Question

A composition teacher wishes to see whether a new grammar program will reduce the number of grammatical errors her student make when writing an essay. the data are shown below. At a=0.05 can it be concluded that the number of errors has been reduced?

Student- 1, 2, 3, 4, 5, 6,

Errors before- 12 9 0 5 4 3

Errors after - 9 6 1 1 2 5

Answer 1

= (3 + 3 + (-1) + 4 + 2 + (-2))/6 = 1.5

s_d = sqrt(((3 - 1.5)^2 + (3 - 1.5)^2 + ((-1) - 1.5)^2 + (4 - 1.5)^2 + (2 - 1.5)^2 + ((-2) - 1.5)^2)/5) = 2.43

H₀: = 0

H₁: > 0

The test statistic t = ( - D)/(s_d/)

                             = (1.5 - 0)/(2.43/)

                            = 1.512

At alpha = 0.05, the critical value is t^* = 2.015

Since the test statistic value is not greater than the critical value(1.512 < 2.015), so we should not reject the null hypothesis.

So at 5% significance level there is not sufficient evidence to conclude that the number of errors has been reduced.