Question

A normal population has a mean of $61 and standard deviation of $13. You select random samples of nine. a. Apply the central limit theorem to describe the sampling distribution of the sample mean with n=9. With the small sample size, what condition is necessary to apply the central limit theorem? Applying the central limit theorem requires the population distribution to be normal. b. What is the standard error of the sampling distribution of sample means? (Round your answer to 2 decimal places.)

Answer 1

The Central Limit Theorem allows us to say that the sampling distribution of the sample mean will be normal due to the original population's normality. The standard error of the sampling distribution with a sample size of 9 and population standard deviation of $13 is $4.33, after rounding to two decimal places.

Step-by-step explanation:

The student's question pertains to the application of the Central Limit Theorem (CLT) for sample means in the case of a random sampling from a normal population. According to the CLT, if the population from which samples are drawn is normal, then the sampling distribution of the sample mean will also be normally distributed, regardless of the sample size. This means, even in the case of a small sample size like n=9, the sampling distribution will remain normal, thanks to the original population's normality.

Furthermore, to determine the standard error of the sampling distribution of the sample means, we use the formula:

Standard Error = \(\frac{\sigma}{\sqrt{n}}\)

Where \(\sigma\) is the population standard deviation and n is the sample size. With the population standard deviation \(\sigma = $13\) and sample size n=9, the calculation is thus:

Standard Error = \(\frac{13}{\sqrt{9}}\) = \(\frac{13}{3}\) = $4.33 (rounded to two decimal places).