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Statistics For Business And Economics 4th Edition Paul Newbold - Solutions
[This problem requires either the material in Appendix A13.1 or a computer program to carry out the multiple regression computations.] In Chapter 12, using the data of Table 12.5, we estimated the regression model Y, Retail sales per household Disposable income per household Use these same data to
A market researcher is interested in the average amount of money per year spent by col- lege students on clothing. From 25 years of annual data the following estimated regression was obtained through least squares. y 50.72 +142x + .027x, +432y-1 (.047) (.021) (136) where y Expenditure per student,
A regression model was estimated to compare performances of students taking a micro- economics principles course either as a standard fourteen-week course, or as an inten- sive three-week-course. The following model was estimated from observations on 3501 students." y= -.7052 +1.4170x+2.1624x:
A consulting group offers courses in financial management for executives. At the end of these courses, participants are asked to provide overall ratings of the value of the course. For a sample of twenty-five courses, the following regression was estimated by least squares. where x y=42.97
[This problem requires a computer program to carry out the multiple regression computa- tions. In a student survey of twenty-seven undergraduates at the University of Illinois, the accompanying results were obtained on grade point average (y), number of hours per week spent on studying (x), average
The following model was fitted to data on fifty states." y 13,472 + 547x + 5.48x + 493x + 32.7x + 5,793 -3,100x (124.3) (1.858) (208.9) (234) R-54 (2,897) (1,761) where y-Annual salary of the chief justice of the state supreme court x Average annual salary of lawyers, in thousands of dollars
A law school dean wanted to assess the importance of factors that might help in predicting success in law school. For a random sample of fifty students, data were obtained when they graduated from law school, and the following model was fitted: where Ya+ Bux+ B+ B+ Y, Score reflecting overall
The following model was fitted to data on thirty-two large bank holding companies." y=7.62-016x + 1.23x R-37 where v=Price-earnings ratio (.008) (.496) Size of total bank assets, in billions of dollars x Dummy variable, taking the value 1 for regional banks, and 0 for money center banks (a)
The following model was fitted, to explain the selling prices of houses, to a sample of 815 sales. - y-1264+48.18x, +3382x, 1859x: +3219% +2005 (0.91) (515) R = .86 (488) (947) (768) where x y Selling price of house, in dollars Square footage of living area x Size of garage, in number of cars 'W.
The following model was fitted to eight annual sets of observations from 1972 to 1979, in an attempt to explain Organization of Petroleum Exporting Countries (OPEC) pricing) behavior. y=37x+5.221) (.029) (.50) where y = Difference between OPEC price in the current year and OPEC price in the
The following regression model was fitted to data on sixty U.S. male professional golfers: y=164,683 + 341.10x + 170.02x + 495.19x-4.23x (100.59) (167.18) (305.48) (90.0) -136,040, 35,549% +202.52x (25,634) (16,240) (106.20) R-516
Based on data from sixty-three countries, the following model was estimated by least squares: y .058-052,-.005x R 17 (.019) (.042) where y x Growth rate in real gross domestic product. Real income per capita Average tax rate, as a proportion of gross national product (a) Test against a two-sided
Based on data on 2,679 college basketball players, the following model was fitted:" Ya+B+ B ++ Box + where Minutes played in season Y x Field goal percentage Free throw percentage x=Rebounds per minute Points per minute xs Fouls per minute No Steals per minute X Blocked shots per minute X Turnovers
Based on 25 years of annual data, an attempt was made to explain savings in India." The model fitted was Ya+B+Bxz + where Y Change in real deposit rate x, Change in real per capita income xChange in real interest rate The least squares parameter estimates (with standard errors in parentheses) were:
Based on 107 students' scores on the first examination in a course on managerial cost ac- counting, the following model" was estimated by least squares: y=2.178+469x, +3.369x + 3.054x (.090) (.456) (1.457) R .686 where y Student's actual score on the examination Student's expected score on the
Based on a sample of sixty-four observations, the following model was estimated by least squares: y=-16.528 + 28.729, +022-023-054x4 -.077x+411% +349x+.028x R-467 where y Index of direct labor efficiency in production plant Ratio of overtime hours to straight-time hours worked by all production
[This exercise requires a computer program to carry out the multiple regression computa- tions.] At the end of classes, professors are rated by their students on a scale from 1 (poor) to 5 (excellent). Students are also asked what course grades they expect, and these are coded as A = 4, B = 3, and
A consulting group offers courses in financial management for executives. At the end of these courses, participants are asked to provide overall ratings of the value of the course. To assess the impact of various factors on ratings, the model Y = a + Bx, + Bxa + Bixs + was fitted for twenty-five
A survey research group conducts regular studies of households through mail question- naires and is concerned about the factors influencing the response rate. In an experiment, thirty sets of questionnaires were mailed to potential respondents. The regression model fitted to the resulting data set
[This exercise requires the material in Appendix A13.1.] Suppose that the regression model Ya+B+B+6 is estimated by least squares. Show that the residuals, e., from the fitted model sum to 11.47. A study was conducted to assess the influence of various factors on the start of new firms in the
A dependent variable is regressed on two independent variables. It is possible that the hy- potheses B. 0 and B. 0 cannot be rejected at low significance levels, yet the hypothe- sis B B 0 can be rejected at a very low significance level. In what circumstances might this result arise?
If an additional independent variable, however inelevant, is added to a multiple regression model, a smaller sum of squared errors will result. Explain why this is so, and discuss the consequences for the interpretation of the coefficient of determination.
State whether each of the following statements is true or false. (a) The error sum of squares must be smaller than the regression sum of squares.(b) Instead of carrying out a multiple regression, we can get the same information from simple linear regressions of the dependent variable on each
It is common practice to compute an analysis of variance table in conjunction with an esti- mated multiple regression. Carefully explain what can be leamed from such a table.
The method of least squares is used far more often than any alternative procedure to esti- mate the parameters of a multiple regression model. Explain the basis for this method of estimation, and discuss why its use is so widespread.
The subjects of a random sample of ninety-three freshmen at the University of Illinois were asked to rate on a scale from 1 (low) to 10 (high) their overall opinion of residence hall life. They were also asked to rate their levels of satisfaction with roommates, with the floor, with the hall, and
In order to assess the effect in a state of banks' economic power on their political power. the following model was hypothesized and fitted to data from all fifty states." Y = + Bx + Bx + Bux + Bux+ Buxe + where Y = Ratio of bank provisions for state and local taxes (in thousands of dollars) to
A real estate agent hypothesizes that in her town, the selling price of a house in dollars (y) depends on its size in square feet of floor space (x), the lot size in square feet (r). the number of bedrooms (x), and the number of bathrooms (x). For a random sample of twenty house sales, the
For the regression on aircraft design effort of Exercise 1.the estimated intercept was a-578 Predict design effort for a plane with top speed Mach 1.0, weighing 7 tons, and having 50% of its parts in common with other models.
Using the information in Exercise. 3.predict the weekly milk consumption of a family of four with an income of $600 per week.
Using the information in Exercise 4, predict the weight gain for a freshman who eats an average of twenty meals per week, exercises an average of 10 hours per week, and con- sumes an average of six beers per week.
In the study of Exercises 3 and 6 on milk consumption, a third independent variable- number of preschool children in the household-was added to the regression model. The sum of squared errors when this augmented model was estimated by least squares was found to be 83.7. Test the null hypothesis
A dependent variable is regressed on K independent variables, using sets of sample ob- servations. We denote by SSE the error sum of squares and by R the coefficient of deter- mination for this estimated regression. We want to test the null hypothesis that K, of these independent variables, taken
Refer to the study on the determinants of community demand for police officers discussed in Exercise18.Test the null hypothesis and interpret your findings.
Refer to the study of Exercise 16.Test the null hypothesis that taken together, consump- tion expenditures and the relative price of imports do not linearly affect the demand for imports in Jamaica.
Refer to Exercise 15.Test the null hypothesis that, taken together, the three independent variables do not linearly influence the price of air conditioners.
Refer to Exercise 14.Test the null hypothesis that, taken together, the four independent variables do not linearly influence revenue generated by state lotteries.
Refer to the study on weight gains, described in Exercises 4 and 7.(a) Test the null hypothesis (b) Set out the analysis of variance table.
Refer to the study on milk consumption, described in Exercises 3 and 6.(a) Test the null hypothesis He: B-B-0 (b) Set out the analysis of variance table.
For the study on the influence of financial institutions on share prices of Exercise 2, forty- eight quarterly observations were used, and the corrected coefficient of determination was found to be R-463. Test the null hypothesis Ho: B-B-0
Refer to the study on aircraft design effort of Exercises I and 5.(a) Test the null hypothesis H: B B B 0 (b) Set out the analysis of variance table for the fitted regression.
Suppose that a dependent variable is related to K independent variables through a multiple regression model. Let R' denote the coefficient of determination and R the corrected coef- ficient. Suppose that sets of observations are used to fit the regression. (a) Show that (n-1)R-K R = (b) Show that
In a study of differences in levels of community demand for police officers," the follow- ing sample regression was obtained, based on data from thirty-nine towns in Delaware County, Pennsylvania. 00024x00002x+.00034x+48122 + .04950 where y00232 (.00010) (.000018) (.00012) (.77954) (.01172)
In a study of the demand for imports in Jamaica," the following model was fitted to 19 years of data: y=-58.9 +20% -.10x (.0092) (.084) where y Quantity of imports x - Personal consumption expenditures Price of imports +Domestic prices R = .96 The figures in parentheses beneath the coefficient
A study was conducted to determine whether certain features could be used to explain variability in the prices of air conditioners." For a sample of nineteen air conditioners, the following regression was estimated: y-68.236.0023x + 19.729 +7.653x) R-84 (.005) (8.992) (3.082) where y Price (in
In a study of revenue generated by state lotteries," the following regression equation was fitted to data from twenty-nine states with lotteries: y-31.323.04045x) +.8772xy-365.01, -9.9298x (.00755) (3107) (263.88) (3.4520) R = .51 where y = Dollars of net revenue per capita per year generated by
Refer to the data of Example 13.2. (a) Find 95% and 99% confidence intervals for the expected change in tender premium resulting from a 1-unit increase in soliciting fee per share, as a percentage of closing market price, when the values of all other independent variables remain unchanged. (b) Test
Refer to the data of Example 13.1. (a) Test against a two-sided alternative the null hypothesis that, all else being equal, me- dian per capita personal income has no influence on the effective property tax rate. (b) Test the null hypothesis that, taken together, the three independent variables do
In the study of Exercise 4, where the sample regression was based on twenty-five observa- tions, the estimated standard errors were S189 -565 -.243 (a) Test against the appropriate one-sided alterative the null hypothesis that, all else be- ing equal, hours of exercise do not linearly influence
In the study of Exercise 3, where the sample regression was based on thirty observations. the estimated standard errors were -.023 3.35 (a) Test against the appropriate one-sided alternative the null hypothesis that for fixed family size, milk consumption does not depend linearly on income. (b)
In the study of Exercise I, the estimated standard errors were -.099 S-032 (a) Find 90% and 95% confidence intervals for B. (b) Find 95% and 99% confidence intervals for B. S-0023 (c) Test against a two-sided alternative the null hypothesis that, all else being equal, the plane's weight has no
In the study of Exercise 4, twenty-five observations were used to calculate the least squares estimates. The regression sum of squares and error sum of squares were found to be SSR 79.2 and SSE = 45.9 (a) Find and interpret the coefficient of determination. (b) Find the corrected coefficient of
In the study of Exercise 3, where the least squares estimates were based on thirty sets of sample observations, the total sum of squares and regression sum of squares were found to be SST 162.1 and SSE = 88.2 (a) Find and interpret the coefficient of determination. (b) Find the corrected
In the study of Exercise 1, where the least squares estimates were based on twenty-seven sets of sample observations, the total sum of squares and regression sum of squares were found to be SST 3.881 and SSR 3.549 (a) Find and interpret the coefficient of determination. (b) Find the error sum of
The following model was fitted to a sample of twenty-five students using data obtained at the end of their freshman year in college. The aim was to explain students' weight gains. Y = a + B+ B+ Bets, + where Y Weight gained, in pounds, during freshman year Average number of meals eaten per week
The following model was fitted to a sample of thirty families in order to explain household milk consumption: Y = x + x + B+Y, Milk consumption, in quarts per week Weekly income, in hundreds of dollars X Family size The least squares estimates of the regression parameters were a--.025 b-.052 by
In a study of the influence of financial institutions on share prices in the United Kingdom. quarterly data over a period of 12 years were analyzed. The postulated model was Ya+ B+ B+ where Y, Change over the quarter in the Financial Times stock price index xChange over the quarter in equity
An aircraft company wanted to predict the number of worker-hours necessary to finish the design of a new plane. Relevant explanatory variables were thought to be the plane's top speed, its weight, and the number of parts it had in common with other models built by the company. A sample of
Comment on the following statement: If a regression of the yield per acre of corn on the quantity of fertilizer used were esti- mated, using fertilizer quantities in the range typically used by farmers, the slope of the es- timated regression line would certainly be positive. However, it is well
An analyst believes that the only important determinant of banks' return on assets (Y) is the ratio of loans to deposits (x). For a random sample of twenty banks, the sample regres sion line Y-.97.47x was obtained, with coefficient of determination .720. (a) Find the sample correlation between
Based on a sample of twenty-five observations, the population regression model was estimated. The least squares estimates obtained were a-15.6 and The total and error sums of squares were SST = 268 and b=1.3 SSE = 204 (a) Find and interpret the coefficient of determination. (b) Test against a
Based on a sample of thirty observations, the population regression model Y = a + Br, +6 was estimated. The least squares estimates obtained were a=10.1 and h 8.4 The regression and error sums of squares were SSR = 128 and SSE 286 (a) Find and interpret the coefficient of determination. (b) Test at
For a sample of 306 students in a basic business communications course, the sample re- gression line was obtained." Here y=58.8132875x y Final student score at the end of the course xScore on a diagnostic writing skills test given at the beginning of the course The coefficient of determination was
The table shows, for eight brands of instant coffee. purchases per buyer (y) and the per- centage buying in a year (x). Y 36 33 28 26 27 29 20 26 A 24 21 22 22 18 13 9 6 (a) Estimate the regression of purchases per buyer on percentage buying. (b) Interpret the slope of the estimated regression
A company routinely administers an aptitude test to all new management trainees. At the end of the first year with the company, these trainees are graded by their immediate super- visors. For a random sample of twelve trainees, the results shown in the accompanying table were obtained. APTITUDE
Based on a sample of n observations, (x, y). ( ). ( ya), the sample regression of y on r is calculated. Show that the sample regression line passes through the point (x=x, yy), where x and y are the sample means.
For a random sample of 192 male employees, a sample correlation of 18 was found he- tween age and a measure of willingness to relocate." Given only this information, derive all the conclusions you can about the regression of willingness to relocate on age.
A study was carried out on the perceptions of problems arising in relationships between advertising agencies and their clients." Agencies were asked the extent to which various agency characteristics caused problems. The table given here shows average responses on a scale from I (seldom a problem)
In a study of the motives for patronage of a used merchandise outlet, the rankings of mo- tives were obtained from samples of light and heavy shoppers, as indicated in the accom panying table. Calculate Spearman's rank correlation coefficient, and test against the al- terative of positive
The accompanying table shows, for a random sample of twenty long-term-growth mutual funds, percentage retum over a period of 12 months and total assets (in millions of dollars).(a) Calculate Spearman's rank correlation coefficient. (b) Carry out a nonparametric test of the null hypothesis of no
Refer to Exercise 71 of Chapter 9.Test against a two-sided alternative the null hypothesis that the variances of absolute percentage forecast errors are the same for the two financial analysts.
For a sample of sixty-six months, the correlation between the returns on U.S. and Japanese 5-year bonds was found to be 293." Test the null hypothesis that the population correla tion is 0 against the alternative that it is positive.
For a random sample of 526 firms, the sample correlation between the proportion of a firm's directors who are outsiders and a risk-adjusted measure of return on the firm's stock was found to be .1398." Test against a two-sided alternative the null hypothesis that the population correlation is zero.
For a random sample of fifty-three specialty stores in a chain, the correlation between annual dollar sales per square foot of floor space and annual dollar rent per square foot of floor space was found to be 37.Test the null hypothesis that these two quantities are uncorre lated in the population
A random sample of five sets of observations on a pair of random variables yielded the re- sults given in the table. 3 A 0 0 (a) Find the sample correlation coefficient. (b) In light of the fact that each y, value is the square of the correspondings, value, com- ment on your answer in part (a).
In this chapter, we discussed two measures of correlation the sample correlation coeffi cient and Spearman's rank correlation coefficient. Discuss the circumstances in which each measure might be appropriate.
What is meant by the statement that a pair of random variables are positively correlated? Give examples of pairs of random variables for which you would expect (a) Positive correlation (b) Negative correlation (c) Zero correlation
A new sales representative for the corporation of Exercise 18 scores 70 on the aptitude test. Find 80% and 90% confidence intervals for the value of weekly sales he will achieve. REVIEW EXERCISES:
Use the data of Exercises 17 and 26 to find 90% and 95% confidence intervals for the ex- pected return on the corporation's stock when the rate of return on the S. and P. 500 index is 1%.
Refer to the data in Exercise 16.For a year in which there is no change in the unemploy- ment rate, find a 90% confidence interval for the actual, and also for the expected, change in mean employee absence rate due to own illness.
Continue the analysis of Exercise 14.If the Dow-Jones index increases by 1.0% in the first five trading days of a year, find 90% confidence intervals for the actual, and also for the expected, percentage changes in the index over the whole year. Discuss the distinction be- tween these intervals.
Refer to the data of Exercise 13.(a) Find a point estimate for the volume of sales when the price of the stereo system is $480 in a given region. (b) If the price of the system is set at $480, find 95% confidence intervals for the actual volume of sales in a particular region and the expected
Refer to the data of Exercise 33.Test at the 5% level against the appropriate one-sided al- terative the null hypothesis that sales do not depend linearly on price for this premium scotch whiskey.
For the problem of Exercise 19, on sales of premium beer in restaurants, it was found that - - 250 Test against a two-sided alternative the null hypothesis that the slope of the population re- gression line is 0.42. For a sample of 74 monthly observations, the regression of the percentage return on
Denote by r the sample correlation between a pair of random variables. (a) Show that 1-7 -2 SST (b) Using the result in part (a), show that V-(-2) b (c) Using the result in part (b), deduce that the test of the null hypothesis of 0 population correlation, given in Section 12.1, is the same as the
Refer to the data of Exercise 15.Test against a two-sided alterative the null hypothesis that mutual fund losses on Friday, November 13, 1989, did not depend linearly on previ- ous gains in 1989.
For the stock rate-of-return problem of Exercises 17 and 26 it was found that 196.2 (a) Test the null hypothesis that the slope of the population regression line is 0 against the alternative that it is positive.(b) Test against a two sided alternative the null hypothesis that the slope of the
Doctors are interested in the relationship between the dosage of a medicine and the time required for a patient's recovery. The following table shows, for a sample of five patients. dosage levels (in grams) and recovery times (in hours). These patients have similar char acteristics except for
A sample of twenty-five blue-collar employees at a production plant was taken. Each em- ployee was asked to assess his or her own job satisfaction (x), on a scale from 1 to 10.In addition, the number of days absent (y) from work during the last year were found for these employees. The sample
Using the data of Exercise 16, test against a two-sided alterative the null hypothesis that change in mean employee absence rate due to own illness does not depend linearly on change in unemployment rate.
A liquor wholesaler is interested in assessing the effect of the price of a premium scotch whiskey on the quantity sold. The results in the accompanying table on price, in dollars. and sales, in cases, were obtained from a sample of 8 weeks of sales records. PRICE SALES 19.2 20.5 19.7 21.3 20.8
A fast-food chain decided to carry out an experiment to assess the influence of advertising expenditure on sales. Different relative changes in advertising expenditure, compared to the previous year, were made in eight regions of the country, and resulting changes in sales levels were observed. The
Continue the analysis of Exercise 14 of the regression of the percentage change in the Dow-Jones index in a year on the percentage change in the index over the first five trading days of the year. (a) Use an unbiased estimation procedure to find a point estimate of the variance of the error terms
Consider the linear regression of stereo system sales on price, based on the data of Exercise 13.(a) Use an unbiased estimation procedure to find an estimate of the variance of the error terms in the population regression. (b) Use an unbiased estimation procedure to find an estimate of the variance
In the study discussed in Exercise 6, we saw that for a sample of 353 college faculty, the correlation was .11 between annual raises and teaching evaluations. What would be the coefficient of determination of a regression of annual raises on teaching evaluations for this sample? Interpret your
(a) Find the predicted values, y., and the residuals,e, for the least squares regression of change in mean employee absence rate due to own illness on change in unemploy- ment rate. (b) Find the sums of squares SST, SSR, and SSE, and verify that SST SSRSSE (c) Using the results in part (b), find
25. Refer to the data on unemployment rate and employee absence rate in Exercise
Compare your answer with the sample correlation found for these data in Exercise 2.24. Find the proportion of the sample variability in mutual fund percentage losses on November 13, 1989, explained by their linear dependence on 1989 percentage gains through November 12, based on the data of Exercise
Find and interpret the coefficient of determination for the regression of the percentage change in the Dow Jones index in a year on the percentage change in the index over the first five trading days of the year, continuing the analysis of Exercise
Find and interpret the coefficient of determination for the regression of stereo system sales on price, using the data of Exercise 13.1
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