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.

Information

For each of Questions 1 to 5, is the given value of the correlation coefficient reasonable?

Hint: think about the strength and the direction of the relationship between the two variables in each case.

Note: It is subjective to decide whether the magnitude of the correlation between two variables should be, for example, 0.7 or 0.8. The below questions don't ask you to make a decision like this.

Question 1 

A correlation of r = +0.7 between gender and height.

Question 2 

Let X = speed of vehicles driving on the highway and Y = distance for the vehicles to stop when the brakes are applied.

A correlation of r = +1.0 between X and Y.

Question 3 

A correlation of r = 0 between number of slurpees (a frozen beverage) sold at 7-Eleven in one day and number of cups of hot chocolate sold at the same store in the same day.

Question 4

A correlation of r = +0.7 between number of pages in a fiction novel and time it takes to read the novel.

Question 5 

Let X = time it takes a marathon runner to finish the race and Y = number of runners who finish ahead of him.

A correlation of r = -0.8 between X and Y.

1. Given  P(A) = 0.3 and P(B) = 0.2, do the following. (For each answer, enter a number.)

(a) If A and B are mutually exclusive events, compute P(A or B).

(b) If P(A and B) = 0.3, compute P(A or B).

2. Given P(A) = 0.8 and P(B) = 0.4, do the following. (For each answer, enter a number.)

(a) If A and B are independent events, compute P(A and B).

(b) If P(A | B) = 0.1, compute P(A and B).

3.  The following question involves a standard deck of 52 playing cards. In such a deck of cards there are four suits of 13 cards each. The four suits are: hearts, diamonds, clubs, and spades. The 26 cards included in hearts and diamonds are red. The 26 cards included in clubs and spades are black. The 13 cards in each suit are: 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King, and Ace. This means there are four Aces, four Kings, four Queens, four 10s, etc., down to four 2s in each deck.

You draw two cards from a standard deck of 52 cards without replacing the first one before drawing the second.

(a) Are the outcomes on the two cards independent? Why?

No. The probability of drawing a specific second card depends on the identity of the first card.

Yes. The probability of drawing a specific second card is the same regardless of the identity of the first drawn card.   

No. The events cannot occur together.

Yes. The events can occur together.

(b)Find P(ace on 1st card and nine on 2nd). (Enter your answer as a fraction.)

(c) Find P(nine on 1st card and ace on 2nd). (Enter your answer as a fraction.)

(d) Find the probability of drawing an ace and a nine in either order. (Enter your answer as a fraction.)

4.  

You draw two cards from a standard deck of 52 cards, but before you draw the second card, you put the first one back and reshuffle the deck.

(a)Are the outcomes on the two cards independent? Why?

Yes. The probability of drawing a specific second card is the same regardless of the identity of the first drawn card.

Yes. The events can occur together.    

No. The events cannot occur together.

No. The probability of drawing a specific second card depends on the identity of the first card.

(b) Find P(ace on 1st card and king on 2nd). (Enter your answer as a fraction.)

(c)  Find P (king on 1st card and ace on 2nd). (Enter your answer as a fraction.)

(d)Find the probability of drawing an ace and a king in either order. (Enter your answer as a fraction.)