A random sample of 8 computer technicians from your chain of computer shops were

ID: 3217501 • Letter: A

Question

A random sample of 8 computer technicians from your chain of computer shops were selected to diagnose a problem. The time in minutes to diagnosis for each technician was recorded: Does the data provided evidence that the true average diagnostic time of your mechanics is less than 30 minutes? (a) Test the hypothesis using a t-test at alpha = 0.01. Make sure you state all assumptions needed to perform the test. (b) Test the hypothesis using the Wilcoxon signed-rank test (WSRT) at alpha = 0.01. Make sure you state all assumptions needed to perform the test. (c) Suppose that the true average diagnostic time is 28 minutes. Compute the type II error for the t-test. Suppose you know that the Type II error for the WSRT test is beta(28) = 0.8. Which test would you prefer under this scenario and what might it suggest to you about the data (make sure you justify your preference)?

Explanation / Answer

Part a

Here, we have to use the one sample t test for the population mean. The null and alternative hypothesis for this test is given as below:

Null hypothesis: H0: True average diagnostic time of mechanics is 30 minutes.

Alternative hypothesis: Ha: True average diagnostic time of mechanics is less than 30 minutes.

H0: µ = 30 versus Ha: µ < 30

This is a one tailed test. This is a lower tailed test or left tailed test.

We are given alpha = 0.01

From the given sample we have

Sample size = n = 8

Sample mean = Xbar = 27.7375

Sample standard deviation = 5.74529

Degrees of freedom = n – 1 = 8 – 1 = 7

Test statistic formula is given as below:

Test statistic = t = = (Xbar - µ) / [S/sqrt(n)] = (27.7375 – 30)/[5.74529/sqrt(8)] = -1.1138

Lower critical value = -2.9980

P-value = 0.1511

Alpha value = 0.01

P-value > Alpha value

So, we do not reject the null hypothesis

So, we concluded that there is sufficient evidence that true average diagnostic time of mechanics is 30 minutes.

Part b

H0: m = 30

Ha: m < 30

Alpha = 0.01

Test statistic = W = (Z*R)

Table for signed rank for the given test is shown below:

Time

Sign

Signed Rank R

30.7

30.2

15.7

-1

26.8

-2

27.2

-4

25.3

-3

35.1

30.9

Total

From above table, we have W = (Z*R) = 26

P-value = 0.243323342

(By using excel or table)

Alpha value = 0.01

P-value > Alpha value

So, we do not reject the null hypothesis

So, we concluded that there is sufficient evidence that true average diagnostic time of mechanics is 30 minutes.

Part c

Here, we have to find the probability for type II error.

P(type II error) = 1 – P(X<28)

Z = (28 – 30) / 5.74529

Z = 0.348111235

P(Z<0.348111235) = 0.348111235

P(type II) = 1 – P(Z<.348111235) = 1 - 0.348111235 = 0.651888765

We would prefer the one sample t test for the given scenario.