A college’s data about the incoming freshmen indicates that the mean of their high school GPAs was 3.4, with a standard deviation of 0.35; the distribution was roughly mound-shaped and only slightly skewed. The students are randomly assigned to freshman writing sem- inars in groups of 25. What might the mean GPA of one of these seminar groups be? Describe the appropriate sam- pling distribution model—shape, center, and spread— with attention to assumptions and conditions. Make a sketch using the 68–95–99.7 Rule

To determine the mean GPA of one of the seminar groups, we need to understand the sampling distribution model and its characteristics.

In this case, the appropriate sampling distribution model is the sampling distribution of the mean. This model assumes that the GPAs of the incoming freshmen are approximately normally distributed.

The shape of the sampling distribution is also approximately normal. This is because the original distribution of high school GPAs was described as mound-shaped and only slightly skewed. According to the Central Limit Theorem, when the sample size is large enough (typically considered to be greater than 30), the sampling distribution is approximately normal, even if the original distribution is not. Since the students are randomly assigned to groups of 25, the sample size is large enough to apply the Central Limit Theorem.

The center of the sampling distribution is the same as the population mean, which is given as 3.4. Therefore, the mean of the seminar group would have a sampling distribution centered around 3.4.

The spread or standard deviation of the sampling distribution, also known as the standard error of the mean, can be calculated using the formula: standard deviation of the original population / square root of the sample size. In this case, the standard deviation of the high school GPAs is 0.35, and the sample size is 25. Therefore, the standard error of the mean would be 0.35 / sqrt(25) = 0.07.

To make a sketch using the 68-95-99.7 Rule, we can start by drawing a normal curve with the center at the mean GPA of 3.4. Then, we can mark the mean and three standard deviations above and below the mean.

- The interval within one standard deviation (0.07) would be from 3.4 - 0.07 to 3.4 + 0.07, which is approximately 3.33 to 3.47.
- The interval within two standard deviations (0.07 * 2) would be from 3.4 - 0.14 to 3.4 + 0.14, which is approximately 3.26 to 3.54.
- The interval within three standard deviations (0.07 * 3) would be from 3.4 - 0.21 to 3.4 + 0.21, which is approximately 3.19 to 3.61.

By using the 68-95-99.7 Rule, we can say that about 68% of the seminar groups would have a mean GPA within the range of 3.33 to 3.47, about 95% would have a mean GPA within the range of 3.26 to 3.54, and about 99.7% would have a mean GPA within the range of 3.19 to 3.61.