Practice Problems for Part VI
1. A company sets different prices for a particular stereo system in eight different regions of the country. The accompanying table shows the numbers of units sold and the corresponding prices (in hundreds of dollars).
SALES / 420 / 380 / 350 / 400 / 440 / 380 / 450 / 420PRICE / 5.5 / 6.0 / 6.5 / 6.0 / 5.0 / 6.5 / 4.5 / 5.0
(a)Plot these data, and estimate the linear regression of sales on price.
(b)What effect would you expect a $100 increase in price to have on sales?
2. On Friday, November 13, 1989, prices on the New York Stock Exchange fell steeply; the Standard and Poors 500-share index was down 6.1% on that day. The accompanying table shows the percentage losses (y) of the twenty-five largest mutual funds on November 13, 1989. Also shown are the percentage gains (x), assuming reinvested dividends and capital gains, for these same funds for 1989, through November 12.
y / x / y / x / y / x4.7 / 38.0 / 6.4 / 39.5 / 4.2 / 24.7
4.7 / 24.5 / 3.3 / 23.3 / 3.3 / 18.7
4.0 / 21.5 / 3.6 / 28.0 / 4.1 / 36.8
4.7 / 30.8 / 4.7 / 30.8 / 6.0 / 31.2
3.0 / 20.3 / 4.4 / 32.9 / 5.8 / 50.9
4.4 / 24.0 / 5.4 / 30.3 / 4.9 / 30.7
5.0 / 29.6 / 3.0 / 19.9 / 3.8 / 20.3
3.3 / 19.4 / 4.9 / 24.6
3.8 / 25.6 / 5.2 / 32.3
(a)Estimate the linear regression of November 13 losses on pre-November 13, 1989, gains.
(b)Interpret the slope of the sample regression line.
3. For a period of 11 years, the figures in the accompanying table were found for annual change in unemployment rate and annual change in mean employee absence rate due to own illness.
Year / Change InUnemployment
Rate / Change In Mean Employee
Absence Rate Due To Own
Illness
1 / -.2 / +.2
2 / -.1 / +.2
3 / +1.4 / +.2
4 / +1.0 / -.4
5 / -.3 / -.1
6 / -.7 / +.2
7 / +.7 / -.1
8 / +2.9 / -.8
9 / -.8 / +.2
10 / -.7 / +.2
11 / -1.0 / +.2
(a)Estimate the linear regression of change in mean employee absence rate due to own illness on change in unemployment rate.
(b)Interpret the estimated slope of the regression line.
4. Refer to the data of Exercise 2. Test against a two-sided alternative the null hypothesis that mutual fund losses on Friday, November 13, 1989, did not depend linearly on previous gains in 1989.
5. An attempt was made to evaluate the forward rate as a predictor of the spot rate in the Canadian treasury bill market. For a sample of seventy-nine quarterly observations, the estimated linear regression:
y = .00027 + .7916x
was obtained, where
y / = Actual change in the spot ratex / = Change in the forward rate
The coefficient of determination was 0.097, and the estimated standard deviation of the estimator of the slope of the population regression line was 0.2759.
(a)Interpret the slope of the estimated regression line.
(b)Interpret the coefficient of determination.
(c)Test the null hypothesis that the slope of the population regression line is 0 against the alternative that the true slope is positive, and interpret your result.
(d)Test against a two-sided alternative the null hypothesis that the slope of the population regression line is 1, and interpret your result.
6. For a sample of 306 students in a basic business communications course, the sample regression line
y = 58.813 + 0.2875x
was obtained, where
y / = Final student score at the end of the coursex / = Score on a diagnostic writing skills test given at the beginning of the course
The coefficient of determination was 0.1158, and the estimated standard deviation of the estimated slope of the population regression line was 0.04566.
(a)Interpret the slope of the sample regression line.
(b)Interpret the coefficient of determination.
(c)The information given allows the null hypothesis that the slope of the population regression line is 0 to be tested against the alternative that it is positive. Carry out this test and state your conclusion.
7. The marketing manager of a large supermarket chain would like to determine the effect of shelf space on the sales of pet food. A random sample of 12 equal-sized stores is selected with the following results:
STORE / SHELF SPACE, X (FEET) / WEEKLY SALES, Y (HUNDREDS OF DOLLARS) / STORE / SHELF SPACE, X (FEET) / WEEKLY SALES, Y (HUNDREDS OF DOLLARS)1 / 5 / 1.6 / 7 / 15 / 2.3
2 / 5 / 2.2 / 8 / 15 / 2.7
3 / 5 / 1.4 / 9 / 15 / 2.8
4 / 10 / 1.9 / 10 / 20 / 2.6
5 / 10 / 2.4 / 11 / 20 / 2.9
6 / 10 / 2.6 / 12 / 20 / 3.1
(a)Set up a scatter diagram.
(b)Assuming a linear relationship, use the least-squares method to find the regression coefficients and .
(c)Interpret the meaning of the slope in this problem.
(d)Predict the average weekly sales (in hundreds of dollars) of pet food for stores with 8 feet of shelf space for pet food.
(e)Suppose that sales in store 12 are 2.6. Do parts (a)-(d) with this value and compare the results,
(f)What shelf space would you recommend that the marketing manager allocate to pet food? Explain.
8. A company that has the distribution rights to home video sales of previously released movies would like to be able to estimate the number of units that it can be expected to sell. Data are available for 30 movies that indicate the box office gross (in millions of dollars) and the number of units sold (in thousands) of home videos. The results are as follows:
MOVIE / BOX OFFICE GROSS ($ MILLIONS) / HOME VIDEO UNITS SOLD / MOVIE / BOX OFFICE GROSS ($ MILLIONS) / HOME VIDEO UNITS SOLD1 / 1.10 / 57.18 / 16 / 9.36 / 190.80
2 / 1.13 / 26.17 / 17 / 9.89 / 121.57
3 / 1.18 / 92.79 / 18 / 12.66 / 183.30
4 / 1.25 / 61.60 / 19 / 15.35 / 204.72
5 / 1.44 / 46.50 / 20 / 17.55 / 112.47
6 / 1.53 / 85.06 / 21 / 17.91 / 162.95
7 / 1.53 / 103.52 / 22 / 18.25 / 109.20
8 / 1.69 / 30.88 / 23 / 23.13 / 280.79
9 / 1.74 / 49.29 / 24 / 27.62 / 229.51
10 / 1.77 / 24.14 / 25 / 37.09 / 277.68
11 / 2.42 / 115.31 / 26 / 40.73 / 226.73
12 / 5.34 / 87.04 / 27 / 45.55 / 365.14
13 / 5.70 / 128.45 / 28 / 46.62 / 218.64
14 / 6.43 / 126.64 / 29 / 54.70 / 286.31
15 / 8.59 / 107.28 / 30 / 58.51 / 254.58
(a)Set up a scatter diagram.
(b)Use the least-squares method to find the regression coefficients and .
(c)State the regression equation.
(d)Interpret the meaning of and in this problem.
(e)Predict the average video unit sales for a movie that had a box office gross of $20 million.
(f)What other factors in addition to box office gross might be useful in predicting video unit sales?
9. An agent for a residential real estate company in a large city would like to be able to predict the monthly rental costs for apartments based on the size of apartment as defined by square footage. A sample of 25 apartments in a particular residential neighborhood was selected and the information gathered revealed the following:
APARTMENT / MONTHLY RENT ($) / SIZE (SQUARE FEET) / APARTMENT / MONTHLY RENT ($) / SIZE (SQUARE FEET)1 / 950 / 850 / 14 / 1,800 / 1,369
2 / 1,600 / 1,450 / 15 / 1,400 / 1,175
3 / 1,200 / 1,085 / 16 / 1,450 / 1,225
4 / 1,500 / 1,232 / 17 / 1,100 / 1,245
5 / 950 / 718 / 18 / 1,700 / 1,259
6 / 1,700 / 1,485 / 19 / 1,200 / 1,150
7 / 1,650 / 1,136 / 20 / 1,150 / 896
8 / 935 / 726 / 21 / 1,600 / 1,361
9 / 875 / 700 / 22 / 1,650 / 1,040
10 / 1,150 / 956 / 23 / 1,200 / 755
11 / 1,400 / 1,100 / 24 / 800 / 1,000
12 / 1,650 / 1,285 / 25 / 1,750 / 1,200
13 / 2,300 / 1,985
(a)Set up a scatter diagram.
(b)Use the least-squares method to find the regression coefficients and .
(c)State the regression equation.
(d)Interpret the meaning of and in this problem.
(e)Predict the average monthly rental cost for an apartment that has 1,000 square feet.
(f)Why would it not be appropriate to use the model to predict the monthly rental for apartments that have 500 square feet?
(g)Your friends Jim and Jennifer are considering signing a lease for an apartment in this residential neighborhood. They are trying to decide between two apartments, one with 1,000 square feet for a monthly rent of $1,250 and the other with 1,200 square feet for a monthly rent of $1,425. What would you recommend to them? Why?
10. If SSR = 36 and SSE = 4, find SST, then compute the coefficient of correlation R and interpret its meaning.
11. In Problem 7 above the marketing manager used shelf space for pet food to predict weekly sales. Use the computer output you obtained to solve that problem.
(a)Compute the coefficient of determination and interpret its meaning.
(b)Compute the standard error of the estimate.
(c)How useful do you think this regression model is for predicting sales?
12. Suppose you are testing the null hypothesis that the slope is not significant. From your sample of n = 18 you determine that
(a)What is the value of the t-test statistic?
(b)At the = 0.05 level of significance, what are the critical values?
(c)On the basis of your answers to (a) and (b), what statistical decision should be made?
(d)Set up a 95% confidence interval estimate of the population slope .
13. Suppose you are testing the null hypothesis that the slope is not significant. From your sample of n = 20, you determine that SSR = 60 and SSE = 40.
(a)What is the value of the F-test statistic?
(b)At the = 0.05 level of significance, what is the critical value?
(c)On the basis of your answers to (a) and (b), what statistical decision should be made?
14. In Problem 8 above a company wanted to predict home video sales based on the box office gross of movies. Use the computer output you obtained to solve that problem.
(a)At the 0.05 level of significance, is there evidence of a linear relationship between box office gross and home video sales?
(b)Set up a 95% confidence interval estimate of the population slope .
15. In Problem 9 above an agent for a real estate company wanted to predict the monthly rent for apartments based on the size of the apartment. Use the computer output you obtained to solve that problem.
(a)At the 0.05 level of significance, is there evidence of a linear relationship between the size of the apartment and the monthly rent?
(b)Set up a 95% confidence interval estimate of the population slope .
16. Management of a soft-drink bottling company wished to develop a method for allocating delivery costs to customers. Although one aspect of cost clearly relates to travel time within a particular route, another type of cost reflects the time required to unload the cases of soft drink at the delivery point. A sample of 20 customers was selected from routes within a territory and the delivery time and the number of cases delivered were measured with the following results:
CUSTOMER / NUMBER OF CASES / DELIVERY TIME (MINUTES) / CUSTOMER / NUMBER OF CASES / DELIVERY TIME (MINUTES)1 / 52 / 32.1 / 11 / 161 / 43.0
2 / 64 / 34.8 / 12 / 184 / 49.4
3 / 73 / 36.2 / 13 / 202 / 57.2
4 / 85 / 37.8 / 14 / 218 / 56.8
5 / 95 / 37.8 / 15 / 243 / 60.6
6 / 103 / 39.7 / 16 / 254 / 61.2
7 / 116 / 38.5 / 17 / 267 / 58.2
8 / 121 / 41.9 / 18 / 275 / 63.1
9 / 143 / 44.2 / 19 / 287 / 65.6
10 / 157 / 47.1 / 20 / 298 / 67.3
Assuming that we wanted to develop a model to predict delivery time based on the number of cases delivered:
(a)Set up a scatter diagram.
(b)Use the least-squares method to find the regression coefficients and .
(c)State the regression equation.
(d)Interpret the meaning of and in this problem.
(e)Predict the average delivery time for a customer who is receiving 150 cases of soft drink.
(f)Would it be appropriate to use the model to predict the delivery time for a customer who is receiving 500 cases of soft drink? Why?
(g)Compute the coefficient of determination and explain its meaning in this problem.
(h)Compute the coefficient of correlation.
(i)Compute the standard error of the estimate.
(j)Perform a residual analysis using either the residuals or the Studentized residuals. Is there any evidence of a pattern in the residuals? Explain.
(k)At the .05 level of significance, is there evidence of a linear relationship between delivery time and the number of cases delivered?
(l)Set up a 95% confidence interval estimate of the average delivery time for customers that receive 150 cases of soft drink.
(m)Set up a 95% prediction interval estimate of the delivery time for an individual customer who is receiving 150 cases of soft drink.
(n)Set up a 95% confidence interval estimate of the population slope.
(o)Explain how the results obtained in (a)-(n) can help allocate delivery costs to customers.
17. A brokerage house would like to be able to predict the number of trade executions per day and has decided to use the number of incoming phone calls as a predictor variable. Data were collected over a period of 35 days with the following results:
DAY / NUMBER OF INCOMING CALLS / TRADE EXECUTIONS / DAY / NUMBER OF INCOMING CALLS / TRADE EXECUTIONS1 / 2,591 / 417 / 18 / 2,237 / 397
2 / 2,146 / 321 / 19 / 2,328 / 365
3 / 2,185 / 362 / 20 / 2,078 / 330
4 / 2,245 / 364 / 21 / 2,134 / 312
5 / 2,600 / 442 / 22 / 2,192 / 340
6 / 2,510 / 386 / 23 / 1,965 / 339
7 / 2,394 / 370 / 24 / 2,147 / 364
8 / 2,486 / 376 / 25 / 2,015 / 295
9 / 2,483 / 463 / 26 / 2,046 / 292
10 / 2,297 / 389 / 27 / 2,073 / 379
11 / 2,106 / 302 / 28 / 2,032 / 294
12 / 2,035 / 266 / 29 / 2,108 / 329
13 / 1,936 / 339 / 30 / 1,923 / 274
14 / 1,951 / 369 / 31 / 2,069 / 326
15 / 2,292 / 403 / 32 / 2,061 / 306
16 / 2,094 / 319 / 33 / 2,010 / 352
17 / 1,897 / 306 / 34 / 1,913 / 290
35 / 1,904 / 283
(a)Set up a scatter diagram.
(b)Use the least-squares method to find the regression coefficients and .
(c)State the regression equation.
(d)Interpret the meaning of and in this problem.
(e)Predict the average number of trades executed for a day in which the number of incoming calls is 2,000.
(f)Would it be appropriate to use the model to predict the average number of executed for a day in which the number of incoming calls is 5,000? Why?
(g)Compute the coefficient of determination and explain its meaning in this problem.
(h)Compute the coefficient of correlation.
(i)Compute the standard error of the estimate.
18. The director of graduate studies at a large college of business would like to be able to predict the grade point index (GPI) of students in an MBA program based on the Graduate Management Aptitude Test (GMAT) score. A sample of 20 students who have completed 2 years in the program is selected; the results are as follows:
OBSERVATION / GMAT SCORE / GPI / OBSERVATION / GMAT SCORE / GPI1 / 688 / 3.72 / 11 / 567 / 3.07
2 / 647 / 3.44 / 12 / 542 / 2.86
3 / 652 / 3.21 / 13 / 551 / 2.91
4 / 608 / 3.29 / 14 / 573 / 2.79
5 / 680 / 3.91 / 15 / 536 / 3.00
6 / 617 / 3.28 / 16 / 639 / 3.55
7 / 557 / 3.02 / 17 / 619 / 3.47
8 / 599 / 3.13 / 18 / 694 / 3.60
9 / 616 / 3.45 / 19 / 718 / 3.88
10 / 594 / 3.33 / 20 / 759 / 3.76
Hint:First determine which are the independent and dependent variables.
(a)Plot a scatter diagram and, assuming a linear relationship, use the least-squares method to find the regression coefficients and .
(b)Interpret the meaning of the Y intercept and the slope in this problem.
(c)Use the regression model developed in (a) to predict the average GPI for a student with a GMAT score of 600.
(d)Compute the standard error of the estimate.
(e)Compute the coefficient of determination and interpret its meaning in this problem.
(f)Compute the coefficient of correlation, R.
(g)Perform a residual analysis on your results and determine the adequacy of the fit of the model.
(h)At the .05 level of significance, is there evidence of a linear relationship between GMAT score and GPI?
(i)Set up a 95% confidence interval estimate for the average GPI of students with a GMAT score of 600.
(j)Set up a 95% prediction interval estimate of the GPI for a particular student with a GMAT score of 600.
(k)Set up a 95% confidence interval estimate of the population slope.
19. The manager of the purchasing department of a large banking organization would like to develop a model to predict the amount of time it takes to process invoices. Data are collected from a sample of 30 days with the following results:
DAY / INVOICES PROCESSED / COMPLETION TIME (HOURS) / DAY / INVOICES PROCESSED / COMPLETION TIME (HOURS)1 / 149 / 2.1 / 16 / 169 / 2.5
2 / 60 / 1.8 / 17 / 190 / 2.9
3 / 188 / 2.3 / 18 / 233 / 3.4
4 / 19 / 0.3 / 19 / 289 / 4.1
5 / 201 / 2.7 / 20 / 45 / 1.2
6 / 58 / 1.0 / 21 / 193 / 2.5
7 / 77 / 1.7 / 22 / 70 / 1.8
8 / 222 / 3.1 / 23 / 241 / 3.8
9 / 181 / 2.8 / 24 / 103 / 1.5
10 / 30 / 1.0 / 25 / 163 / 2.8
11 / 110 / 1.5 / 26 / 120 / 2.5
12 / 83 / 1.2 / 27 / 201 / 3.3
13 / 60 / 0.8 / 28 / 135 / 2.0
14 / 25 / 0.4 / 29 / 80 / 1.7
15 / 173 / 2.0 / 30 / 29 / 0.5
Hint:Determine which are the independent and dependent variables.
(a)Set up a scatter diagram.
(b)Assuming a linear relationship, use the least-squares method to find the regression coefficients and .
(c)Interpret the meaning of the Y intercept and the slope in this problem.
(d)Use the regression model developed in (b) to predict the average amount of time would take to process 150 invoices.
(e)Compute the standard error of the estimate.
(f)Compute the coefficient of determination and interpret its meaning.
(g)Compute the coefficient of correlation R.
(h)Plot the residuals against the number of invoices processed and also against time.
(i)Based on the plots in (h), does the model seem appropriate?
(j)At the .05 level of significance, is there evidence of a linear relationship between amount of time and the number of invoices processed?
(k)Set up a 95% confidence interval estimate of the average amount of time taken process 150 invoices.
(l)Set up a 95% prediction interval estimate of the amount of time it takes to process 150 invoices on a particular day.
B01.13051Prof. Juran