Answer to Question #350635 in Statistics and Probability for Channah

Question

The average number of milligrams (mg) of cholesterol in a cup of a certain brand of ice cream is 660 mg, and the standard deviation is 35 mg. Assume the variable is normally distributed.

a. If a cup of ice cream is selected, what is the probability that the cholesterol content will be more than 690?

b. If a sample of 10 cups of ice cream is selected, what is the probability that the mean of the sample will be larger than 690 mg?

c. Why is the probability in part A larger than part B? 

Answer 1

a.

$$P(X>690)=1-P(Z\le\dfrac{690-660}{35})$$

$$\approx1-P(Z\le 0.857143)\approx0.195683$$

b.

$$P(X>690)=1-P(Z\le\dfrac{690-660}{35/\sqrt{10}})$$

$$\approx1-P(Z\le 2.710524)\approx0.003359$$

c. By the Central Limit Theorem the standard deviation decreases in $\sqrt{n}=\sqrt{10}$ times, and the mean does not change.

Since $z-$ value is positive in part A and part B, then $z-$ value $z_A$ in part A is less than $z-$ value $z_B$ in part B.

So

$$P(Z>z_A)>P(Z>z_B), 0<z_A<z_B$$