Question

Suppose we are about to draw a random sample of n values from a distribution where μ=10 and σ=4. The shape of the distribution is not known, but it's not normal. Which of the following statements are true? There may be more than one correct statement; check all that are true.

a) The sampling distribution of the sample mean has a mean of 4 .
b) If n is very large, the distribution of the sample mean would be approximately normal.
c) The sample mean is an unbiased estimator of the population mean, regardless of the value of n.
d) The standard deviation of the sample mean is 4/√n
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Answer 1

If n is large, the distribution of the sample mean is approximately normal. The sample mean is an unbiased estimator of the population mean. The standard deviation of the sample mean is 4/√n.

Step-by-step explanation:

b) If n is very large, the distribution of the sample mean would be approximately normal.

c) The sample mean is an unbiased estimator of the population mean, regardless of the value of n.

d) The standard deviation of the sample mean is 4/√n.

The Central Limit Theorem states that if the sample size, n, is sufficiently large, the distribution of the sample mean will be approximately normal, regardless of the shape of the population. This aligns with statement b.

The sample mean is an unbiased estimator of the population mean, regardless of the sample size. This aligns with statement c.

The standard deviation of the sample mean is equal to the population standard deviation divided by the square root of the sample size (n). Thus, the standard deviation of the sample mean is 4/√n, which aligns with statement d.