# Expert Answers

1. An automobile dealer wants to estimate the proportion of customers who still own the cars they purchased six years ago. A random sample of 200 customers selected from the automobile dealer’s records indicates that 88 still own cars that were purchased six years earlier.

a. Construct a 95% confidence interval estimate of the population proportion of all customers who still own the cars they purchased six years ago.

b. Is the estimated proportion statistically significantly different from 50% still owning their cars six years later?

c. Construct a 95% confidence interval for the population proportion if the results of sampling had been 176 out of 400 rather than 88 out of 200.

2. A large factory analyzes the relationship between annual salaries (Y) and the number of years employees have worked (X) at the factory. Data was collected for a sample of 27 employees. The Excel output is shown below:

Summary measures

Multiple R 0.7952

R-Square 0.6324

Std Err of Estimate 6595.19

ANOVA table

Source Df SS MS F p-value

Explained 1 1870854328 1870854328 43.012 0.0000

Unexplained 25 1087414471 43496579

Regression coefficients

Coefficient Std Err t-value p-value 95% Interval

Constant 28326.88 2096.929 13.5087 0.0000

Years Employed 1067.03 162.699 6.5583 0.0000 843 to 1281

a. What is the equation describing the relationship between salary and years worked?

b. Is the relationship statistically significant? That is, does the data indicate there is actually a linear relationship between salary and seniority? Why or why not?

c. Give a 95% confidence interval for the average annual raise employees get at this factory.

d. What percentage of the variation in salaries at the factory is explained by variation in the number of years worked by employees?

e. How well correlated are the predicted salaries from employees using this model and the actual salaries of employees?

3. A company wants to investigate the relationship between annual salaries and the following explanatory variables: number of years an employee has worked at the company, whether the employee is male or female (gender, 1 if female, 0 if male), and the interaction between gender and years at the company. These data have been collected for a sample of 28 employees and the regression output is shown below.

Summary measures

Multiple R 0.8065

R-Square 0.6504

Adj R-Square 0.6067

StErr of Estimate 6572.3

Regression coefficients

Coefficient Std Err t-value p-value

Constant 29831.68 3904.56 7.640 0.0000

Years Employed 869.04 266.78 3.258 0.0033

Gender -2396.54 4620.04 -0.519 0.6087

Years & Gender 403.93 350.38 1.153 0.2603

a. What proportion of variation in salaries is explained by differences in seniority (years at the company) and gender?

b. How well do the predicted salaries from this model correlate with the actual salaries of employees at the company?

c. Does gender have a statistically significant impact on salary at this company?

d. According to this model, do men’s and women’s salaries change at different rates for each additional year employed and, if so, how different are the changes each year?

4. For a new auction site for car parts for “classic” cars, the price received for a particular item increases with its age (i.e., the age of the car on which the item can be used, measured in years), the number of bidders, whether the car is a “classic” (measured as 1 if it is, 0 if not), and if the car is a General Motors product (“GM,” measured as 1 if it is, 0 if not). The Excel multiple regression output is shown below.

Summary measures

Multiple R 0.8391

R-Square 0.7041

Adj R-Square 0.6783

StErr of Estimate 148.828

ANOVA Table

Source df SS MS F p-value

Explained 4 1212039.4 303009.9 13.68 0.0000

Unexplained 23 509444.9 22149.8

Regression coefficients

Coefficient Std Err t-value p-value

Constant -1242.99 331.204 -3.7529 0.0010

Age of Item 75.017 10.65 7.0459 0.0000

Number of Bidders 13.973 10.44 1.3380 0.1940

“Classic” 17.838 6.73 2.65 0.009

“GM” x “Classic” 8.88 9.22 0.96 0.38

a. What is the regression equation estimated by this analysis for the relationship between the average price of an item as a function of the independent variables shown in the analysis:

b. Do the independent variables taken together (the overall model as shown in the analysis) explain a statistically significant proportion of the variation in the price of items put up for auction?

c. The average number of bidders for an item now is 10; if the number of bidders doubled to 20, what would this do for the average price of items put up for auction?

d. Suppose an item is for a “Classic” non-GM auto – how much does that impact the price of the item?

e. Suppose an item is for a car that is both a Classic and also a General Motors (GM) product – does this have a statistically significant impact on the price of the item and, if so, what is that impact?

f. Give a 95% confidence interval for the price of an item that is for a car that is 20 years old, has 10 bidders, is a Classic and is a Ford product, using the regression relationship shown in the analysis above. Use a t statistic of 2 for this calculation. Show your calculation.

g. Give a 95% confidence interval for the impact that a car classified as a classic GM has on the price of an item.

h. If you were going to develop a second version of this model, what variables would you include and why?

5. The scanning equipment at the airport buzzes (spots metal) 95% of the time it is present on a passenger and indicates metal present (buzzes) 10% of the time it isn’t present (false positives). Five percent of passengers have metal objects on them detectable by the scanning equipment.

a. what is the probability that the scanning equipment misses a piece of metal that is present on a passenger?

b. What is the probability that a passenger going through the scanner does not set the scanner buzzer off?

c. The buzzer goes off. What is the probability the passenger is carrying metal?

a. Construct a 95% confidence interval estimate of the population proportion of all customers who still own the cars they purchased six years ago.

b. Is the estimated proportion statistically significantly different from 50% still owning their cars six years later?

c. Construct a 95% confidence interval for the population proportion if the results of sampling had been 176 out of 400 rather than 88 out of 200.

2. A large factory analyzes the relationship between annual salaries (Y) and the number of years employees have worked (X) at the factory. Data was collected for a sample of 27 employees. The Excel output is shown below:

Summary measures

Multiple R 0.7952

R-Square 0.6324

Std Err of Estimate 6595.19

ANOVA table

Source Df SS MS F p-value

Explained 1 1870854328 1870854328 43.012 0.0000

Unexplained 25 1087414471 43496579

Regression coefficients

Coefficient Std Err t-value p-value 95% Interval

Constant 28326.88 2096.929 13.5087 0.0000

Years Employed 1067.03 162.699 6.5583 0.0000 843 to 1281

a. What is the equation describing the relationship between salary and years worked?

b. Is the relationship statistically significant? That is, does the data indicate there is actually a linear relationship between salary and seniority? Why or why not?

c. Give a 95% confidence interval for the average annual raise employees get at this factory.

d. What percentage of the variation in salaries at the factory is explained by variation in the number of years worked by employees?

e. How well correlated are the predicted salaries from employees using this model and the actual salaries of employees?

3. A company wants to investigate the relationship between annual salaries and the following explanatory variables: number of years an employee has worked at the company, whether the employee is male or female (gender, 1 if female, 0 if male), and the interaction between gender and years at the company. These data have been collected for a sample of 28 employees and the regression output is shown below.

Summary measures

Multiple R 0.8065

R-Square 0.6504

Adj R-Square 0.6067

StErr of Estimate 6572.3

Regression coefficients

Coefficient Std Err t-value p-value

Constant 29831.68 3904.56 7.640 0.0000

Years Employed 869.04 266.78 3.258 0.0033

Gender -2396.54 4620.04 -0.519 0.6087

Years & Gender 403.93 350.38 1.153 0.2603

a. What proportion of variation in salaries is explained by differences in seniority (years at the company) and gender?

b. How well do the predicted salaries from this model correlate with the actual salaries of employees at the company?

c. Does gender have a statistically significant impact on salary at this company?

d. According to this model, do men’s and women’s salaries change at different rates for each additional year employed and, if so, how different are the changes each year?

4. For a new auction site for car parts for “classic” cars, the price received for a particular item increases with its age (i.e., the age of the car on which the item can be used, measured in years), the number of bidders, whether the car is a “classic” (measured as 1 if it is, 0 if not), and if the car is a General Motors product (“GM,” measured as 1 if it is, 0 if not). The Excel multiple regression output is shown below.

Summary measures

Multiple R 0.8391

R-Square 0.7041

Adj R-Square 0.6783

StErr of Estimate 148.828

ANOVA Table

Source df SS MS F p-value

Explained 4 1212039.4 303009.9 13.68 0.0000

Unexplained 23 509444.9 22149.8

Regression coefficients

Coefficient Std Err t-value p-value

Constant -1242.99 331.204 -3.7529 0.0010

Age of Item 75.017 10.65 7.0459 0.0000

Number of Bidders 13.973 10.44 1.3380 0.1940

“Classic” 17.838 6.73 2.65 0.009

“GM” x “Classic” 8.88 9.22 0.96 0.38

a. What is the regression equation estimated by this analysis for the relationship between the average price of an item as a function of the independent variables shown in the analysis:

b. Do the independent variables taken together (the overall model as shown in the analysis) explain a statistically significant proportion of the variation in the price of items put up for auction?

c. The average number of bidders for an item now is 10; if the number of bidders doubled to 20, what would this do for the average price of items put up for auction?

d. Suppose an item is for a “Classic” non-GM auto – how much does that impact the price of the item?

e. Suppose an item is for a car that is both a Classic and also a General Motors (GM) product – does this have a statistically significant impact on the price of the item and, if so, what is that impact?

f. Give a 95% confidence interval for the price of an item that is for a car that is 20 years old, has 10 bidders, is a Classic and is a Ford product, using the regression relationship shown in the analysis above. Use a t statistic of 2 for this calculation. Show your calculation.

g. Give a 95% confidence interval for the impact that a car classified as a classic GM has on the price of an item.

h. If you were going to develop a second version of this model, what variables would you include and why?

5. The scanning equipment at the airport buzzes (spots metal) 95% of the time it is present on a passenger and indicates metal present (buzzes) 10% of the time it isn’t present (false positives). Five percent of passengers have metal objects on them detectable by the scanning equipment.

a. what is the probability that the scanning equipment misses a piece of metal that is present on a passenger?

b. What is the probability that a passenger going through the scanner does not set the scanner buzzer off?

c. The buzzer goes off. What is the probability the passenger is carrying metal?

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