Question

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.
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.
\\( \mu _ { \overline { x } } = \\)
\\( \sigma _ { \overline { x } } = \\)

Explanation:

Step1: Mean of sample mean formula

The mean of the sample mean ($\mu_{\bar{x}}$) is equal to the population mean ($\mu$). So, $\mu_{\bar{x}}=\mu$.

Step2: Standard deviation of sample mean formula

The standard deviation of the sample mean ($\sigma_{\bar{x}}$) is given by the formula $\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}$, where $\sigma$ is the population standard deviation and $n$ is the sample size.

Assuming the population mean $\mu = 17$ (from the first part where for $n = 13$, $\mu_{\bar{x}}=17$) and population standard deviation $\sigma=1.39\times\sqrt{13}\approx5$ (since for $n = 13$, $\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{13}} = 1.39$, so $\sigma=1.39\times\sqrt{13}\approx5$)

For $n = 35$:

Mean of sample mean

Since $\mu_{\bar{x}}=\mu$, and $\mu = 17$, so $\mu_{\bar{x}}=17$

Standard deviation of sample mean

Using the formula $\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}$, with $\sigma\approx5$ and $n = 35$
$\sigma_{\bar{x}}=\frac{5}{\sqrt{35}}\approx\frac{5}{5.92}\approx0.84$

Answer:

$\mu_{\bar{x}} = 17$
$\sigma_{\bar{x}}\approx0.84$