Question

Chapter 14 Problem F: A random sample of 15 automobile mechanics certified to work on a certain type of car was selected, and the time in minutes) necessary for each one to diagnose a particular problem was determined, resulting in the following data: 30.6 30.1 15.6 26.7 27.1 25.4 35.0 30.8 31.9 53.2 12.5 23.2 8.8 24.9 30.2 Use the Wilcoxon signed-rank test at significance level a = .10 (approximately) to decide whether the data suggests that true average diagnostic time is less than 30 minutes. Let denote the true average diagnostic time. Then the appropriate hypotheses are: Hou = 30 versus H: 4 < 30. 15. Refer to Problem F. What is the rejection region for this test? (1) 8+ < 45 (3) 8+ > 84 (2) 8+ <37 (4) 8+ > 83

Answer 1

Problem: A random sample of 15 automobile mechanics certified to work on a certain type of car was selected, and the time (in minutes)for each one to diagnose a particular problem was determined,resulting in the following data

Time(in min) to diagnose

30.6

30.1

15.6

26.7

27.1

25.4

35

30.8

31.9

53.2

12.5

23.2

8.8

24.9

30.2

Use Wilcoxon signed-rank test at significance level α=0.10 to decide whether the above data suggests that average diagnostic time is less than 30 minuets. Let u denote the true average diagnostic time. The hypothesis are:

Ho: u=30 V/s Ha: u<30.

Answer:

We have to test that whether the given data suggests that average diagnostic time is less than 30 minuets.

Let u be the true average diagnostic time.

We have to test the null and alternative hypothesis,

Ho: u=30 verses Ha=u<30.

First we calculate the difference between each value of sample and average diagnostic time in minuets.

Time(in min) to diagnose	30.6	30.1	15.6	26.7	27.1	25.4	35	30.8	31.9	53.2	12.5	23.2	8.8	24.9	30.2
Difference	0.6	0.1	-14.4	-3.3	-2.9	-4.6	5	0.8	1.9	23.2	-17.5	-6.8	-21.2	-5.1	0.2

Ignoring the sign of difference, we rank the difference from smallest to largest(smallest rank=1)

Time(in min) to diagnose	30.6	30.1	15.6	26.7	27.1	25.4	35	30.8	31.9	53.2	12.5	23.2	8.8	24.9	30.2
Difference	0.6	0.1	-14.4	-3.3	-2.9	-4.6	5	0.8	1.9	23.2	-17.5	-6.8	-21.2	-5.1	0.2
Absolute difference	0.6	0.1	14.4	3.3	2.9	4.6	5	0.8	1.9	23.2	17.5	6.8	21.2	5.1	0.2
Rank	3	1	12	7	6	8	9	4	5	15	13	11	14	10	2

Separate the positive and negative ranks,

Time(in min) to diagnose	30.6	30.1	15.6	26.7	27.1	25.4	35	30.8	31.9	53.2	12.5	23.2	8.8	24.9	30.2
Difference	0.6	0.1	-14.4	-3.3	-2.9	-4.6	5	0.8	1.9	23.2	-17.5	-6.8	-21.2	-5.1	0.2
+ Rank	3	1					9	4	5	15					2
- Rank			12	7	6	8					13	11	14	10

The sum of + ve rank is, S+ =39

The sum of + ve rank is, S- = 81

The test statistic for Wilcoxon singed rank test for testing Ho verses Ha is smaller of sum of positive rank and sum of negative rank.

W=min(S+,S-)

W=S+ =39.

The critical value of one tailed Wilcoxon sign rank test at α=0.10 and n=15 is 36.

Since test statistic value (39) is greater than the critical value(36), hence accept the null hypothesis and conclude that the average diagnostic time of mechanics is equals to the 30 minuets

There is no significant evident that suggest average diagnostic time is less than 30 minuets.

The rejection or critical region for this test is,

• The rejection region for this test is S+≤ 36.