Question

The life expectancy in the United States is 75 with a standard deviation of 7 years. A random sample of 49 individuals is selected. Round all probabilities to four decimal places.

a.What is the probability that the sample mean will be larger than 77 years?

b.What is the probability that the sample mean will be within 1 year of the population mean?

c.What is the probability that the sample mean will be within 2.5 years of the population mean?

Answer 1

The life expectancy in the United States is 75 with a standard deviation of 7 years. A random sample of 49 individuals is selected.

Given:

Population Mean

Population standard deviation

Sample size n = 49

a. What is the probability that the sample mean will be larger than 77 years?

If the population standard deviation is known, we can transform the sample mean to an approximately standard normal variable, z score, Z:

So the probability that the sample mean will be larger than 77 years is 0.0228.

Lookup the standard normal distribution table for probability associated with respective z score

b.What is the probability that the sample mean will be within 1 year of the population mean?

The sample mean within 1 year of the population mean can be represented as

The corresponding z-score will be

So the probability that the sample mean will be within 1 year of the population mean will be

is 0.6826.

c.What is the probability that the sample mean will be within 2.5 years of the population mean?

The sample mean within 2.5 years of the population mean can be represented as

The corresponding z-score will be

So the probability that the sample mean will be within 2.5 years of the population mean will be

is 0.9876.