both groups (for a total of 100 students).
-What is the probability that the average GPA in the sample of students who are not so involved is higher than the average GPA in the sample of students who are very involved?
A) < 0.0001
B) 0.0136
C) 0.0582
D) 0.9864

Quizplus · Accepted Answer

The probability that the average GPA of group 2 is higher than group 1 can be found by calculating the difference in their means and the standard deviation of this difference. The mean difference is $2.7 - 2.9 = -0.2$. The standard deviation of the difference in sample means is $\sqrt{\frac{0.4^2}{50} + \frac{0.5^2}{50}}$. Using the Z-score formula, we find the probability corresponding to this Z-score, which is approximately 0.0136, indicating a low but possible chance that the average GPA of group 2 is higher than that of group 1.

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