INSTRUCTIONS:
I. Assignments are to be uploaded to the course website on CULEARN as a single legible pdf file by the above due date and time. No late assignments will be accepted without sufficient advanced notice and a legitimate, documented reason.
II. You must show all of your work. No credit will be given for answers without justification. No credit will be given for illegible work.
III. Do not use MINITAB for any part of a question unless it specifically says to do so. For questions that require MINITAB, you must include all relevant output with your assignment. The lab for this assignment will take place during the week of January 25, 2016.
IV. This assignment is intended to represent your individual knowledge. It is not a group assignment.
PART A: MINITAB QUESTIONS
1. In an experiment to investigate the effect of colour of paper (blue, green, orange) on response rates for questionnaires distributed by the "windshield method" in supermarket parking lots, 15 lots were chosen, and each colour was assigned at random to five of the lots. The response rates (in %) are given below.
Let ??1, ??2, and ??3 be the population mean response rates for blue, green, and orange questionnaires, respectively. Enter the data for the "Blue", "Green", and "Orange" questionnaires into columns c1, c2, and c3, respectively, in a MINITAB worksheet, and label the columns accordingly.
| Blue: |
27 |
25 |
30 |
26 |
34 |
| Green: |
34 |
29 |
25 |
31 |
29 |
| Orange: |
28 |
22 |
24 |
26 |
25 |
(A) Define (1) the response variable, (2) the factor, (3) the treatments, and (4) the experimental units.
(B) USE MINITAB to produce output for testing whether there is sufficient evidence at the 10% level of significance to indicate that there is a difference in the mean response rates under the different colours, along with 90% Tukey simultaneous confidence intervals in the event that there is evidence of a difference. See BOX A and BOX B on the next page.
(C) Is there sufficient evidence at the 10% level of significance to indicate that there is a difference in the mean response rates under the different colours? Use a p-value approach.
(D) Use the 90% Tukey intervals from MINITAB to discuss the nature of any differences in the mean response rates under the different colours. Is there a particular colour that is a clear winner when it comes to mean response rate?
(E) What assumption is necessary to validate the test result in (C)?
(F) USE MINITAB to conduct a Kruskal-Wallis test at the 10% level of significance to determine if the three populations differ in terms of location using a p-value approach. See BOX C on the next page.
(G) What assumption is necessary to validate the test result in (F)?
(H) Does your conclusion in (F) agree with your finding in (C)? Explain briefly.
Box A
Click Data -> Stack -> Columns...
In the Stack the following columns box, type c1 c2 c3.
Click Column of current worksheet:. Type c4 in the box.
In the Store subscripts in: box, type c5.
Deselect (unclick) Use variable names in subscript column.
Click OK.
Give column c4 the label Response and column c5 the label Colour.
Box B
Click Stat -> ANOVA -> One-Way...
Click in the Response: box, type c4.
Click in the Factor: box and type c5.
Click Comparisons...
In the Error rate for comparisons: box, change the value to 10.
Under Comparison procedures assuming equal variances, select Tukey.
Under Results, deselect Interval plot for differences of means and Grouping information, and select Tests.
Click OK.
Click Graphs...
Under Data plots, deselect (unclick) Interval plot.
Click OK.
Click OK again.
Box C
Click Stat -> Nonparametrics -> Kruskal-Wallis...
In the Response: box, type c4.
Click in the Factor: box and type c5.
Click OK.
2. A city road official studied the wear characteristics of four different paints on eight roads in the city. The standard, currently-used paint (Paint 1) and three experimental paints (Paints 2, 3, 4) were considered. The eight roads were randomly selected to reflect variations in traffic densities. For each road, a random ordering of the paints to the road surface was employed. After a suitable period of exposure to weather and traffic, a measure of durability was obtained; the higher the score, the better the paint. The results are summarized below. Enter the data for "Paint 1", "Paint 2", "Paint 3" and "Paint 4" into columns c1, c2, c3, and c4, respectively, in a MINITAB worksheet, and label the columns accordingly.
| PAINT |
Road 1 |
Road 2 |
Road 3 |
Road 4 |
Road 5 |
Road 6 |
Road 7 |
Road 8 |
| 1 |
11 |
20 |
8 |
30 |
14 |
25 |
43 |
13 |
| 2 |
10 |
14 |
8 |
27 |
14 |
26 |
41 |
12 |
| 3 |
18 |
31 |
16 |
41 |
22 |
34 |
54 |
20 |
| 4 |
15 |
17 |
12 |
28 |
16 |
25 |
42 |
13 |
(A)Explain why this is a randomized block design.
(B) Define (1) the response variable, (2) the treatment factor, and (3) the block factor.
(C) USE MINITAB to produce output for testing whether there is sufficient evidence at the 5% level of significance to indicate that there is a difference in the mean durability of the different paints, and to determine if the mean durability differs on different roads. See BOX D and BOX E on the next page.
(D)Is there sufficient evidence at α = 0.05 to indicate that there is a difference in the mean durability of the different paints? Use a p-value approach, and make sure to define each mean in your hypotheses.
(E) Is there sufficient evidence at α = 0.05 to indicate that there is a difference in the mean durability on different roads? Use a p-value approach, and make sure to define each mean in your hypotheses.
(F) Suppose we wish to compute 95% Tukey simultaneous confidence intervals to compare the mean durability of the different paints. Use the MINITAB output to help you to calculate the margin of error for these intervals. DO NOT CALCULATE THE INTERVALS!
(G)What assumption is necessary to validate the test result in (D)?
(H)USE MINITAB to conduct a Friedman test at the 5% level of significance to determine if the three populations differ in terms of location using a p-value approach. See BOX F below.
(I) Does your conclusion in (H) agree with your finding in (D)? Explain briefly.
Box D
Click Data -> Stack -> Columns...
In the Stack the following columns box, type c1 c2 c3 c4.
Click Column of current worksheet:. Type c5 in the box. In the Store subscripts in: box, type c6.
Deselect (unclick) Use variable names in subscript column.
Click OK.
Click Calc -> Make Patterned Data -> Simple Set of Numbers...
In the Store patterned data in: box, type c7.
Click in the From first value: box, and type 1.
Click in the To last value: box, and type 8.
Click in the Number of times to list the sequence: box, and change the value to 4.
Click OK.
Give column c5 the label Durability, column c6 the label Paint, and column c7 the label Road.
Box E
Click Stat -> ANOVA -> Balanced ANOVA...
In the Responses: box, type c5.
Click in the Model: box and type c6 c7.
Click Results...
Click in the Display means corresponding to the terms: box, and type c6 c7.
Click OK.
Click OK again.
Box F
Click Stat -> Nonparametrics -> Friedman...
In the Response: box, type c5.
Click in the Treatment: box and type c6.
Click in the Blocks: box and type c7.
Click OK.
PART B: WRITTEN QUESTIONS
1. An economist compiled data on productivity improvement (measured on a scale from 0 to 100) last year for a sample of firms producing electronic computing equipment. The firms were classified by the amount (large, medium, small) spent on research and development (R&D) in the past. The values for productivity improvement are given below. Assume that the distributions for productivity improvement for the different levels of the amount spent on R&D are normally distributed.
| Large: |
9.7 |
7.8 |
9.6 |
9.5 |
9.9 |
| Medium: |
6.8 |
8 |
9.4 |
8.5 |
7.8 |
| Small: |
8.2 |
6.8 |
5.8 |
7 |
6.7 |
(A)Define (1) the response variable, (2) the factor, (3) the treatments, and (4) the experimental units.
(B) Compute the sample means and variances for each treatment, and the overall sample mean.
(C) Use the sample means and variances computed in (B) to calculate ???????? = i=1∑ρ ????(??¯?? -??¯)2 and ?????? = i=1∑ρ (???? - 1)????2 . Use your answers to obtain SSTO.
(D)Refer to the handout entitled "An Alternative Way to Calculate the Sum of Squares in a One-Way Analysis of Variance". Compute SSTO, SSTR, and SSE using the approach in the handout.
(E) Do you prefer the method in (C) or (D)? You must make a choice to get 1 mark for this question.
(F) Give the ANOVA table.
(G)Test at α = 0.01 if there is a difference in the mean productivity improvement for firms that are classified at different levels of R&D expenditures. Use a critical value approach, and make sure to define each mean in your hypotheses.
(H)Compute 99% simultaneous Tukey confidence intervals to discuss the nature of any differences in mean productivity improvement for firms that are classified at different levels of R&D expenditures.
(I) Suppose that it can only be assumed that the distributions for productivity improvement for different levels of R&D expenditure have the same shape. Use a nonparametric procedure to test at α = 0.01 if there is a difference in the location of the three treatments. Does your conclusion agree with your finding in (G)? Explain.
2. Safety in motels is a growing concern among travellers. In order to compare four national motel chains, four people were randomly selected who had stayed overnight in a motel in each of the four chains in the past two years. Each traveller was asked to rate each motel chain on a scale from 0 to 100 to indicate how safe he or she felt at that motel; the higher the score, the safer the traveller felt. The results are below. Assume that the scores given by any traveller for any motel follow a normal distribution.
| MOTEL |
Traveller 1 |
Traveller 2 |
Traveller 3 |
Traveller 4 |
| Discomfort Inn |
40 |
60 |
60 |
20 |
| Scandals Resort |
30 |
50 |
50 |
40 |
| Worst Eastern |
50 |
80 |
60 |
60 |
| Stupor Eight |
40 |
70 |
60 |
50 |
(A)Define (1) the response variable, (2) the treatment factor, and (3) the block factor.
(B) What additional assumption is needed to describe this study as a randomized block design?
(C) Compute the treatment sample means, block sample means, and overall sample mean.
(D) Refer to the handout entitled "An Alternative Way to Calculate the Sum of Squares in a Randomized Block Design". Compute SSTR, SSBL, SSTO, and SSE using the approach in the handout.
(E) Give the ANOVA table.
(F)Is there sufficient evidence at α = 0.05 to indicate that there is a difference in the mean safety scores of the motels? Use a critical value approach, and make sure to define each mean in your hypotheses.
(H)Is there sufficient evidence α = 0.05 to indicate that there is a difference in the mean safety scores assigned by the different travellers? Use a critical value approach, and make sure to define each mean in your hypotheses.
(I) Compute 95% simultaneous Tukey confidence intervals to discuss the nature of any differences in the mean safety scores of the different motels.
(J) Assuming that this study can be treated as a randomized block design, use a nonparametric procedure to test at α = 0.05 if there is a difference in the location of the four treatments. Ignore the fact that the number of treatments and blocks are both less than five so that you may answer the question. Does your conclusion agree with your finding in (G)? Explain.