Question

a random variable is normally distributed. it has a mean of 248 and a standard deviation of 27.
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 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 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 normally distributed since the variable is normally distributed.
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.
$\mu_{\overline{x}} =$
$\sigma_{\overline{x}} =$
for a sample of size 13, find the probability that the sample mean is more than 242.
$p(\overline{x}>242)=$
if you take a sample of size 38, can you say what the shape of the distribution of the sample mean is? why?
○ if the sample size is 38, 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 38, then you can say the sampling distribution of the sample mean is normally distributed since the variable is normally distributed.
○ if the sample size is 38, 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: Determine the shape of the sampling distribution for sample size 13

According to the Central Limit Theorem, if the population is normally distributed, then the sampling distribution of the sample mean is also normally distributed for any sample size \(n\). Here, the population (random variable) is normally distributed. So, for \(n = 13\), the sampling - distribution of the sample mean is normally distributed.

Step2: Calculate the mean of the sample mean (\(\mu_{\bar{x}}\))

The mean of the sampling distribution of the sample mean is equal to the population mean. Given \(\mu=248\), so \(\mu_{\bar{x}}=\mu = 248\)

Step3: Calculate the standard deviation of the sample mean (\(\sigma_{\bar{x}}\))

The formula for 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 \(\sigma = 27\) and \(n = 13\), then \(\sigma_{\bar{x}}=\frac{27}{\sqrt{13}}\approx\frac{27}{3.60556}\approx7.49\)

Step4: Calculate \(P(\bar{x}>242)\)

First, we calculate the \(z\) - score using the formula \(z=\frac{\bar{x}-\mu_{\bar{x}}}{\sigma_{\bar{x}}}\). Substitute \(\bar{x} = 242\), \(\mu_{\bar{x}}=248\), and \(\sigma_{\bar{x}}\approx7.49\) into the formula: \(z=\frac{242 - 248}{7.49}=\frac{-6}{7.49}\approx - 0.80\)
Then, \(P(\bar{x}>242)=P(Z>-0.80)\)
Since \(P(Z > - 0.80)=1 - P(Z\leq - 0.80)\)
From the standard normal table, \(P(Z\leq - 0.80)=0.2119\)
So \(P(Z>-0.80)=1 - 0.2119 = 0.7881\)

Step5: Determine the shape of the sampling distribution for sample size 38

Again, since the population (random variable) is normally distributed, for \(n = 38\), the sampling distribution of the sample mean is normally distributed.

Answer:

• For the shape of the sampling distribution when \(n = 13\): The third option “If the sample size is 13, then you can say the sampling distribution of the sample mean is normally distributed since the variable is normally distributed.”
• \(\mu_{\bar{x}}=248\)
• \(\sigma_{\bar{x}}\approx7.49\)
• \(P(\bar{x}>242)\approx0.79\)
• For the shape of the sampling distribution when \(n = 38\): The second option “If the sample size is 38, then you can say the sampling distribution of the sample mean is normally distributed since the variable is normally distributed.”