Question

5. A random variable is not normally distributed, but it is mound shaped. It has a mean of 14 and a standard deviation of 3. (Section 6.5)

a.) If you take a sample of size 10, can you say what the shape of the sampling

distribution for the sample mean is? Why?

b.) For a sample of size 10, state the mean of the sample mean and the standard deviation of the sample mean.

c.) If you take a sample of size 35, can you say what the distribution of the

sample mean is?

Answer 1

a.) If you take a sample of size 10, can you say what the shape of the sampling distribution for the sample mean is? Why?

We know that sampling distribution of the sample mean follows an approximately normal distribution even if the random variable is not normally distributed. So, if we take a sample of size 10, we can say that the shape of the sampling distribution for the sample mean is approximately bell-curved or normal curve.

b.) For a sample of size 10, state the mean of the sample mean and the standard deviation of the sample mean.

We know that the estimate for a mean of the sampling distribution of the sample means is given as the population mean and standard deviation of the sampling distribution of the sample mean is given as the standard error of the population mean.

Estimate for mean for sampling distribution of sample means = 14

We are given sample size = n = 10

Estimate for standard deviation for sampling distribution of sample means = Standard error = σ/sqrt(n)

Estimate for standard deviation for sampling distribution of sample means = 3/sqrt(10) = 0.948683298

Estimate for standard deviation for sampling distribution of sample means = 0.948683298

c.) If you take a sample of size 35, can you say what the distribution of the sample mean is?

The estimate for the mean for sampling distribution of sample means is same as the population mean even if we change the samples with different sizes.

So, estimate for mean of the sampling distribution = 14

We are given sample size = n = 35

Estimate for standard deviation of sampling distribution = σ/sqrt(n) = 3/sqrt(35) = 0.507092553