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 $\sigma=\sqrt{5}$ ?
$\mu_{\mathrm{X}}=5$ (Round to three decimal places as needed)
$\sigma_{\bar{x}}=0.316$ (Round to three decimal places as needed.)
(c) What is the probability a simple random sample of 50 ten-gram portions of the food item results in a mean of at least 5.8 insect fragments?
$P(\bar{x} \geq 5.8)=\square$ (Round to four decimal places as needed.)

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 \(\mu_{\mathrm{X}}=5\) and the standard deviation is \(\sigma_{\bar{x}}=0.316\).

Step 3 ：We need to find the probability that the sample mean is greater than or equal to 5.8. This is a question about the sampling distribution of the mean. The Central Limit Theorem tells us that the sampling distribution of the mean is approximately normal if the sample size is large enough. In this case, the sample size is 50, which is generally considered large enough for the Central Limit Theorem to apply.

Step 4 ：The mean of the sampling distribution of the mean is equal to the population mean, which is given as 5. The standard deviation of the sampling distribution of the mean is equal to the population standard deviation divided by the square root of the sample size. The population standard deviation is given as the square root of 5.

Step 5 ：To find the probability that the sample mean is greater than or equal to 5.8, we need to standardize 5.8 by subtracting the mean of the sampling distribution and dividing by the standard deviation of the sampling distribution. This will give us a z-score, which we can look up in a standard normal distribution table to find the corresponding probability.

Step 6 ：Final Answer: The probability that a simple random sample of 50 ten-gram portions of the food item results in a mean of at least 5.8 insect fragments is \(\boxed{0.0057}\).