Answer to Question #328308 in Statistics and Probability for Gibbie

Question

Suppose the mean amount of cholesterol in eggs labeled “large” is 186 milligrams, with standard deviation 7 milligrams. Find the probability that the mean amount of cholesterol in a sample of 144 eggs will be within 1.5 milligrams of the population mean.

Answer 1

We have a normal distribution, $μ=186,σ=7,n=144,$

$\bar{x}_1=186-1.5=184.5,\\ \bar{x}_2=186+1.5=187.5.$

Let's convert it to the standard normal distribution,

$\bar{z}=\cfrac{\bar{x}-\mu}{\sigma/\sqrt{n}},\\ \bar{z}_1=\cfrac{184.5-186}{7/\sqrt{144}}=-2.57,\\ \bar{z}_2=\cfrac{187.5-186}{7/\sqrt{144}}=2.57,\\ P(184.5<\bar{X}<187.5)=P(-2.57<\bar{Z}<2.57)=\\ =P(\bar{Z}<2.57)-P(\bar{Z}<-2.57)=\\ =0.9949-0.0051=0.9898\text{ (from z-table).}$