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, $\mu=186, \sigma=7,n=144.$

Let's convert it to the standard normal distribution,

$z=\cfrac{\bar{x}-\mu}{\sigma/\sqrt{n}}.$

$\bar{x}_1=186-1.5=184.5;\\\bar{x}_2=186+1.5=187.5;\\ z_1=\cfrac{184.5-186}{7/\sqrt{144}}=-2.57;\\ z_2=\cfrac{187.5-186}{7/\sqrt{144}}=2.57;\\ P(184.5<\bar{X}<187.5)=P(-2.57<Z<2.57)=\\ =P(Z<2.57)-P(Z<-2.57)=\\ =0.9949-0.0051=0.9898 \text{ (from z-table).}$