The acceptable level for insect filth in a certain food item is 5 insect fragments (larvae, eggs, body parts, and so on) per 10 grams. A simple random sample of 50 ten-gram portions of the food item is obtained and results in a sample mean of $\bar{x}=5.8$ insect fragments per ten-gram portion. Complete parts (a) through (c) below.
(a) Why is the sampling distribution of $\bar{x}$ approximately normal?
A. The sampling distribution of $\bar{x}$ is approximately normal because the population is normally distributed and the sample size is large enough.
B. The sampling distribution of $\bar{x}$ is approximately normal because the sample size is large enough.
C. The sampling distribution of $\bar{x}$ is assumed to be approximately normal.
D. The sampling distribution of $\bar{x}$ is approximately normal because the population is normally distributed.
(b) What is the mean and standard deviation of the sampling distribution of $\bar{x}$ assuming $\mu=5$ and $\sigma=\sqrt{5}$ ?
$\mu_{\mathrm{x}}=\square$ (Round to three decimal places as needed.)
Answer
Final Answer: The mean of the sampling distribution of \(\bar{x}\) is \(\mu_{x}=5\) and the standard deviation is \(\sigma_{x}=\boxed{0.316}\).
Step 1 ：The sampling distribution of \(\bar{x}\) is approximately normal because the sample size is large enough.
Step 2 ：The mean of the sampling distribution of \(\bar{x}\) is equal to the population mean, which is given as \(\mu=5\).
Step 3 ：The standard deviation of the sampling distribution of \(\bar{x}\), also known as the standard error, is equal to the population standard deviation divided by the square root of the sample size.
Step 4 ：The population standard deviation is given as \(\sigma=\sqrt{5}\) and the sample size is 50.
Step 5 ：Calculate the standard deviation of the sampling distribution of \(\bar{x}\) using the formula \(\sigma_{x} = \frac{\sigma}{\sqrt{n}}\), where \(\sigma\) is the population standard deviation and \(n\) is the sample size.
Step 6 ：Substitute the given values into the formula: \(\sigma_{x} = \frac{\sqrt{5}}{\sqrt{50}}\).
Step 7 ：Simplify to get \(\sigma_{x} = 0.316\).
Step 8 ：Final Answer: The mean of the sampling distribution of \(\bar{x}\) is \(\mu_{x}=5\) and the standard deviation is \(\sigma_{x}=\boxed{0.316}\).
