Question

We are interested in estimating the proportion of graduates at a mid-sized university who found a job within one year of completing their undergraduate degree. We can do so by creating a 95% confidence interval for the true proportion p. Suppose we conduct a survey and find out that 340 of the 430 randomly sampled graduates found jobs within one year. Assume that the size of the population of graduates at this university is large enough so that all our conditions for normality are satisfied.

(a) What is the value of the point estimate for the proportion of graduates who found a job within one year? Round your answer to two decimal places.

(b) Use the point estimate in part (a) to calculate the standard error.

(c) For a 95% confidence interval, determine the margin of error. (Be sure that you show all your steps.) Then determine the lower and upper bounds of the interval.

(d) Which of the following is an appropriate way to interpret the confidence interval obtained in part (c)?

A. There is a 95% probability that the true proportion of people who found a job within one year of graduation is between the lower bound and the upper bound.

B. We are 95% confident that the true proportion of people who found a job within one year of graduation is between the lower bound and the upper bound.

C. We are 95% confident that the proportion of people in our sample that found a job within one year of graduation is the true proportion.

D. If we were to create many 95% confidence intervals in the same way with the same sample sizes, then all of them would contain the true proportion of the graduates who found a job within one year of graduation.

Answer 1

Solution :

Given that,

n = 430

x = 340

a) Point estimate = sample proportion = = x / n = 340 / 430 = 0.79

1 - = 1 - 0.791 = 0.21

b) =  [p( 1 - p ) / n] = [(0.79 * 0.21) / 430 ] = 0.0196

c) At 95% confidence level

= 1 - 95%

=1 - 0.95 =0.05

/2 = 0.025

Z/2 = Z_{0.025 = 1.960}

Margin of error = E = Z _{/ 2} *

= 1.96 * 0.0196

= 0.038

A 95% confidence interval for population proportion p is ,

± E  

= 0.79  ± 0.038

= ( 0.752, 0.828 )

d) B. We are 95% confident that the true proportion of people who found a job within one year of graduation is between the lower bound and the upper bound.