"Insect Fragments The Food and Drug Administration sets Food Defect Action Levels (FDALs) for some of the various foreign substances that inevitably end up in the food we eat and liquids we drink. For example, the FDAL for insect filth in peanut butter is 3 insect fragments (larvae, eggs, body parts, and so on) per 10 grams. A random sample of 50 ten-gram portions of peanut butter is obtained and results in a sample mean of x_bar=3.6 insect, fragments per ten-gram portion.
a. Why is the sampling distribution of x_bar approximately normal?"

Verified step by step guidance

1. Recall the Central Limit Theorem (CLT), which states that the sampling distribution of the sample mean \(\overline{x}\) will be approximately normal if the sample size is sufficiently large, regardless of the shape of the population distribution.
2. Identify the sample size in the problem, which is \(n = 50\). This is generally considered large enough for the CLT to apply.
3. Understand that even if the distribution of insect fragments per 10-gram portion is not normal, the distribution of the sample mean \(\overline{x}\) will tend to be normal because it is an average of many independent observations.
4. Note that the sample portions are assumed to be independent and identically distributed, which is a key condition for the CLT to hold.
5. Conclude that because the sample size is large and the observations are independent, the sampling distribution of \(\overline{x}\) is approximately normal.

Key Concepts

Sampling Distribution of the Sample Mean

The sampling distribution of the sample mean is the probability distribution of all possible sample means from samples of the same size drawn from a population. It describes how the sample mean varies from sample to sample and is fundamental for making inferences about the population mean.

Central Limit Theorem (CLT)

The Central Limit Theorem states that, for sufficiently large sample sizes, the sampling distribution of the sample mean will be approximately normal regardless of the population's distribution. This allows us to use normal probability models even when the original data are not normal.

Sample Size and Normality Approximation

A key condition for the CLT to hold is having a large enough sample size, often n ≥ 30. In this question, the sample size of 50 is large enough to ensure the sampling distribution of the mean insect fragments per portion is approximately normal, enabling valid statistical inference.