Question

According to U.S. News & World Report's publication America's Best Colleges, the average cost to attend the University of Southern California (USC) after deducting grants based on need is $24,675. Assume the population standard deviation is $7,100. Suppose that a random sample of 70 USC students will be taken from this population.

a. What is the value of the standard error of the mean?

(to nearest whole number)

b. What is the probability that the sample mean will be more than 24,675?

(to 2 decimals)

c. What is the probability that the sample mean will be within $1500 of the population mean?

(to 4 decimals)

d. How would the probability in part (c) change if the sample size were increased to 120?

(to 4 decimals)

Answer 1

a. What is the value of the standard error of the mean?

= sd(x)/sqrt(n) =7100/sqrt(70) = 848.6123

b. What is the probability that the sample mean will be more than 24,675?

P(Xbar > 24675)

= P(Z > 0 )

= 0.5

c. What is the probability that the sample mean will be within $1500 of the population mean?

P(|Xbar - 24675| < 1500)

= P( - 1500/ 848.6123 < Z < 1500/ 848.6123 )

= P(-1.76759 < Z < 1.76759)

= 0.9229

d. How would the probability in part (c) change if the sample size were increased to 120?

when n = 120

standard error decreases

z-score increases

probability will increase

P(-2.31432 < Z < 2.31432)

= 0.9793