Question

The following table shows the grades (out of 100) for a sample of 8 randomly chosen students on two consecutive exams. Construct the 90% confidence interval for the difference of mean grades on the two exams. Round your answer to 2 decimal places. Can it be concluded that the grades on the second test were significantly different from the first one?

Student #: 1 2 3 4 5 6 7 8

Test 1: 67 88 76 88 91 95 79 71

Test 2: 69 78 76 84 86 96 70 68

Answer 1

Confidence interval for difference between two population means of paired samples is given as below:

Confidence interval = Dbar ± t*S_D/sqrt(n)

From given data, we have

Dbar = 3.5

S_d = 4.440077219

n = 8

df = n – 1 = 7

Confidence level = 90%

Critical t value = 1.8946

(by using t-table)

Confidence interval = Dbar ± t*S_D/sqrt(n)

Confidence interval = 3.5 ± 1.8946*4.440077219/sqrt(8)

Confidence interval = 3.5 ± 1.8946*1.569804355

Confidence interval = 3.5 ± 2.9741

Lower limit = 3.5 - 2.9741 = 0.53

Upper limit = 3.5 + 2.9741 =6.47

0.53 < µ_d < 6.47

It can be concluded that the grades on the second test were significantly different from the first one, because the above interval do not contain the value zero.