Instructions: Answer each question, and when required explain your answer. Your explanation must be clear and complete. You may refer to your book, your notes and your homework papers.

[1] Which is the best deal over a 3-year period: investing at 7.2% compounded annually, investing at 7% compounded monthly or investing at 6.8% compounded daily?

Solution using EAR approach: Compute the interest on a dollar over a year (the 3-year period is irrelevant). Or even simpler just find out how much \$1 grows to over the course of a year (if you subtract that \$1 out at the end you have the EAR). For an annual rate of 7.2% compounded annually, \$1 grows to (1+(.072/1))^1=1.072. For an annual rate of 7% compounded monthly, \$1 grows to (1+(.07/12))^12=1.0722. For an annual rate of 6.8% compounded daily, \$1 grows to (1+(.068/365))^365=1.07036. Thus 7% compounded monthly is best.

Solution using doubling time: the deal with the shortest doubling time is best. For an annual rate of 7.2% compounded annually, the doubling time is t = log(2)/(1*log(1.072)) = 9.970 years. For an annual rate of 7% compounded monthly, the doubling time is t = log(2)/(12*log(1+(.07/12))) = 9.931 years. For an annual rate of 6.8% compounded daily, the formula for doubling time is t = log(2)/(365*log(1+(.068/365))), which gives t = 10.194 years. Thus 7% compounded annually is best since it gives the shortest doubling time.

[2] Problem 21ab on p. 814: How much money would have to be invested in an account at 4.25% annual interest to achieve a balance of \$50,000 in 20 years in each of the following cases?

(a) The account pays simple interest? Solve for P in F = P(1+rt) where F = \$50,000, r = .0425 and t = 20. This gives \$50,000 = P(1+.0425*20) or 50,000 = P(1.85) so P = 50,000/1.85 = \$27,027.03.

(b) The account compounds interest semi-annually? Solve for P in F = P(1+(r/m))mt where F = \$50,000, r = .0425, m = 2 and t = 20. This gives \$50,000 = P(1+.0425/2)40 or 50,000 = P(1.02125)40 so P = 50,000/(1.02125)40 = 50,000/2.3189 = \$21,561.91.

[3] Problem 24 p. 831.

(a) Use the monthly payment formula to determine the monthly payment for a 60-month amortized loan of \$25,495 at 4.5% interest. The formula is that the payment is PMT = P(r/12)(1 + (r/12))12t/((1 + (r/12))12t - 1), where P = \$25,495, r = 0.045 and t = 5 years, which gives PMT = \$475.30.

(b) Use an amortization table to find the monthly payment for the loan from part (a), and compare the result with the monthly payment found in part (a). The entry in the table (on p. 824) for r = 4.5% and 5 years is 18.643019 is the payment for \$1000. Multiply by 25.495 to get our payment of \$475.30, which is what we got using the formula.

[4] Explain how to select a simple random sample of 7 elements from the whole numbers running from 1 to 100, using the table on page 570. What sample do you get? Explain in enough detail that I can verify that your sample is the one you should have gotten.

Answer: Randomly pick a starting entry in the table, say the entry in row 5 column 3. Then read down and pick the last two digits of each entry, skipping an entry if it gives a number already chosen. (If the two digits are 00 then that counts as 100.) Here is the simple random sample I get: 26, 6, 59, 32, 25, 10, 20.

[5] Explain how to select a 40% independent sample from the whole numbers running from 1 to 10, using the table on page 570. What sample do you get? Explain in enough detail that I can verify that your sample is the one you should have gotten.

Answer: Randomly pick a starting entry in the table, say the entry in row 2 column 4 (which is 64569). Then read down that column, counting from 1 to 10 as you go. Every time the last two digits of the entry gives a number between 1 and 40 inclusive, the number you counted is selected. The results are given in the following table, where the first column gives the count from 1 to 10, the second column gives the corresponding table entry, the third column gives its last two digits and the fourth column indicates whether we select the number in the first column or not:
1     64569   69   do not select 1
2     17707   07   do select 2
3     60638   38   do select 3
4     93608   08   do select 4
5     78545   45   do not select 5
6     39445   45   do not select 6
7     50784   84   do not select 7
8     33358   58   do not select 8
9     36246   46   do not select 9
10     17068   68   do not select 10
Our 40% independent sample is thus {2, 3, 4}.

[6] Do Problem 37 on page 592.

Answer: The book gives a solution on page 920.