# MATHEMATICS

Sampling Distributions MCQs Quiz Solution

1.) During the years 2005–2009, the world record for the 100-meter dash was repeatedly broken as never before. The years and corresponding record times are in data file XR17008 and shown in the table. Using x = 1 through 5 to represent years 2005–2009, fit a second-order polynomial model to the data. (SOURCE: Karen Rosen, “9.58: Bolt Strikes Again at Worlds,” USA Today, August 17, 2009, p. 1C.)

Please round your answers to 2 decimal places.

a. The estimate of the world record time for the 100-meter dash in 2014 is seconds

2.) This exercise requires a computer and statistical software. The data mentioned below can be found in several formats on your Premium Website under “Chapter datasets”. An early edition of Car and Driver magazine reported the engine horsepower, top speed (mph), curb weight (pounds), and 0–60 mph acceleration time (seconds) for a selection of automobiles that ranged from large American “muscle cars” to small economy imports. The data are in file XR17009. Fit a thirdorder polynomial model (using top speed (mph) as the dependent variable and horsepower as the independent variable) to predict the top speed for a car with 170 horsepower. (SOURCE: 1964 Car and Driver Yearbook, pp. 14-63.)

Please round (a) through (e) to 4 decimals, and (f) to the nearest whole number.

a. The p-value of the hypothesis test of whether the regression equation is significant is .

b. The estimated intercept is .

c. The estimated coefficient of horsepower is .

d. The estimated coefficient of horsepower2 is .

e. The estimated coefficient of horsepower3 is .

f. The estimated top speed of a car with 170 advertised horsepower is .

3.) The regression model

= 200 + 5×1 – 2×2 has been fitted to a set of data.

a. If x2 = 10, the effect on of an increase in x1 of 1 unit is .

b. If x2 = 20, the effect on of an increase in x1 of 1 unit is

4.) This exercise requires a computer and statistical software. The data mentioned below can be found in several formats on your Premium Website under “Chapter datasets”. Auto Rental News has reported U.S. operations data for 15 top rental-car companies. Using data file XR17016, with y = rental revenue (millions of dollars) for the most recent year of record, x1 = thousands of cars in service, and x2 = number of rental locations, fit a first-order model of the form E(y) = 0 + 1×1 + 2×2 (SOURCE: Wall Street Journal Almanac 1999, p. 649.)

Please round (a) through (c) to 2 decimals, and (d) and (e) to 3 decimals.

a. The estimated intercept is .

b. The estimated coefficient of “Cars” is .

c. The estimated coefficient on “Location” is .

d. The p-value of the hypothesis test of whether the regression equation is significant is .

e. The amount of variation in “Revenue” that is explained by the equation is . (express your answer as a decimal)

5.) This exercise requires a computer and statistical software. The data mentioned below can be found in several formats on your Premium Website under “Chapter datasets”. Auto Rental News has reported U.S. operations data for 15 top rental-car companies. Using data file XR17016, with y = rental revenue (millions of dollars) for the most recent year of record, x1 = thousands of cars in service, and x2 = number of rental locations, fit the model E(y) = 0 + 1×1 + 2×2 + 3x1x2 (include an interaction term).

Please round (a) to 4 decimals, and (b) to 3 decimals.

a. The estimated coefficient on the interaction term is .

b. The amount of variation in “Revenue” that is explained by the equation is .

6.) The Air Transport Association has reported the following kinds of information describing 23 individual U.S. airlines for the most recent year of record: y = total operating revenue ($millions), x1 = number of aircraft and x2 = number of employees. The data are in file XR17017. (SOURCE: Wall Street Journal Almanac 1999, p. 644.)

Fit a first-order model of the form E(y) = 0 + 1×1 + 2×2

Please round (a) to 1 decimal, and (b) through (e) to 3 decimals.

a. The estimated intercept is .

b. The estimated coefficient of x1 is .

c. The estimated coefficient on x2 is .

d. The p-value of the hypothesis test of whether the regression equation is significant is .

e. The amount of variation in “Revenue” that is explained by the equation is . (express your answer as a decimal)

7.) The Air Transport Association has reported the following kinds of information describing 23 individual U.S. airlines for the most recent year of record: y = total operating revenue ($millions), x1 = number of aircraft and x2 = number of employees. The data are in file XR17017.

Fit the model E(y) = 0 + 1×1 + 2×2 + 3x1x2 (include an interaction term).

Please round (a) to 5 decimals, and (b) to 3 decimals.

a. The estimated coefficient on the interaction term is .

b. The amount of variation in y that is explained by the equation is

8.) For the Car and Driver information described in Exercise 17.9, it would seem reasonable that the number of seconds required to accelerate to 60 mph would depend on both a car’s weight and its horsepower. Further, it would be logical to assume that horsepower and weight might interact. Using data file XR17009, with y = 0-60 mph acceleration time (seconds), x1 = horsepower, and x2 = curb weight (pounds), fit a model of the form E(y) = 2

2 0 + 1×1 + 2×2 + 3x1x2 + 4×1 + 5×2

Please round your answers to 3 decimal places.

a. The p-value of the hypothesis test of whether the regression is significant is .

b. The R2 of the regression is . (express your answer as a decimal)

9.) The dean of a law school has developed a regression equation for estimating the starting salary (thousands of dollars) of a new graduate on the basis of two independent variables: x1 = the score achieved on the Law School Admission Test (LSAT) at the time of application, and x2 = whether the new graduate’s position is in the private sector (coded as 1) or the public sector (coded as 0). For the estimation = 25 + 0.1×2 + 30×2, the estimated the starting salary (in thousands of dollars) for a new graduate with an LSAT score of 160 who is entering the private sector is

10.) A researcher has observed 50 automatic teller machine (ATM) users outside the gift shop of a museum, and the data are in file XR17028. With y = amount withdrawn (dollars), x1 = time required for the transaction (seconds), and x2 = gender (1 = female, 0 = male), determine the linear regression equation for estimating the

withdrawal amount as a function of the two predictor variables.

Please round your answers to 2 decimal places.

a. The estimated intercept is .

b. The estimated coefficient of x1 is .

c. The estimated coefficient of x2 (gender variable) is

11.) For a small sample of suburban homes recently sold, the selling price ($), living space (square feet), and suburban area (Sunny Haven, Easy Acres, or Rolling Hills) have been recorded. Given the data shown here, determine the linear regression equation for estimating selling price based on the other variables.

Please round (a) to 2 decimals and round (b) and (c) to the nearest integer.

a. An additional foot of living space will tend to cost . (Please round your answer to 2 decimal places)

b. A home in Sunny Haven will tend to cost more than a home in Rolling Hills.

c. A home in Easy Acres will tend to cost less than a home in Rolling Hills.

12.) Following data transformation, regression analysis results in the estimation equation log = 3.15 + 0.473log x1 + 0.354log x2. Transform this equation to the equivalent multiplicative model with estimation equation = b0x1b1x2b2 (The base of the logarithms is 10.)

Please round your answers to 3 decimal places

a. b0 = .

b. b1 = .

c. b2 = .

13.) Note: During the years 2005–2009, the world record for the 100meter dash was repeatedly broken as never before. The years and corresponding record times are in data file XR17008 and shown in the table. Using x = 1 through 5 to represent years 2005–2009, fit an exponential model. (SOURCE: Karen Rosen, “9.58: Bolt Strikes Again at Worlds,” USA Today,August 17, 2009, p. 1C.)

Please round your answers to 2 decimal places.

a. The estimate the world record time for the 100meter dash in the year 2014 is seconds.

14.) This exercise requires a computer and statistical software. The data mentioned below can be found in several formats on your Premium Website under “Chapter datasets”. Use logarithms with base 10. Auto Rental News has reported U.S. operations data for 15 top rental-car companies. Using data file XR17016, with y = rental revenue (millions of dollars) for the most recent year of record, x1 = thousands of cars in service, and x2 = number of rental locations, fit a multiplicative model.

(SOURCE: Wall Street Journal Almanac 1999, p. 649.) The estimated rental revenue for a company with 80,000 cars in service and 2500 rental locations is .

Please round your answer to 2 decimal places

15.) Note: This exercise requires a computer and statistical software. The data mentioned below can be found in several formats on your Premium Website under “Chapter datasets”. Auto Rental News has reported U.S. operations data for 15 top rental-car companies. The data file is XR17016, with y = rental revenue (millions of dollars) for the most recent year of record, x1 = thousands of cars in service, and x2 = number of rental locations. (SOURCE: Wall Street Journal Almanac 1999, p. 649.)

Complete the following correlation matrix for the independent variables.

Please round to 3 decimals. If there is no entry enter “0”. Cars

16.) The sales director for an industrial supplies firm has collected information describing the performance and personal characteristics of 80 members of her sales force. The data in file XR17066 include the percent of sales quota the salesperson achieved last year, age, number of years with the firm, and personal appearance

(very neat, about average, or a little sloppy). In the data file, a very neat person is coded as neat = 1, average = 0, a person of average neatness is coded as neat = 0, average = 1, and a sloppy person is coded as neat = 0, average = 0. With y = percent of sales quota, perform a linear regression using the other variables as

predictors.

Please round (a) through (d) to 2 decimals, and (e) and (f) to 3 decimals.

a. The coefficient of “Age” is .

b. The coefficient of “Years with the firm” is .

c. A person that is “very neat” will tend to have a quota performance that is percentage points greater than a person who is “sloppy.

d. A person that is “average” will tend to have a quota performance that is percentage points greater than a person who is sloppy.

e. The amount of variation that the regression equation explains is .

f. The p-value of the hypothesis test of whether the regression is significant is .

17.) The Dow Jones Global Indexes describe over 3000 companies worldwide that represent more than 80% of the equity capital in the world’s stock markets. Data file XR17068 shows the quarterly indexes for the world as a whole, and indexes for the United States, the Americas, Europe, and the Asia/Pacific region over 23 consecutive quarters. (SOURCE: Wall Street Journal Almanac 1999, p. 264.) Given that the dependent variable will be y = world index, construct a correlation matrix for the four possible predictor variables.

Please round to 3 decimals.

US