MATHEMATICS 150
2001/2002
PROBLEM SET 4

Solutions are due on Monday, December 3. Solutions may be submitted in class or may be delivered by 4:00 pm to the instructor's office (CC F30). Print your name on the upper right-hand corner of the front page.

1. A prediction model is to be developed to predict daily coffee consumption in an office complex. The prediction model is to be based on the average number of people in the complex (a methodology has been developed to determine the average number), the number of hours of operation and the price of the coffee. Sample values from a number of complexes are listed below. The data are also available in a file coffee.dat in the usual way. (The data are at    http://euclid.trentu.ca/math/courses/stat/files/coffee.dat. )

1. Use multiple regression to develop a linear prediction equation to predict coffee consumption from number of people, hours of operation and price. Predict consumption for a complex with 100 people that operates 10 hours per day with the price of coffee as \$1.10.
2. It was decided that a multiplicative model should be used instead of a linear model. This is accomplished by converting all data to new values which are logarithms of original values. Use multiple regression to develop a linear prediction equation to predict the logarithm of the coffee consumption from the logarithms of the number of people, hours of operation and price. How much improvement in fit is there? Predict the logarithm of the consumption and, hence, the consumption for a complex with 100 people that operates 10 hours per day with the price of coffee as \$1.10.

 number ofpeople hours ofoperation price number of cupsconsumed 316 361 270 304 268 289 127 190 268 183 279 304 150 129 153 212 289 186 87 241 187 283 247 170 356 169 237 242 154 249 267 255 117 367 9 9 12 12 10 10 9 9 12 9 9 11 11 9 9 9 12 10 9 9 10 12 12 9 10 9 11 10 9 9 9 10 9 9 1.00 1.25 1.25 1.00 1.00 1.00 1.50 1.35 1.25 1.15 1.25 1.50 1.25 1.05 1.25 1.00 1.25 1.15 1.00 1.25 1.00 1.35 1.05 1.25 1.00 1.15 1.25 1.25 1.25 1.50 1.25 1.50 1.25 1.25 1094 1088 946 1285 956 1251 280 545 914 604 756 904 470 404 473 689 973 738 312 606 651 933 1016 377 1438 559 842 806 470 611 720 821 306 984
2. Suppose that a record was kept of the number of reported cases of a disease over several years and the annual rates per 100,000 population were as follows:

 year 1981 1982 1983 1984 1985 1986 1987 1988 rate 4.2 6.5 6.5 6.9 5.1 6.7 8.2 8.8 year 1989 1990 1991 1992 1993 1994 1995 rate 6.9 8.7 8.8 12.0 15.4 18.4 19.6

1. What was the percentage increase/decrease i) from 1983 to 1984? ii) from 1988 to 1989? iii) from the start to the end of the record?
2. Determine the actual annual average rate of increase from 1981 to 1995.
3. Using a transformed regression, determine the average rate of increase of the underlying trend and determine what would have been the predicted rates for 1996 and for 1997. (Note: it may be preferable to work with a time variable such as t= year - 1980.)
1. A software development firm has examined its recent experience to determine the number of hours of employee time that a 'typical' project requires for programming, for testing, for documentation preparation and for marketing strategy planning. Different professionals are involved in these stages of the project and are paid at different hourly rates. The numbers of hours required are tabulated below with the hourly rates (dollars) paid in the initial study year (year 0) and the first and second follow-up years (years 1 and 2.)

 software development hours and hourly rates
 rate stage hours year 0 year 1 year 2 programming 1350 30.00 31.00 33.00 testing 270 18.00 19.00 21.00 documentation 180 24.00 24.00 25.00 marketing 64 20.00 23.00 27.00

The CPI for years 0, 1 and 2 for the company's area of operation were 138.1, 141.2 and 143.6
1. Determine a software development cost index for this firm for years 1 and 2 using year 0 as the base year.
2. Using year 0 as the base year, what was the real cost of a 'typical' project for each of years 0, 1 and 2?
2. The firm also works on contracts for other organizations. The hours of staff time per quarter assigned to contract work has been estimated to follow a trend line y = 960 + 32t where yis the quarterly number of hours of staff time and tis time in quartersfrom an initial time point five years earlier. The starting time is set at t= 0 and subsequent quarters produce increments of 1 so that quarter one of year 1 produces t = 1, quarter two of year 1 produces t= 2, quarter one of year 2 produces t= 5 etc . It is assumed that quarters 1, 2, 3, and 4 of any year have seasonal factors of 1.38, 0.96, 0.55, and 1.11, respectively.

1. Predict demand for each quarter of year 6. (Assume no cyclical effect.)
2. If the centred moving average for quarter one of year 5 was 1512 and the actual number of hours was 2115, what was the individual empirical seasonal factor for that quarter?
3. If the actual number of hours for quarter 3 of year 3 was 707, what was the seasonally adjusted value?

3. A major-equipment service facility has experienced seasonal variation according to the quarter of the year. A five-year record of the numbers of service calls is shown in the following table
 Year Quarter 1 2 3 4 5 I 272 288 336 392 384 II 144 152 184 200 213 III 88 88 120 128 122 IV 244 284 268 324 341

1. Sketch a graph of this time series.
2. Convert these values to seasonally adjusted values. In the determination of the seasonal factors, use centredmoving averages and, unless you are using Minitab, use ordinary means (not modified means or medians) to average the individual empirical seasonal factors.
3. Plot the seasonally adjusted values on the graph in a).
4. Determine the forecasts for the four quarters of year 6.

### PROTOCOL FOR MATH 150 PROBLEM SET SOLUTION SUBMISSION

TRENT MATH
 E.A. Maxwell   2001-11-15