A. The GPAs of all 5540 students enrolled at a University have an approximately normal distribution with a mean of 3.02 and a standard deviation of 0.29. Let x be the mean GPA of a random sample of 48 students selected from this University. Find the mean and standard deviation of x , and comment on the shape of its sampling distribution.
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To find the mean (μ) and standard deviation (σ) of the sample mean (x), we can use the formulas for the sampling distribution.
The mean of the sample mean is the same as the population mean:
Mean of x = μ = 3.02
The standard deviation of the sample mean (also known as the standard error) can be calculated using the formula:
Standard deviation of x = σ / √n
where σ is the standard deviation of the population and n is the sample size.
Standard deviation of x = 0.29 / √48 ≈ 0.0417
So, the mean of the sample mean (x) is 3.02 and the standard deviation is approximately 0.0417.
Now, let's comment on the shape of the sampling distribution. According to the Central Limit Theorem, when the sample size is large enough (typically 30 or more), the sampling distribution of the sample mean will be approximately normal, regardless of the shape of the population from which the samples are taken. In this case, the sample size is 48, which is considered large enough. Therefore, the sampling distribution of the sample mean will be approximately normal.