Questions 3-6 are each weighted twice as much as each of the other questions.

Transcription

1 Mathematics 107 Professor Alan H. Stein December 1, 005 SOLUTIONS Final Examination Questions 3-6 are each weighted twice as much as each of the other questions. 1. A savings account is opened with a deposit of $500 in an account paying interest at an annual rate of 3.1 percent, compounded continuously. (a) What is the effective annual yield? In one year, $100 would grow to 100e , so the effective annual yield is approximately %. (b) What will the balance be in 15 years if the principle is left undisturbed? The balance will be 500e Since banks don t generally credit fractions of cents, the balance will be $ A $8, 000 loan is taken out, to be repaid with monthly payments over a period of 5 years. If the annual interest rate is 6.3 percent, what will the monthly payment be? You may P r use the formula m = 1 (1 + r). n In that formula, m represents the monthly payment, P the initial loan balance (in this case 8, 000), r the monthly interest rate (in this case ), and n the number 1 of payments (in this case 5 1 = 60) /1 We thus have m = The monthly payment will 1 ( /1) 60 have to be $155.79, with the final payment being slightly less. Page 1 of 5

2 Page of 5 3. The following table gives the consumer price index for the years , with the base period corresponding to a of 100. t (year) P Actual P (t) Predicted SSE Average Error Error Error Squared (a) Assuming a linear model, determine a linear function of the form P = mt + b which agrees with the actual Consumer Price Index in the years 001 and The slope is m = = 59. We can use the first point in the 15 Point-Slope Formula to get P = 59 (t 001), or P = t 7, (b) Find the predicted Consumer Price Index for the year 010 based on your model. P = (c) In what year does your model predict the will reach 00? We need to solve t = 00. We proceed as follows: t = t = / t The should reach 00 during 006 or 007. (d) Fill in the empty boxes in the table above. t (year) P Actual P (t) Predicted Error Error Squared SSE.189 Average Error

3 Page 3 of 5 4. The following is the same table given in the previous question. t (year) P Actual P (t) Predicted SSE Average Error Error Error Squared (a) Use your calculator to determine the linear regression model for this data. The calculator gives a linear regression P = 3.95t (b) Fill in the empty boxes in the table above. t (year) P Actual P (t) Predicted Error Error Squared SSE Average Error (c) Which model better fits this data, the model you created in the first question or the model obtained by your calculator? Justify your conclusion. The model determined by the calculator is a better one. It has a smaller SSE and a smaller average error.

4 Page 4 of 5 5. The following table gives DVD sales in the United States in billions of dollars for the years Year Actual Sales (a) Use your calculator to determine the best exponential model fitting this data. Note: Your calculator will overflow if you let t be the calendar year, so you will need to define t differently. Be explicit regarding the meaning of t. We will let t = 0 in the year 1999, so the actual calendar year will be t The calculator gives the regression P = t, where P represents sales. (b) What does your model predict sales will be this year (005)? In 005, t = 6, so the prediction is P = , so it predicts sales will be approximately $51, 688, 37, (c) When does your model predict sales will reach $30 billion? We need to solve t = 30. We may proceed as follows: ln( t ) = ln 30 ln ln( t ) = ln 30 t ln = ln 30 ln t = ln 30 ln ln t This corresponds to the year 004, since t = 0 in 1999.

5 Page 5 of 5 6. The following table gives the same data as in the previous question. Year Actual Sales (a) Use your calculator to determine the best logistic model fitting this data. Note: Your calculator will overflow if you let t be the calendar year, so you will need to define t differently. Be explicit regarding the meaning of t. We will let t = 0 in the year 1999, so the actual calendar year will be t The calculator gives the regression P = e, t where P represents sales. (b) What does your model predict sales will be this year (005)? P = , so the model e predicts sales of $15, 97, 084, (c) When does your model predict sales will reach $30 billion? According to the model, sales will never reach $30 billion. (d) What does your model indicate as the ultimate level of annual sales as the market gets saturated and sales level out? The ultimate, maximal level is the numerator, , representing sales of $ billion, i.e. $18, 309, 145, Iterate the Babylonian Square Root Algorithm four times to get an estimate for 73, starting with an estimate of 10. Recall that in this algorithm, given one estimate x for x + N N, the next estimate is x / / / /