1. Dr. Rasp just finished re-reading Charles Dickens' A Christmas Carol, a tragic novel in which a prosperous, hard-working businessman named Scrooge degenerates into a sentimental softie. This has prompted Dr. Rasp to think about how the cost of Christmas presents has changed over time. Some relevant data, for this year and for ten years ago:
|
1993 quantity |
1993 cost |
2003 quantity |
2003 cost |
| Lumps of coal |
6 |
5 cents |
3 |
8 cents |
| Switches |
6 |
2 cents |
4 |
5 cents |
a) Use these data to compute a Laspeyres "Cost of Christmas Index."
b) Back in 1993, Dr. Rasp devoted 42 cents (a large portion of his life savings) to Christmas. Now, due to the high rate of inflation, he feels that he must cut back on Christmas purchases. How much would he need to spend today, just to keep that original 42-cent investment apace with inflation (as measured by your index)?
2. Recent Rollins College computer science graduate Mortimer Byttemapper is employed by Spam-R-Us, a direct marketing firm that uses email solicitation to sell a variety of products, including investments in Nigerian financial markets, mortgage refinancing and re-refinancing, and medical procedures for ... uh, "enhancement" of various body parts. The data table below gives, for the past four months, the number of customers the company has and the number of email solicitations (in millions) that the company has sent.
|
Month:
|
Aug.
|
Sep.
|
Oct.
|
Nov.
|
|
# of customers
|
9
|
6
|
8
|
9
|
|
Million emails sent
|
13
|
9
|
11
|
15
|
Note that, for these data, the regression equation is Y = .5X + 2.
a) Which is the "X" variable and which is the "Y" variable in this situation?
b) Find the error variance (se2) for these data.
c) Compute a 95% confidence interval for the slope of the line. Interpret this result, in the context of the problem.
3. The expected sales of aproduct in a city are assumed to be affected by the per capitadiscretionary income and the population of the city. Per capitadiscretionary income will be referred to as PCDI in all the questions. InQuestions a-f examine only the effect of per capita discretionary incomeon the mean sales. Thus the following model is hypothesized:
E(Y) = B0 +B1X1 where
Y = Sales (in thousands of dollars)
X1 = Per Capita Discretionary Income (in dollars)
A sample of 15 cities, along with their sales, per capita discretionary income, and the population of the city (in thousands) is given in the table below. The 15 values and a printout follow:
OBS INCOME SALES
1 2450 162
2 3254 120
3 3802 223
4 2838 131
5 2347 67
6 3782 169
7 3008 81
8 2450 192
9 2137 116
10 2560 55
11 4020 252
12 4427 232
13 2660 144
14 2088 103
15 2605 212
16 2500 .
17 3500 .
a) Give a 95% confidence interval for the mean sales of all cities with PCDI= 2500
b)Test the null hypothesis that the slope equals to zero versusthe alternative hypothesis that the slope does not equal to zero.
c) Does the PCDI help predict the sales of the product?
d) What is the interpretation of the coefficient of determination?
e) What table value would you use in the calculation of a 90% confidenceinterval for a value of Y given a value of X?
f) How many estimated standard errors is the point estimate of the slopeaway from zero? Slope is the change in the mean sales for each dollar increase in PCDI.
4. The "line" in football betting is a procedure for turning all games, even the most lopsided ones, into equally-likely propositions. The "line" is an amount that is added or subtracted from one team's point total in the game to determine the winner of the bet. Thus, for example, if Stetson were playing FSU in football, the line might be "FSU minus 200." To determine who wins the bet in this case, you take FSU's point total and subtract 200 points. If FSU still outscores Stetson, FSU wins the bet. If Stetson outscores FSU's adjusted total, Stetson wins the bet. Hence, for example, a score of 220-0 means people who bet on FSU win, while a score of 186-2 means people who bet on Stetson win. In theory, each team should now have a 50% chance of winning the bet.
In most casinos you may bet not only on individual games but also on "parley" bets - series of several games. These offer higher payoffs - but, of course, reduced chance of winning.
a) Casinos in Boravia use an unusual system of parley betting, in which you are expected to both win and lose a specified number of bets. For example, in a "Twelve Game Mixed Parley" in Boravia, you bet on twelve games against the ‘line.' To win the bet, you must call nine of the twelve games correctly and three of them incorrectly. What is the probability that you win a "Twelve Game Mixed Parley" in a Boravian casino?
b) Twelve Game Mixed Parley bets pay off at 16-to-1. That is, for every $1 you bet, if you win you receive $16 net (your $1 back, plus $16 more). Of course, if you lose, your $1 is lost. Find the expected value and variance of the net return on a $100 bet on a Twelve Game Mixed Parley.
c) Boravia's casino typically sees 10,000 people play $100 Twelve Game Mixed Parley bets in a given weekend. What is the probability that the casino makes money overall, on these bets?
5. The sales of a company (in million dollars) for each year are shown in the table below.
|
x (year)
|
2005
|
2006
|
2007
|
2008
|
2009
|
|
y (sales)
|
12
|
19
|
29
|
37
|
45
|
a) Find the least square regression line y = ax + b.
b) Use the least squares regression line as a model to estimate the sales of the company in 2012.
6. IsmereldaTempusfugit knows that 42 is The Answer to Life, the Universe, and Everything. She wonders whether it is also the Secret to winning the lottery. In other words, she wonders whether 42 is more likely to be a winning lottery number than would be expected from chance alone.
She knows that in the Florida lottery, six out of 53 numbers are selected as "winners" each week. She obtains historical data from the Florida Lottery Commission. From those data, she notes that out of the past 500 drawings in the Florida Lottery, 42 has been one of the winning numbers fifty times.
a) State Ismerelda's null and alternative hypotheses, in words and in symbols.
b) Compute an appropriate test statistic. Give the p-value.
c) State an appropriate conclusion, both in statistical terms (reject/don't reject) and in the context of the problem.
7. Horatio Wajberlinski, while a student at Mad Hatter Vocational College, was enrolled in that school's Ronald McGeorge McDonald Investment Program and School of French Fry Technology. He has used the information he gained there to invest in a wide range of fast food franchises. Total value of his portfolio over the years is given below.
Year Portfolio value
1983 $10,000
1988 $20,000
1993 $60,000
1998 $300,000
2003 $3,000,000
a) What do the data indicate about the growth of Horatio's portfolio? Sketch an appropriate graph. Interpret your graph.
b) What has been Horatio's average annual rate of growth, over the 1983-2003 time period?
c) Suppose that in the future Horatio maintains that same average growth rate (from Part B). When will his portfolio be worth $1,000,000,000 (making Horatio a billionaire)?