Question

a random variable is not normally distributed, but it is mound shaped. it has a mean of 17 and a standard deviation of 5.
if you take a sample of size 13, can you say what the shape of the sampling distribution for the sample mean is? why?
if the sample size is 13, then you cant say anything about the sampling distribution of the sample mean, since the population of the random variable is not normally distributed and the sample size is less than 30.
if the sample size is 13, then you can say the sampling distribution of the sample mean is not normally distributed since the sample size is less than 30.
if the sample size is 13, then you can say the sampling distribution of the sample mean is normally distributed since the sample size is less than 30.
for a sample of size 13, state the mean of the sample mean and the standard deviation of the sample mean. round to two decimal places if necessary.
\\( \mu _ { \overline { x } } = \\)
\\( \sigma _ { \overline { x } } = \\)
if you take a sample of size 35, can you say what the shape of the distribution of the sample mean is? why?
if the sample size is 35, then you can say the sampling distribution of the sample mean is normally distributed since the sample size is greater than 30.
if the sample size is 35, then you cant say anything about the sampling distribution of the sample mean, since the population of the random variable is not normally distributed and the sample size is greater than 30.
if the sample size is 35, then you can say the sampling distribution of the sample mean is not normally distributed since the sample size is greater than 30.

Explanation:

Step1: Central Limit Theorem

The Central Limit Theorem states that if the population is not normally distributed, the sampling distribution of the sample mean will be approximately normal if the sample size \(n\geq30\). For \(n = 13\) (less than 30), since the population is not normal, we cannot assume the sampling distribution of the sample mean is normal.

Step2: Mean of the sample mean

The mean of the sampling distribution of the sample mean \(\mu_{\bar{x}}\) is equal to the population mean \(\mu\). Given \(\mu = 17\), so \(\mu_{\bar{x}}=\mu = 17\)

Step3: Standard deviation of the sample mean

The standard deviation of the sampling distribution of the sample mean (also known as the standard error) is \(\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}\). Given \(\sigma = 5\) and \(n = 13\), then \(\sigma_{\bar{x}}=\frac{5}{\sqrt{13}}\approx\frac{5}{3.6056}\approx1.39\)

Step4: Sampling distribution for \(n = 35\)

Since \(n=35\geq30\), by the Central Limit Theorem, the sampling distribution of the sample mean is approximately normal.

Answer:

• For the first multiple - choice question: If the sample size is 13, then you can't say anything about the sampling distribution of the sample mean, since the population of the random variable is not normally distributed and the sample size is less than 30.
• \(\mu_{\bar{x}} = 17\)
• \(\sigma_{\bar{x}}\approx1.39\)
• For the second multiple - choice question: If the sample size is 35, then you can say the sampling distribution of the sample mean is normally distributed since the sample size is greater than 30.