Question

The average student loan debt of a U.S. college student at the end of 4 years of college is estimated to be about $23,300. You take a random sample of 136 college students in the state of Vermont and find the mean debt is $24,500 with a standard deviation of $2,700. You want to construct a 99% confidence interval for the mean debt for all Vermont college students.

(a) What is the point estimate for the mean debt of all Vermont college students?

(b) Construct the 99% confidence interval for the mean debt of all Vermont college students. Round your answers to the nearest whole dollar.

(c) Are you 99% confident that the mean debt of all Vermont college students is greater than the quoted national average of $23,300 and why?

a. Yes, because $23,300 is above the lower limit of the confidence interval for Vermont students.

b. No, because $23,300 is above the lower limit of the confidence interval for Vermont students.

c. No, because $23,300 is below the lower limit of the confidence interval for Vermont students.

d. Yes, because $23,300 is below the lower limit of the confidence interval for Vermont students.

d) We are never told whether or not the parent population is normally distributed. Why could we use the above method to find the confidence interval?

a. Because the margin of error is positive.

b. Because the margin of error is less than 30.

c. Because the sample size is greater than 30.

d. Because the sample size is less than 100.

Answer 1

Solution :

Given that,

a) Point estimate = sample mean = = $ 24,500

sample standard deviation = s = $ 2,700

sample size = n = 136

Degrees of freedom = df = n - 1 = 136 - 1 = 135

b) At 99% confidence level

= 1 - 99%

=1 - 0.99 =0.01

/2 = 0.005

t/2,df = t_0.005,135 = 2.613

Margin of error = E = t_/2,df * (s /n)

= 2.613 * ( 2700 / 136)

Margin of error = E = $ 605

The 99% confidence interval estimate of the population mean is,

  ± E

= $ 24,500 ± $ 605

= ( $ 23,895, $ 25,105 )

c) a. Yes, because $23,300 is above the lower limit of the confidence interval for Vermont students.

d) c. Because the sample size is greater than 30.