Question

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 40 ten - gram portions of the food item is obtained and results in a sample mean of $overline{x}=5.4$ insect fragments per ten - gram portion. complete parts (a) through (c) below
(a) why is the sampling distribution of $overline{x}$ approximately normal?
○ a. the sampling distribution of $overline{x}$ is approximately normal because the sample size is large enough
○ b. the sampling distribution of $overline{x}$ is approximately normal because the population is normally distributed
○ c. the sampling distribution of $overline{x}$ is assumed to be approximately normal.
○ d. the sampling distribution of $overline{x}$ is approximately normal because the population is normally distributed and the sample size is large enough.

Explanation:

According to the Central Limit Theorem, if the sample size \(n\) is large (\(n\geq30\)), the sampling distribution of the sample mean \(\bar{x}\) is approximately normal, regardless of the shape of the population distribution. Here, the sample size \(n = 40\geq30\).

Answer:

A. The sampling distribution of \(\bar{x}\) is approximately normal because the sample size is large enough.