Question

The results of a (20 questions) multiple choice questions exam are recorded. Random samples of 10 and 12 students are investigated. For the 10 student sample, the individual performances are 11, 8, 8, 3, 7, 5, 9, 5, 1, and 3 errors. For the 12 student sample, the individual performances are 10, 11, 9, 7, 2, 11, 12, 3, 6, 7, 8, and 12 errors.

1. Assuming the population standard deviations are equal, use the 0.10 level of significance in testing whether the two samples average performances are the same.
2. Construct a 90% confidence interval for the difference between average performances.

Answer 1

= 6, s1 = 3.13, n1 = 10

= 8.17, s2 = 3.33, n2 = 12

a) H₀:

    H₁:

The pooled sample variance(s_p²) = ((n1 - 1)s1^2 + (n2 - 1)s2^2)

                                                    = (9 * (3.13)^2 + 11 * (3.33)^2)/(10 + 12 - 2)

                                                    = 10.5075

The test statistic t = ()/sqrt(s_p²/n1 +s_p²/n2)

                            = (6 - 8.17)/sqrt((10.5075)^2/10 + (10.5075)^2/12)

                            = -0.48

df = 10 + 12 - 2 = 20

At alpha = 0.10, the critical values are t_{0.05, 20} = +/- 1.725

Since the test statistic value is not less than the negative critical value(-1.57 > -1.725), so we should not reject the null hypothesis.

b) At 90% confidence interval the critical value is t^* = 1.725

The 90% confidence interval for is

() +/- t^* * sqrt(s_p²/n1 +s_p²/n2)

= (6 - 8.17) +/- 1.725 * sqrt((10.5075)^2/10 + (10.5075)^2/12)

= -2.17 +/- 7.76

= -9.93, 5.59