Assume that Fancy shoe warehouse has 500 pairs of women's shoes in stock. The distribution of the women's shoe sizes is approximately normal. The average size of women's shoes is size 8, with a standard deviation size of 4. Suppose that a random sample of 250 pairs of shoes is selected. What can be assumed about the distribution of the sample mean?

a) The distribution of the sample mean is skewed by the mean distribution theorem.
b) The distribution of the sample mean is approximately normal by the mean distribution theorem.
c) There is not enough information to make assumptions regarding the distribution of the sample mean.
d) The distribution of the sample mean is approximately normal by the central limit theorem.
e) The distribution of the sample mean is non-normal by the central limit theorem.



Answer :

Using the Central Limit Theorem, the correct option regarding the shape of the sample mean is given by:

d) The distribution of the sample mean is approximately normal by the central limit theorem.

What does the Central Limit Theorem state?

It states that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].

For a skewed variable, the sampling distribution is also approximately normal, as long as n is at least 30.

For this problem, we have a skewed underlying distribution, however the sample size is greater than 30, hence the sampling distribution of sample means is normal and option d is correct.

More can be learned about the Central Limit Theorem at https://brainly.com/question/16695444

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