Discuss the below:
Hypothesis Testing With Two Samples
Q1 A problem with a telephone line that prevents a customer from receiving or making calls is disconcerting to both the customer and the telephone company. The data in the file PHONE represent samples of 20 problems reported to two different offices of a telephone company and the time to clear these problems (in minutes) from the customers' lines:
| Time |
Location |
| 1.48 |
1 |
| 1.75 |
1 |
| 0.78 |
1 |
| 2.85 |
1 |
| 0.52 |
1 |
| 1.60 |
1 |
| 4.15 |
1 |
| 3.97 |
1 |
| 1.48 |
1 |
| 3.10 |
1 |
| 1.02 |
1 |
| 0.53 |
1 |
| 0.93 |
1 |
| 1.60 |
1 |
| 0.80 |
1 |
| 1.05 |
1 |
| 6.32 |
1 |
| 3.93 |
1 |
| 5.45 |
1 |
| 0.97 |
1 |
| 7.55 |
2 |
| 3.75 |
2 |
| 0.10 |
2 |
| 1.10 |
2 |
| 0.60 |
2 |
| 0.52 |
2 |
| 3.30 |
2 |
| 2.10 |
2 |
| 0.58 |
2 |
| 4.02 |
2 |
| 3.75 |
2 |
| 0.65 |
2 |
| 1.92 |
2 |
| 0.60 |
2 |
| 1.53 |
2 |
| 4.23 |
2 |
| 0.08 |
2 |
| 1.48 |
2 |
| 1.65 |
2 |
| 0.72 |
2 |
Central Office I Time to Clear Problems (minutes)
1.48 1.75 0.78 2.85 0.52 1.60 4.15 3.97 1.48 3.10
1.02 0.53 0.93 1.60 0.80 1.05 6.32 3.93 5.45 0.97
Central Office II Time to Clear Problems (minutes)
7.55 3.75 0.10 1.10 0.60 0.52 3.30 2.10 0.58 4.02
3.75 0.65 1.92 0.60 1.53 4.23 0.08 1.48 1.65 0.72
a. Assuming that the population variances from both offices are equal, is there evidence of a difference in the mean waiting time between the two offices? (use α = 0.05. )
b. Find the p-value in (a) and interpret its meaning.
c. What other assumption is necessary in (a)?
Q2 Digital cameras have taken over the majority of the point-and-shoot camera market. One of the important features of a camera is the battery life as measured by the number of shots taken until the battery needs to be recharged. The data in the file Digitalcameras contain the battery life of 31 subcompact cameras and 15 compact cameras (data extracted from "Cameras," Consumer Reports, November 2006, pp. 20-21).
| Battery Life |
Camera Type |
Subcompact |
Compact |
| 320 |
Subcompact |
320 |
520 |
| 520 |
Subcompact |
520 |
260 |
| 160 |
Subcompact |
160 |
400 |
| 160 |
Subcompact |
160 |
200 |
| 300 |
Subcompact |
300 |
300 |
| 120 |
Subcompact |
120 |
150 |
| 520 |
Subcompact |
520 |
360 |
| 440 |
Subcompact |
440 |
200 |
| 300 |
Subcompact |
300 |
260 |
| 170 |
Subcompact |
170 |
80 |
| 150 |
Subcompact |
150 |
200 |
| 300 |
Subcompact |
300 |
260 |
| 180 |
Subcompact |
180 |
400 |
| 100 |
Subcompact |
100 |
260 |
| 150 |
Subcompact |
150 |
200 |
| 400 |
Subcompact |
400 |
|
| 170 |
Subcompact |
170 |
|
| 180 |
Subcompact |
180 |
|
| 160 |
Subcompact |
160 |
|
| 240 |
Subcompact |
240 |
|
| 320 |
Subcompact |
320 |
|
| 260 |
Subcompact |
260 |
|
| 150 |
Subcompact |
150 |
|
| 320 |
Subcompact |
320 |
|
| 180 |
Subcompact |
180 |
|
| 140 |
Subcompact |
140 |
|
| 220 |
Subcompact |
220 |
|
| 160 |
Subcompact |
160 |
|
| 340 |
Subcompact |
340 |
|
| 200 |
Subcompact |
200 |
|
| 130 |
Subcompact |
130 |
|
| 520 |
Compact |
|
|
| 260 |
Compact |
|
|
| 400 |
Compact |
|
|
| 200 |
Compact |
|
|
| 300 |
Compact |
|
|
| 150 |
Compact |
|
|
| 360 |
Compact |
|
|
| 200 |
Compact |
|
|
| 260 |
Compact |
|
|
| 80 |
Compact |
|
|
| 200 |
Compact |
|
|
| 260 |
Compact |
|
|
| 400 |
Compact |
|
|
| 260 |
Compact |
|
|
| 200 |
Compact |
|
|
a. Assuming that the population variances from both types of digital cameras are equal, is there evidence of a difference in the mean battery life between the two types of digital cameras
b. Determine the p-value in (a) and interpret its meaning.
Q3 In industrial settings, alternative methods often exist for measuring variables of interest. The data in the file Measurement (coded to maintain confidentiality) represent measurements in-line that were collected from an analyzer during the production process and from an analytical lab (extracted from M. Leitnaker, "Comparing
Measurement Processes: In-line Versus Analytical Measurements," Quality Engineering, 13, 2000-2001, pp. 293-298).
| Sample |
In-Line |
Analytical lab |
| 1 |
8.01 |
8.01 |
| 2 |
7.56 |
7.29 |
| 3 |
7.47 |
7.54 |
| 4 |
7.4 |
7.42 |
| 5 |
7.83 |
7.8 |
| 6 |
7.5 |
7.65 |
| 7 |
6.86 |
6.93 |
| 8 |
7.31 |
7.46 |
| 9 |
7.45 |
7.6 |
| 10 |
7.23 |
7.4 |
| 11 |
7.37 |
7.5 |
| 12 |
7.49 |
7.41 |
| 13 |
6.21 |
6.25 |
| 14 |
6.68 |
6.54 |
| 15 |
5.12 |
5.2 |
| 16 |
4.84 |
4.7 |
| 17 |
4.84 |
4.82 |
| 18 |
5.21 |
5.33 |
| 19 |
5.35 |
5.3 |
| 20 |
5.6 |
5.4 |
| 21 |
5.32 |
5.39 |
| 22 |
5.16 |
5.17 |
| 23 |
5.66 |
5.5 |
| 24 |
6.31 |
6.24 |
a. At the 0.05 level of significance, is there evidence of a difference in the mean measurements in-line and from an analytical lab?
b. What assumption is necessary about the population distribution in order to perform this test?
Q4 A newspaper article discussed the opening of a Whole Foods Market in the Time-Warner building in New York City. The following data (stored in the file Wholefoods1 ) compared the prices of some kitchen staples at the new Whole Foods Market and at the Fairway supermarket located about 15 blocks from the Time-Warner building:
| Item |
Whole Foods |
Fairway |
| Half-gallon milk |
2.19 |
1.35 |
| Dozen eggs |
2.39 |
1.69 |
| Tropicana orange juice (64 oz.) |
2.00 |
2.49 |
| Head of Boston lettuce |
1.98 |
1.29 |
| Ground round 1lb. |
4.99 |
3.69 |
| Bumble Bee tuna 6 oz. can |
1.79 |
1.33 |
| Granny Smith apples (1 lb.) |
1.69 |
1.49 |
| Box DeCecco linguini |
1.99 |
1.59 |
| Salmon steak 1 lb. |
7.99 |
5.99 |
| Whole chicken per pound |
2.19 |
1.49 |
a. At the 0.01 level of significance, is there evidence that the mean price is higher at Whole Foods Market than at the Fairway supermarket?
b. Interpret the meaning of the p-value in (a).
c. What assumption is necessary about the population distribution in order to perform the test in (a)?
Q5 The director of training for a company that manufactures electronic equipment is interested in determining whether different training methods have an effect on the productivity of assembly-line employees. She randomly assigns 21 of the 42 recently hired employees to a computerassisted, individual-based training program. The other 21 are assigned to a team-based training program. Upon completion of the training, the employees are evaluated on the time (in seconds) it takes to assemble a part. The results are in the data file Training.
| Time |
Training |
| 19.4 |
1 |
| 16.7 |
1 |
| 20.7 |
1 |
| 19.3 |
1 |
| 21.8 |
1 |
| 16.8 |
1 |
| 14.1 |
1 |
| 17.7 |
1 |
| 16.1 |
1 |
| 19.8 |
1 |
| 16.8 |
1 |
| 19.3 |
1 |
| 14.7 |
1 |
| 16.0 |
1 |
| 16.5 |
1 |
| 17.7 |
1 |
| 16.2 |
1 |
| 17.4 |
1 |
| 16.4 |
1 |
| 16.8 |
1 |
| 18.5 |
1 |
| 22.4 |
0 |
| 13.8 |
0 |
| 18.7 |
0 |
| 18.0 |
0 |
| 19.3 |
0 |
| 20.8 |
0 |
| 15.6 |
0 |
| 17.1 |
0 |
| 18.0 |
0 |
| 28.2 |
0 |
| 21.7 |
0 |
| 20.8 |
0 |
| 30.7 |
0 |
| 24.7 |
0 |
| 23.7 |
0 |
| 17.4 |
0 |
| 23.2 |
0 |
| 20.1 |
0 |
| 12.3 |
0 |
| 15.2 |
0 |
| 16.0 |
0 |
a. Using a 0.05 level of significance, is there evidence of a difference between the variances in assembly times (in seconds) of employees trained in a computer-assisted, individual-based program and those trained in a teambased program?
b. On the basis of the results in (a), which t test defined should you use to compare the means of the two training programs? Discuss.
APPLYING THE CONCEPTS
Q6 A professor in the accounting department of a business school claims that there is much more variability in the final exam scores of students taking the introductory accounting course who are not majoring in accounting than for students taking the course who are majoring in accounting. Random samples of 13 non-accounting majors and 10 accounting majors are taken from the professor's class roster in his large lecture, and the following results are computed based on the final exam scores:
Non-Accounting: n = 13 S2 = 210.2
Accounting: n = s2 = 36.5
a. At the 0.05 level of significance, is there evidence to support the professor's claim?
b. Interpret the p-value.
c. What assumption do you need to make in (a) about the two populations in order to justify your use of the F test?