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Answer :
Final answer:
The mean of the sampling distribution is equal to the population mean, which is 163. The standard deviation, or the standard error, is found using the formula σ/√n, which in this case is 25/√52. The closest option to the calculated standard error is Option D.
Explanation:
The question is asking for the mean and standard deviation of the sampling distribution of the sample means for a population with a certain mean and standard deviation, given a specific sample size. For any population with a mean (μ) and a standard deviation (σ), the mean of the sampling distribution of the sample means will always be equal to the population mean. This concept is based on the central limit theorem. Therefore, the mean of the sampling distribution is 163.
However, the standard deviation of the sampling distribution (often called the standard error) is calculated by dividing the population standard deviation by the square root of the sample size (n). The formula is σ/√n. Plugging the values in, we get:
Standard Deviation = 25/√52 = 25/√52 ≈ 3.47
Since none of the provided options exactly match this calculated value, we would choose the closest option, which suggests that the standard deviation of the sampling distribution is 25/√52. Option D appears to be the closest approximation to our calculated value.
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