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 _ { \bar { x } } = \\)
\\( \sigma _ { \bar { 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.
for a sample of size 35, state the mean of the sample mean and the standard deviation of the sample mean. round to two decimal places if necessary.

Explanation:

Step1: Central Limit Theorem (CLT)

The CLT states that for a sample of size \(n\), if \(n\geq30\), the sampling distribution of the sample mean \(\bar{X}\) is approximately normal, regardless of the population distribution. If \(n < 30\) and the population is not normal, we cannot assume the sampling - distribution of the sample mean is normal.
For the first multiple - choice question:
Since the population is not normal and \(n = 13<30\), we cannot say anything about the sampling distribution of the sample mean.
For the formula of the mean and standard deviation of the sample mean:
The mean of the sample mean \(\mu_{\bar{x}}\) is equal to the population mean \(\mu\). The standard deviation of the sample mean (also known as the standard error) is \(\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}\), where \(\sigma\) is the population standard deviation and \(n\) is the sample size.
Given \(\mu = 17\) and \(\sigma = 5\), \(n = 13\)
\(\mu_{\bar{x}}=\mu=17\)
\(\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}=\frac{5}{\sqrt{13}}\approx1.39\)
For the second multiple - choice question:
Since \(n = 35>30\), by the CLT, the sampling distribution of the sample mean is approximately normal.
Given \(n = 35\), \(\mu = 17\) and \(\sigma = 5\)
\(\mu_{\bar{x}}=\mu = 17\)
\(\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}=\frac{5}{\sqrt{35}}\approx0.84\)

Answer:

• First multiple - choice: 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.
• \(\mu_{\bar{x}} = 17\), \(\sigma_{\bar{x}}\approx1.39\)
• Second multiple - choice: 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.
• \(\mu_{\bar{x}} = 17\), \(\sigma_{\bar{x}}\approx0.84\)