Answer to Question #321032 in Statistics and Probability for Mark

Question

1. The average number of milligrams (mg) of cholesterol in a cup of a certain brand of ice cream is 660 mg, 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 670 mg?

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 670 mg?

2.In a study of the life expectancy of 400 people in a certain geographic region, the mean age at death was 70 years, and the standard deviation was 5.1 years. If a sample of 50 people from this region is selected, what is the probability that the mean life expectancy will be less than 68 years?

Answer 1

1. We have a normal distribution, $\mu=600, \sigma=35.$

a.

Let's convert it to the standard normal distribution:

$z=\cfrac{x-\mu}{\sigma}=\cfrac{670-600}{35}=2,$

$P(X>670)=1-P(X<670)=\\ =1-P(Z<2)=$

$=1-0.9772=0.0228$ ​(from z-table)

b.

Let's convert it to the standard normal distribution,

$z=\cfrac{\bar{x}-\mu}{\sigma/\sqrt{n}}=\cfrac{670-600}{35/\sqrt{10}}=6.32.$

$P(\bar{X}>670)=1-P(\bar{X}<670)=\\ =1-P(Z<6.32)=$

$=1-1=0$ ​(from z-table)

2. We have a normal distribution, $\mu=70, \sigma=5.1,n=50.$

Let's convert it to the standard normal distribution,

$z=\cfrac{\bar{x}-\mu}{\sigma/\sqrt{n}}=\cfrac{68-70}{5.1/\sqrt{50}}=-2.77.$

$P(\bar{X}<68)=P(Z<-2.77)=0.0028$ (from z-table)