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 x = 5.4 insect fragments per ten - gram portion. complete parts (a) through (c) below(a) why is the sampling distribution of x approximately normal?a. the sampling distribution of x is approximately normal because the sample size is large enough.b. the sampling distribution of x is approximately normal because the population is normally distributed.c. the sampling distribution of x is assumed to be approximately normal.d. the sampling distribution of 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 distribution of x assuming μ = 5 and σ = √5?μₓ = 5.000 (round to three decimal places as needed.)σₓ = (round to three decimal places as needed.)

Explanation:

Step1: Recall the formula for the standard deviation of the sampling distribution

The formula for the standard deviation of the sampling distribution of the sample mean \(\sigma_{\bar{x}}\) is \(\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}\), where \(\sigma\) is the population standard deviation and \(n\) is the sample size.

Step2: Substitute the given values into the formula

Given that \(\sigma = \sqrt{5}\approx2.236\) and \(n = 40\).

$$ \sigma_{\bar{x}}=\frac{\sqrt{5}}{\sqrt{40}}=\frac{\sqrt{5}}{2\sqrt{10}}=\frac{\sqrt{50}}{20}=\frac{5\sqrt{2}}{20}=\frac{\sqrt{2}}{4}\approx0.354 $$

Answer:

\(\sigma_{\bar{x}}\approx0.354\)