the take-home section of the next exam

252x0831 4/24/08

ECO252 QBA2

THIRD EXAM

April 21, 2008

TAKE HOME SECTION

Name: _________________________

Student Number: _________________________

Class days and time : _________________________

Please Note: Computer problems 2 and 3 should be turned in with the exam (2). In problem 2, the 2 way

ANOVA table should be checked. The three F tests should be done with a 5% significance level and you should note whether there was (i) a significant difference between drivers, (ii) a significant difference between cars and (iii) significant interaction. In problem 3, you should show on your third graph where the regression line is. Check what your text says about normal probability plots and analyze the plot you did.

Explain the results of the t and F tests using a 5% significance level. You will not get full credit for the work unless you do.

(2)

III Do the following. (25+ points) Note: Look at 252thngs ( 252thngs ) on the syllabus supplement part of the website before you start (and before you take exams). Show your work! State H

0

and H

1

where appropriate. You have not done a hypothesis test unless you have stated your hypotheses, run the numbers and stated your conclusion.

(Use a 95% confidence level unless another level is specified.)

Answers without reasons or accompanying calculations usually are not acceptable. Neatness and clarity of explanation are expected. This must be turned in when you take the in-class exam. Note that from now on neatness means paper neatly trimmed on the left side if it has been torn, multiple pages stapled and paper written on only one side.

1. (McClave et. al) Three anticoagulant drugs are studied to compare how effective they are in dissolving blood clots. Each of 10 volunteers is given a drug in random order and the amount of time (in seconds) it takes for a uniform cut to stop bleeding is recorded. After adequate time has passed to eliminate the drug from the system, the experiment is repeated with another drug. The data is recorded below.

Drug

Volunteer A* B* C*

1

2

118.8 116.0 129.2

126.9 121.7 136.8

3 137.8 128.6 135.3

4 150.0 137.0 159.4

5 181.4 172.3 186.2

6 115.6 112.8 126.0

7 138.9 133.7 148.8

8 131.4 122.2 128.9

9 138.0 125.0 147.4

10 179.0 169.9 183.8

To personalize these data, let the last digit of your student number be a . Mark your problem ‘VERSION a

.’ If a is zero, use the original numbers. Add a to all numbers in the B* column. For example, if the second-to-last digit of Seymour’s student number is 6, he changes the numbers in column 2 to 122.0, 127.7 etc. For your convenience the following sums and sums of squares are given for columns. Remember that the longer clotting takes the better the drug.

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Column Sum of Column

 i x ij

Sum of Squares

 i x ij

2

A*

B*

C*

B*+ 1

1417.8

1339.2

1481.8

1349.2

205619

183285

223934

185973

B*+ 2

B*+ 3

B*+ 4

B*+ 5

B*+ 6

1359.2

1369.2

1379.2

1389.2

188681

191410

194158

196927

1399.2 199715

B*+ 7

B*+ 8

1409.2

1419.2

202523

205352

B*+ 9 1429.2 208200 a) State your null hypothesis and test it by doing a 1-way ANOVA, treating each column as a random sample and explain whether the test tells us that there is a difference in clotting times between the three drugs.(4) b) Using your results from a) present individual and Tukey intervals for the difference between average times for all three possible contrasts. Explain (i) under what circumstances you would use each interval and

(ii) whether the intervals show a significant difference. (2) Do not do this if you intend to do a 2-way

ANOVA later. If you do both b) and g) your total credit will be limited.

c) What other method could we use on these data to see if the clotting drug makes a difference? Under what circumstances would we use it? Try it and tell what it tests and what it shows. (3) [9] d) (Extra Credit) Do a Levene test on these data and explain what it tests and shows. (4) e) (Extra credit) Check you results on the computer. The Minitab setup for 1-way ANOVA can either be unstacked (in columns) or stacked (one column for data and one for column name – this is identical to the setup for a 2-way ANOVA, but you only tell the computer about the column on your spreadsheet that contains column labels. ). There is a Tukey option. The corresponding non-parametric methods require the stacked presentation. You may want to use and compare the Statistics pull-down menu and ANOVA or

Nonparametrics to start with. You may also want to try the Mood median test and the test for equal variances which are on the menu categories above. Even better, do a normality test on your columns. Use the Statistics pull-down menu to find the normality tests. The Kolmogorov-Smirnov option is actually

Lilliefors. The ANOVA menu can check for equality of variances. In light of these tests was the 1-way

ANOVA appropriate? You can get descriptions of unfamiliar tests by using the Help menu and the alphabetic command list or the Stat guide. (Up to 7) f) Do a 2-way ANOVA on these data and explain what hypotheses you test and what the conclusions are.

Show your work. (6) Explain why the results have changed so dramatically. (1) [16] g) Using your results from f) present two different confidence intervals for the difference between clotting times for the best and worst volunteer. Present two different kinds of confidence intervals for the three possible differences between drugs. Explain which of the intervals show a significant difference and whether this enables you to order the drugs by effectiveness. (4) h) What other method could we use on these data to see if city makes a difference while allowing for crossclassification? Under what circumstances would we use it? Try it and tell what it tests and what it shows.

(3)[21] h) (Extra credit) Check you results on the computer. The Minitab setup for both 2-way ANOVA and the corresponding non-parametric method is the same as the extra credit in the third graded assignment or the third computer problem. You may want to use and compare the Statistics pull-down menu and ANOVA or

Nonparametrics to start with. Explain your results. (4)

2

252x0831 4/24/08

2. (McClave et al.) In order to estimate labor costs for the next year, a shipping department tries to put together a regression explaining labor hours/week as a function of (1) Pounds of Product Shipped in thousands of pounds, (2) Percent of Product shipped by Truck and (3) Average shipment weight in pounds.

The data are as below.

Row Labor Pounds Truck AvWgt

1 100 5.1 90 20

2 85 3.8 99 22

3 108 5.3 58 19

4 116 7.5 16 15

5 92 4.5 54 20

6 63 3.3 42 26

7 79 5.3 12 25

8 101 5.9 32 21

9 88 4.0 56 24

10 71 4.2 64 29

11 122 6.8 78 10

12 85 3.9 90 30

In order to personalize the data, let the second-to-last digit of your student number be b and remove row b .

If b



0 , change it to 10. Clearly mark your paper with Version b . In order to simplify your work, the following sums have been computed. They are labeled by the version number. Please do not waste our time by computing means or other statistics from entire columns below.

Version Sum SSQ Sum SSQ Sum SSQ Sum SSQ Sum Sum Sum

Labor Labor Pounds

Labor Labor Pounds Pounds Truck Truck AvWgt AvWgt Pounds Truck Truck

1 1010 96114 54.5 287.91 601 40685 241 5629 54967 2811.8

2 1025 98889 55.8 299.48 592 38984 239 5545 55552 2894.6

3 1002 94450 54.3 285.83 633 45421 242 5668 57703 2963.4

4 994 92658 52.1 257.67 675 48529 246 5804 62111 3150.8

5 1018 97650 55.1 293.67 637 45869 241 5629 58999 3027.8

6 1047 102145 56.3 303.03 649 47021 235 5353 61321 3132.2

7 1031 99873 54.3 285.83 679 48641 236 5404 63019 3207.2

8 1009 95913 53.7 279.11 659 47761 240 5588 60735 3082.0

9 1022 98370 55.6 297.92 635 45649 237 5453 59039 3046.8

10 1039 101073 55.4 296.28 627 44689 232 5188 59423 3002.0

In the above, for example ‘Sum Truck’ is the sum of the numbers in the ‘Truck’ column, ‘SSQ Truck’ is the sum of the squared numbers in the ‘Truck’ column and ‘Sum Pounds Truck’ is the sum of the products of the ‘Pounds’ and ‘Truck.’ The numbers above are rounded as much as is safe. Excess rounding will be penalized. a) For a simple regression, you only need the data in the ‘Labor’ and the ‘Pounds’ columns.



X ,



Y ,



X

2

, and



Y

2 have been computed for you. the table, so your first job is to compute



XY . Show your work!

(1) n



11 and



XY , however, has been deleted from b) Compute the regression equation Y

  b

0

 b x to predict the number of labor hours on the basis of pounds of products shipped. (3).You may check your results on the computer, but let me see real or simulated hand calculations. [25] c) Compute e) Compute

R

2

. (2) s b

1

and do a significance test on b

1



  d) Compute

.

05



(2) s e

. (2) f) For each version of this problem I have confidence and prediction intervals for the following possible values of the 3 independent variables. Use the ‘pounds’ numbers to compute three 95% intervals for the number of hours worked on particular weeks in which these data are observed.

Observation Pounds Truck AvWgt

1 5.0 58.0 22.0

2 7.5 99.0 30.0

3 33.0 12.0 10.0

3

252x0831 4/24/08 g) Make a graph of the data. Show the regression line and the data points clearly. If you are not willing to do this neatly and accurately, don’t bother. (2)

[33] h) Do an ANOVA for the simple regression. Remember that your ANOVA should duplicate your significance test in results. (1) i) Replace your regression with a multiple regression showing how labor hours depend on both (thousands of) pounds shipped and percentage shipped by truck. (6) j) Find the new R

2

(2) k) Do an ANOVA to show that the variable that you have added has additional explanatory power. (2) l) After doing the regression that I just asked you to do, I did the set of regressions that appear on the next few pages. (You may want to only print the page with your version on it.) What the conclusion that you should draw from the ANOVA for your regression with three independent variables? (1) [45] m) As you may have showed by the ANOVA in k), the coefficients of labor hours and percent shipped by truck in the equation with two independent variables are significant at the 5% level. What has happened to the coefficients when the third independent variable was added? Does their sign make sense? (Don’t just say yes or no.) (2) n) Compute R-squared adjusted for degrees of freedom for the regression with two independent variables.

You can do this without having the results in j) through k) if you use the sequential sum of squares in the printout you used in l) and m). I have said that R-squared adjusted doesn’t usually tell us much. What about here? (2) o) Each of the printouts has ‘ Predicted Values for New Observations ’ based on the values of the three independent variables shown in f). This part of the printout shows confidence and prediction intervals for each of the three observations. (3) [52] i) Predict a value of Labor hours based on your results in i) and the data in observation 2). In percentage terms how much change has there been from adding the last independent variable. ii) Use the formula given in class for confidence and prediction intervals with the standard error in the printout. Compare the size of the interval using this sloppy formula with the size of the intervals in the printout. Notice that one of the intervals is extremely large. Why is it so gigantic?

4

252x0831 4/24/08

MTB > Regress c11 3 c12 c13 c14;

SUBC> Constant;

SUBC> Predict c120 c121 c122;

SUBC> Brief 2.

Version 1.

Regression Analysis: Labor_1 versus Pounds_1, Truck_1, AvWgt_1

The regression equation is

Labor_1 = 55.0 + 9.88 Pounds_1 + 0.205 Truck_1 - 1.07 AvWgt_1

Predictor Coef SE Coef T P

Constant 55.04 33.06 1.66 0.140

Pounds_1 9.880 3.304 2.99 0.020

Truck_1 0.20535 0.09769 2.10 0.074

AvWgt_1 -1.0676 0.6747 -1.58 0.158

S = 6.97482 R-Sq = 89.9% R-Sq(adj) = 85.6%

Analysis of Variance

Source DF SS MS F P

Regression 3 3037.1 1012.4 20.81 0.001

Residual Error 7 340.5 48.6

Total 10 3377.6

Source DF Seq SS

Pounds_1 1 2444.6

Truck_1 1 470.7

AvWgt_1 1 121.8

Predicted Values for New Observations

New

Obs Fit SE Fit 95% CI 95% PI

1 92.86 2.16 ( 87.76, 97.96) ( 75.60, 110.12)

2 117.44 16.23 ( 79.06, 155.82) ( 75.66, 159.22)XX

3 372.86 83.75 (174.82, 570.90) (174.14, 571.58)XX

XX denotes a point that is an extreme outlier in the predictors.

Values of Predictors for New Observations

New

Obs Pounds_1 Truck_1 AvWgt_1

1 5.0 58.0 22.0

2 7.5 99.0 30.0

3 33.0 12.0 10.0


SUBC> Constant;


SUBC> Brief 2.

Version 2.





Constant 61.60 32.44 1.90 0.099

Pounds_2 9.097 3.277 2.78 0.027

Truck_2 0.21031 0.09076 2.32 0.054

AvWgt_2 -1.1913 0.6725 -1.77 0.120

S = 6.79148 R-Sq = 90.4% R-Sq(adj) = 86.3%


Source DF SS MS F P

Regression 3 3054.8 1018.3 22.08 0.001


Total 10 3377.6

Source DF Seq SS

Pounds_2 1 2451.8

Truck_2 1 458.3

AvWgt_2 1 144.8


New


1 93.07 2.08 ( 88.16, 97.99) ( 76.28, 109.87)

2 114.91 15.36 ( 78.59, 151.23) ( 75.20, 154.62)XX

3 352.41 83.31 (155.40, 549.42) (154.75, 550.07)XX


5

252x0831 4/24/08


SUBC> Constant;


SUBC> Brief 2.

Version 3.





Constant 51.42 28.13 1.83 0.110

Pounds_3 9.991 2.793 3.58 0.009

Truck_3 0.19618 0.07692 2.55 0.038

AvWgt_3 -0.9515 0.5825 -1.63 0.146

S = 5.98061 R-Sq = 92.1% R-Sq(adj) = 88.7%


Source DF SS MS F P

Regression 3 2926.53 975.51 27.27 0.000


Total 10 3176.91

Source DF Seq SS

Pounds_3 1 2352.84

Truck_3 1 478.26

AvWgt_3 1 95.44


New


1 91.82 1.81 ( 87.53, 96.11) ( 77.04, 106.59)

2 117.22 13.46 ( 85.40, 149.05) ( 82.40, 152.05)XX

3 373.96 70.76 (206.64, 541.27) (206.04, 541.87)XX



New


1 5.0 58.0 22.0

2 7.5 99.0 30.0

3 33.0 12.0 10.0


SUBC> Constant;


SUBC> Brief 2.

Version 4.





Constant 56.32 33.69 1.67 0.138

Pounds_4 9.797 3.618 2.71 0.030

Truck_4 0.19163 0.09234 2.08 0.077

AvWgt_4 -1.0782 0.6832 -1.58 0.159

S = 7.02313 R-Sq = 87.8% R-Sq(adj) = 82.6%


Source DF SS MS F P

Regression 3 2491.28 830.43 16.84 0.001


Total 10 2836.55

Source DF Seq SS

Pounds_4 1 1934.71

Truck_4 1 433.73

AvWgt_4 1 122.84


New


1 92.69 2.23 ( 87.41, 97.98) ( 75.27, 110.12)

2 116.42 16.46 ( 77.49, 155.35) ( 74.09, 158.74)XX

3 371.14 93.87 (149.17, 593.10) (148.55, 593.72)XX


6

252x0831 4/24/08


SUBC> Constant;


SUBC> Brief 2.

Version 5.





Constant 49.68 35.76 1.39 0.207

Pounds_5 10.354 3.524 2.94 0.022

Truck_5 0.20465 0.09241 2.21 0.062

AvWgt_5 -0.9520 0.7250 -1.31 0.231

S = 6.91855 R-Sq = 90.3% R-Sq(adj) = 86.1%


Source DF SS MS F P

Regression 3 3103.7 1034.6 21.61 0.001


Total 10 3438.7

Source DF Seq SS

Pounds_5 1 2494.6

Truck_5 1 526.5

AvWgt_5 1 82.5


New


1 92.38 2.09 ( 87.45, 97.32) ( 75.30, 109.47)

2 119.04 16.78 ( 79.36, 158.73) ( 76.12, 161.97)XX

3 384.32 88.82 (174.29, 594.35) (173.66, 594.98)XX



New


1 5.0 58.0 22.0

2 7.5 99.0 30.0

3 33.0 12.0 10.0


SUBC> Constant;


SUBC> Brief 2.

Version 6.





Constant 82.54 36.28 2.28 0.057

Pounds_6 6.925 3.716 1.86 0.105

Truck_6 0.12905 0.09681 1.33 0.224

AvWgt_6 -1.4240 0.6725 -2.12 0.072

S = 6.36527 R-Sq = 88.6% R-Sq(adj) = 83.7%


Source DF SS MS F P

Regression 3 2206.02 735.34 18.15 0.001


Total 10 2489.64

Source DF Seq SS

Pounds_6 1 1647.73

Truck_6 1 376.64

AvWgt_6 1 181.66


New


1 93.33 1.94 (88.74, 97.91) (77.59, 109.06)

2 104.54 17.09 (64.13, 144.95) (61.42, 147.66)XX

3 298.39 94.01 (76.08, 520.69) (75.58, 521.20)XX


7

252x0831 4/24/08


SUBC> Constant;


SUBC> Brief 2.

Version 7.





Constant 55.30 31.38 1.76 0.121

Pounds_7 9.956 3.143 3.17 0.016

Truck_7 0.16110 0.09453 1.70 0.132

AvWgt_7 -0.9630 0.6635 -1.45 0.190

S = 6.70801 R-Sq = 90.3% R-Sq(adj) = 86.1%


Source DF SS MS F P

Regression 3 2925.20 975.07 21.67 0.001


Total 10 3240.18

Source DF Seq SS

Pounds_7 1 2601.67

Truck_7 1 228.72

AvWgt_7 1 94.80


New


1 93.24 2.09 ( 88.30, 98.17) ( 76.62, 109.85)

2 117.03 15.12 ( 81.27, 152.78) ( 77.91, 156.14)XX

3 376.15 79.86 (187.31, 565.00) (186.64, 565.66)XX



New


1 5.0 58.0 22.0

2 7.5 99.0 30.0

3 33.0 12.0 10.0


SUBC> Constant;


SUBC> Brief 2.

Version 8.





Constant 59.36 32.34 1.84 0.109

Pounds_8 9.304 3.267 2.85 0.025

Truck_8 0.19941 0.08893 2.24 0.060

AvWgt_8 -1.1457 0.6702 -1.71 0.131

S = 6.86252 R-Sq = 90.2% R-Sq(adj) = 86.0%


Source DF SS MS F P

Regression 3 3030.5 1010.2 21.45 0.001


Total 10 3360.2

Source DF Seq SS

Pounds_8 1 2395.6

Truck_8 1 497.3

AvWgt_8 1 137.6


New


1 92.24 2.11 ( 87.24, 97.24) ( 75.26, 109.22)

2 114.51 15.67 ( 77.46, 151.55) ( 74.07, 154.95)XX

3 357.32 83.19 (160.61, 554.03) (159.94, 554.70)XX


8

252x0831 4/24/08


SUBC> Constant;


SUBC> Brief 2.

Version 9.





Constant 45.56 30.46 1.50 0.178

Pounds_9 10.904 3.059 3.56 0.009

Truck_9 0.21444 0.08217 2.61 0.035

AvWgt_9 -0.9352 0.6147 -1.52 0.172

S = 6.26833 R-Sq = 92.0% R-Sq(adj) = 88.5%


Source DF SS MS F P

Regression 3 3141.9 1047.3 26.65 0.000


Total 10 3416.9

Source DF Seq SS

Pounds_9 1 2499.6

Truck_9 1 551.3

AvWgt_9 1 90.9


New


1 91.95 1.90 ( 87.46, 96.44) ( 76.46, 107.43)

2 120.52 14.47 ( 86.30, 154.74) ( 83.23, 157.81)XX

3 398.63 77.50 (215.38, 581.87) (214.78, 582.47)XX



New


1 5.0 58.0 22.0

2 7.5 99.0 30.0

3 33.0 12.0 10.0

MTB > Regress c101 3 c102 c103 c104;

SUBC> Constant;


SUBC> Brief 2.

Version 10.





Constant 39.55 31.11 1.27 0.244

Pounds_10 11.055 3.019 3.66 0.008

Truck_10 0.21929 0.08120 2.70 0.031

AvWgt_10 -0.6294 0.6711 -0.94 0.380

S = 6.15670 R-Sq = 91.0% R-Sq(adj) = 87.1%


Source DF SS MS F P

Regression 3 2669.39 889.80 23.47 0.000


Total 10 2934.73

Source DF Seq SS

Pounds_10 1 2139.99

Truck_10 1 496.08

AvWgt_10 1 33.33


New


1 93.70 1.94 ( 89.11, 98.29) ( 78.44, 108.96)

2 125.29 15.22 ( 89.31, 161.27) ( 86.48, 164.10)XX

3 400.70 76.12 (220.70, 580.70) (220.11, 581.28)XX


9

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