Question

1. For each of the 2 majors consider the ‘School Type’ column. Construct a 92% confidence interval for the proportion of the schools that are ‘Private’.  

2. For each of the 2 majors, construct a 98% confidence interval for the mean of the column ‘Annual % ROI’.

3.In a highlighted box, discuss the following. How do these confidence intervals provide information about the which is the better major in terms of Annual ROI? Compare the confidence intervals. Which is narrower? Wider? What does the width have to do with advising someone on which major has the better ROI?  Why is it important to know the proportion of schools that are private? What factors differ in public and private schools that could influence ROI? Is a 98% confidence interval a guarantee that the ROI will be in the interval? Why or why not?

Engineer

School Type	Cost	30 Year ROI	Annual ROI
Private	$221,700.00	$2,412,000.00	8.70%
Private	$213,000.00	$2,064,000.00	8.30%
Private	$230,100.00	$1,949,000.00	7.90%
Private	$222,600.00	$1,947,000.00	8.00%
Private	$225,800.00	$1,938,000.00	8.00%
Public	$87,660.00	$1,937,000.00	11.20%
Private	$224,900.00	$1,915,000.00	7.90%
Private	$221,600.00	$1,878,000.00	7.90%
Public	$125,100.00	$1,854,000.00	9.80%
Private	$215,700.00	$1,794,000.00	7.90%
Public	$92,530.00	$1,761,000.00	10.60%
Private	$217,800.00	$1,752,000.00	7.70%
Public	$89,700.00	$1,727,000.00	10.70%
Private	$229,600.00	$1,716,000.00	7.50%
Public	$101,500.00	$1,703,000.00	10.20%
Public	$115,500.00	$1,694,000.00	9.70%
Public	$104,500.00	$1,690,000.00	10.10%
Public	$69,980.00	$1,685,000.00	11.50%
Private	$219,400.00	$1,676,000.00	7.60%
Public	$64,930.00	$1,668,000.00	11.70%

Business

School Type	Cost	30 Year ROI	Annual ROI
Private	$222,700.00	$1,786,000.00	7.70%
Private	$176,400.00	$1,758,000.00	8.40%
Private	$212,200.00	$1,714,000.00	7.80%
Public	$125,100.00	$1,535,000.00	9.10%
Private	$212,700.00	$1,529,000.00	7.40%
Public	$92,910.00	$1,501,000.00	10.10%
Private	$214,900.00	$1,485,000.00	7.30%
Private	$217,800.00	$1,483,000.00	7.20%
Private	$225,600.00	$1,444,000.00	7.00%
Private	$217,300.00	$1,442,000.00	7.10%
Private	$226,500.00	$1,441,000.00	7.00%
Private	$215,500.00	$1,438,000.00	7.20%
Private	$223,500.00	$1,428,000.00	7.00%
Private	$226,600.00	$1,414,000.00	7.00%
Private	$189,300.00	$1,397,000.00	7.50%
Public	$89,700.00	$1,382,000.00	9.90%
Public	$87,030.00	$1,376,000.00	10.00%
Private	$218,200.00	$1,343,000.00	6.90%
Private	$229,900.00	$1,339,000.00	6.70%
Private	$148,800.00	$1,321,000.00	8.10%

Answer 1

1. Let X be the no. of private schools

For confidence interval of proportion we have

Where

• n is the total no.of schools
• is the z-score inverse of normal dist and the values can be found using normal percentage point table or online
• p =

For Engineer major

x = 11 n = 20 (1-92%) Therefore

92% confidence interval for proportion of private school for engineer major

(0.355, 0.7448)

For Business major

x = 16 n = 20 (1-92%) Therefore

92% confidence interval for proportion of private school for business major

(0.6434, 0.9566)

2.

We are going to use t-dist for the confidence interval since we don't have the population S.D.

Confidence interval for mean is

Where

• y is the values of annual Roi
• is the sample mean
• is the sample S.D.
• is the t-score

For engineer major

98% confidence interval for mean Roi for engineer major is (

(0.08216, 0.1008) = (8.216%, 10.08%)

For business major

98% confidence interval for mean Roi for business major is

(0.07116, 0.08524) = (7.116%, 8.52%)

c.

How do these confidence intervals provide information about the which is the better major in terms of Annual ROI?: Looking at the CI we can see that both upper and lower limits are higher for engineer major than business. This means that mean range values of engineer are higher than business. Therefore engineer is a better major than business.

Compare the confidence intervals. Which is narrower? Wider?: The width of engineer is 1.864 and business is 1.404. Engineer has wider width than business.

What does the width have to do with advising someone on which major has the better ROI?: Greater width indicates greater variabilility. Therefore Roi for engineer major is more variable (riskier) than business major. So if a person is looking for more secure Roi,it would be advisable to choose business.

Why is it important to know the proportion of schools that are private? What factors differ in public and private schools that could influence ROI?: The private school are costlier than public. It is also imp to check the avg. ROi for the schools to see whether what we are spending will be recovered in the future or not.

Is a 98% confidence interval a guarantee that the ROI will be in the interval? Why or why not?: CI does not guarantee that the ROI will lie within the range. But it says that we can be 98% sure that it lie within the range.