Question

the acceptable level for insect filth in a certain food item is 3 insect fragments (larvae, eggs, body parts, and so on) per 10 grams. a sample of 50 ten-gram portions of the food item is obtained and results in a sample mean of x̄ = 3.2 insect fragments per ten-gram portion. complete parts (a) through (c) below. click here to view the standard normal distribution table (page 1). click here to view the standard normal distribution table (page 2). (a) why is the sampling distribution of x̄ approximately normal? a. the sampling distribution is approximately normal because the sample size is large enough. b. the sampling distribution is approximately normal because the population is normally distributed. c. the sampling distribution is assumed to be approximately normal. d. the sampling distribution is approximately normal because the population is normally distributed and the sample size is large enough.

Explanation:

To determine why the sampling distribution of \(\bar{x}\) is approximately normal, we use the Central Limit Theorem (CLT). The CLT states that for a sample size \(n\) (usually \(n \geq 30\) is considered large enough), the sampling distribution of the sample mean \(\bar{x}\) is approximately normal, regardless of the population distribution, as long as the sample is random and independent. Here, the sample size is 50 (which is large enough), so the sampling distribution of \(\bar{x}\) is approximately normal because the sample size is large enough (Option A). Option B assumes the population is normal, but the CLT doesn't require that. Option C is incorrect as we don't "assume" it's normal without reason, and Option D incorrectly combines population normality (not required by CLT) with sample size.

Answer:

A. The sampling distribution is approximately normal because the sample size is large enough.