The acceptable level for insect filth in a certain food item is 4 insect fragments (larvae, eggs, body parts, and so on) per 10 grams. A simple random sample of 40 ten-gram portions of the food item is obtained and results in a sample mean of $\bar{x}=4.3$ 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 assumed to be approximately normal B. The sampling distribution of $\bar{x}$ is approximately normal because the population is normally distributed. C. The sampling distribution of $x$ is approximately normal because the sample size is large enough D. The sampling distribution of $\bar{x}$ is approximately normal because the population is normally distributed and the sample size is large enough. (b) What is the mean and standard deviation of the sampling Clstribution of $\bar{x}$ assuming $\mu=4$ and $\sigma=\sqrt{4}$ ? $\mu_{x}=\square$ (Round to three decimal places as needed)

Step 1 ：The problem is asking for the mean and standard deviation of the sampling distribution of the sample mean, denoted as $\bar{x}$. The mean of the sampling distribution of $\bar{x}$ is equal to the population mean, which is given as $\mu=4$.

Step 2 ：The standard deviation of the sampling distribution of $\bar{x}$ is equal to the population standard deviation divided by the square root of the sample size. The population standard deviation is given as $\sigma=\sqrt{4}$, which simplifies to $\sigma=2$. The sample size is 40.

Step 3 ：We can calculate the standard deviation of the sampling distribution of $\bar{x}$ using these values. The formula is $\sigma_{x} = \frac{\sigma}{\sqrt{n}}$, where $\sigma$ is the population standard deviation and $n$ is the sample size.

Step 4 ：Substituting the given values into the formula, we get $\sigma_{x} = \frac{2}{\sqrt{40}}$.

Step 5 ：Solving this expression gives $\sigma_{x} \approx 0.316$.

Step 6 ：So, the mean of the sampling distribution of $\bar{x}$ is 4 and the standard deviation of the sampling distribution of $\bar{x}$ is approximately 0.316.

Step 7 ：Final Answer: $\mu_{x}=\boxed{4}$ and $\sigma_{x}=\boxed{0.316}$