Cincinnati Paint Company sells quality brands of paints through hardware stores throughout the United States. The company maintains a large sales force who call on existing customers and look for new business. The national sales manager is investigating the relationship between the number of sales calls made and the miles driven by the sales representative. Also, do the sales representatives who drive the most miles and make the most calls necessarily earn the most in sales commissions? To investigate, the vice president of sales selected a sample of 25 sales representatives and determined:

• The amount earned in commissions last month (y)
• The number of miles driven last month (x₁)
• The number of sales calls made last month (x₂)

The information is reported below.

Develop a regression equation including an interaction term. (Negative amount should be indicated by a minus sign. Round your answers to 3 decimal places.)

Commissions= ___________ + __________________ calls + _______________________miles + ___________________x1x2

Complete the following table. (Negative amounts should be indicated by a minus sign. Round your answers to 3 decimal places.)

Compute the value of the test statistic corresponding to the interaction term. (Negative amount should be indicated by a minus sign. Round your answer to 2 decimal places.)

Value of the test statistic ___________________

At the 0.05 significance level is there a significant interaction between the number of sales calls and the miles driven?

This is STATISTICALLY SIGNIFICANT or NOT SIGNIFICANT (choose), so we conclude that there IS INTERACTION or IS NO INTERACTION (choose).

Computers in some vehicles calculate various quantities related to performance. One of these is the fuel efficiency, or gas mileage, usually expressed as miles per gallon (mpg). For one vehicle equipped in this way, the miles per gallon were recorded each time the gas tank was filled, and the computer was then reset. In addition to the computer's calculations of miles per gallon, the driver also recorded the miles per gallon by dividing the miles driven by the number of gallons at each fill-up. The following data are the differences between the computer's and the driver's calculations for that random sample of 20 records. The driver wants to determine if these calculations are different. Assume that the standard deviation of a difference is

σ = 3.0.

(a) State the appropriate

H₀

and

H_a

to test this suspicion.

H₀: μ = 3 mpg;    H_a: μ ≠ 3 mpg

H₀: μ > 0 mpg;    H_a: μ < 0 mpg

H₀: μ > 3 mpg;    H_a: μ < 3 mpg

H₀: μ = 0 mpg;    H_a: μ ≠ 0 mpg

H₀: μ < 0 mpg;    H_a: μ > 0 mpg

(b) Carry out the test. Give the P-value. (Round your answer to four decimal places.)

Interpret the result in plain language.

We conclude that μ = 3 mpg; that is, we have strong evidence that the computer's reported fuel efficiency does not differ from the driver's computed values.

We conclude that μ ≠ 0 mpg; that is, we have strong evidence that the computer's reported fuel efficiency differs from the driver's computed values.

We conclude that μ ≠ 3 mpg; that is, we have strong evidence that the computer's reported fuel efficiency differs from the driver's computed values.

We conclude that μ ≠ 0 mpg; that is, we have strong evidence that the computer's reported fuel efficiency does not differ from the driver's computed values.

We conclude that μ = 0 mpg; that is, we have strong evidence that the computer's reported fuel efficiency differs from the driver's computed values.

1. You plan to get a new Honda Civic Coupe when you graduate and are evaluating four alternatives: buy new; buy used, 3-yr lease new; and 3-yr lease new with optional purchase at end of year 3. Use the following data for your analysis of alternatives.
   • Study period is 6 years. (Assume you can make a second 3-yr lease at same terms)
   • Car loan interest rate is 3% APR. (MARR)
   • Planned annual usage is 10,000 miles/yr.  
   • New LX-P model purchase price is $20,500. Salvage (trade-in) value at end of year-6 is $8,200. (Note, also applies to lease with optional purchase alternative).
   • Pre-owned EX-T model (better features than the LX-P) purchase price is $17,500; but it is two years old with 20,000 miles. Salvage value at end of year-6 is $6,000.
   • End-to-End warrantee period is 10 years/100,000 miles.                            
   • 3-year lease new LX-P contract terms: $2,500 initial payment (end of years 0 & 3); $2,300/yr payments (end of years 1 to 6); additional $0.15/mile for usage over 10,000 miles/yr (payable end of years 1 to 6). No salvage value (leasing company owns car).
   • Option to purchase leased LX-P at end of year 3 for $12,500. Salvage value at end of             year-6 is $8,200 (same as new LX-P model buy).    
   • Assume the title, registration, & tax costs; insurance cost; and maintenance costs are identical for all four alternatives

                 A) Draw cash flow diagrams for all four alternatives.

                 B) Determine best alternative using the Annual Equivalent Value on Total Investment                                    evaluation method. (Show your decision analysis work).

                 C) If the loan interest rate increases to 5%, what would be the best alternative?

     D) Is your decision sensitive to driving over 10,000 miles/yr with the added $0.15/mile wear    & tear cost for the 3-yr lease. Note, this cost is refunded if you purchase the car at the             end of the lease period). Provide the rationale for your answer.

The linear model below explores a potential association between property damage and wind speed based on observational data from 94 hurricanes that hit the United States between 1950 and 2012. The variables are

Damage: property damage in millions of U.S. dollars (adjusted for inflation to 2014) for each hurricane

Landfall.Windspeed: Maximum sustained windspeed in miles per hour measured along U.S. coast for each hurricane

* Assume that the sample data satisfies all assumptions for linear regression.

Level of significance = 0.05.   

> summary(model)

Call:

lm(formula = Damage ~ Landfall.Windspeed)

Residuals:

   Min 1Q Median 3Q Max

-9294 -4782 -1996 -531 90478

Coefficients:

          Estimate Std. Error t value Pr(>|t|)

(Intercept) -10041.78 6064.29 -1.656 0.1012

Landfall.Windspeed 142.07 56.65 2.508 0.0139 *

---

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 12280 on 92 degrees of freedom

Multiple R-squared: [ A ], Adjusted R-squared: 0.05381

F-statistic: 6.289 on 1 and 92 DF, p-value: 0.01391

(a) Write the equation for the linear model using the variables Damages and Landfall Windspeed, taking the results of the t-tests into account.

(b) A hurricane is defined as a storm with wind speeds greater than 74 miles per hour. Interpret the value of the intercept in connection to the real-life context of this model (two or three sentences). Hint: Is the intercept truly meaningful, given the definition of a hurricane?

(c) The value of Pearson’s correlation coefficient for Damages and Landfall.Windspeed is 0.2529438. Calculate and interpret the value of R2 , denoted [A] in the table, in relation to the predictor and response variables.

(d) The range of observed maximum wind speeds in the sample data is 75 – 190 miles per hour. Is it appropriate to use the linear model to predict the cost of damage for a hurricane with a maximum wind speed of 150 miles per hour? Why or why not? If so, estimate the typical value of damages (specifying units).

(e) Would it be appropriate to use the linear model to predict the cost of damage for a hurricane with a maximum wind speed of 225 miles per hour? Why or why not? If so, estimate the typical value of damages (specifying units).

4. Interpretation of simple linear regression

The linear model below explores a potential association between property damage and windspeed based on observational data from 94 hurricanes that hit the United States between 1950 and 2012. The variables are

Damage: property damage in millions of U.S. dollars (adjusted for inflation to 2014) for each hurricane

Landfall.Windspeed: Maximum sustained windspeed in miles per hour measured along U.S. coast for each hurricane

* Assume that the sample data satisfies all assumptions for linear regression.

Level of significance = 0.05.   

> summary(model)

Call:

lm(formula = Damage ~ Landfall.Windspeed)

Residuals:

   Min     1Q Median     3Q    Max

-9294 -4782 -1996   -531 90478

Coefficients:

                                                Estimate           Std. Error        t value             Pr(>|t|)

(Intercept)                                -10041.78        6064.29          -1.656              0.1012

Landfall.Windspeed    142.07             56.65               2.508               0.0139 *

---

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 12280 on 92 degrees of freedom

Multiple R-squared: [ A ],      Adjusted R-squared: 0.05381

F-statistic: 6.289 on 1 and 92 DF, p-value: 0.01391

(a)   Write the equation for the linear model using the variables Damages and Landfall.Windspeed, taking the results of the t-tests into account.

(b) A hurricane is defined as a storm with windspeeds greater than 74 miles per hour. Interpret the value of the intercept in connection to the real-life context of this model (two or three sentences). Hint: Is the intercept truly meaningful, given the definition of a hurricane?

(c) The value of Pearson’s correlation coefficient for Damages and Landfall.Windspeed is 0.2529438. Calculate and interpret the value of R² , denoted [A] in the table, in relation to the predictor and response variables.

(d) The range of observed maximum windspeeds in the sample data is 75 – 190 miles per hour. Is it appropriate to use the linear model to predict the cost of damage for a hurricane with a maximum windspeed of 150 miles per hour? Why or why not? If so, estimate the typical value of damages (specifying units).

(e) Would it be appropriate to use the linear model to predict the cost of damage for a hurricane with a maximum windspeed of 225 miles per hour? Why or why not? If so, estimate the typical value of damages (specifying units).

Suppose Mary can have good health with probability 0.8 and bad health with probability 0.2. If the person has a good health her wealth will be $256, if she has bad health her wealth will be $36. Suppose that the utility of wealth come from the following utility function: U(W)=W^0.5 Answer each part:

A. Find the reduction in wealth if Mary bad health.

B. Find the expected wealth of Mary if she has no insurance.

C. Find her utility if she has bad health and she has no insurance.

D. Find her utility if she has good health and she has no insurance.

E. Find the expected utility of Mary if she has no insurance.

F. Find the certain equivalent of the lottery.

G. If she has full insurance, find the payment the insurance company made to her if she has bad health.

H. Find the maximum premium she is willing to pay for full insurance.

I. Find the fair premium if she is full insured.

J. Find her expected utility if she paid the fair premium and has full insurance.