Statistics 343

Homework 1

Due Thursday, 12 October, 1995

1. At the 1987 World Championship in Rome, Ben Johnson set a new world's record in the 100 meter dash. Carl Lewis finished second, tying the previous record. (Johnson's record has since been revoked.) The following data were taken from a newspaper article. They show the elapsed time from the start of the race in 10-meter increments.

```     Meters johnson lewis
1     10    1.86  1.94
2     20    2.87  2.97
3     30    3.80  3.92
4     40    4.66  4.77
5     50    5.55  5.67
6     60    6.38  6.50
7     70    7.21  7.33
8     80    8.11  8.23
9     90    8.98  9.09
10    100    9.83  9.93
```
The reaction time of each athlete was also measured; this is the amount of time from the time the starter's gun was fired until the athlete left the starting block. These times were 0.129 seconds for Johnson and 0.196 seconds for Lewis.

(a) For each runner, fit the elapsed time as a function of distance, using linear regression.

(b) Comment on the interpretation of the parameters in the model.

(c) For each runner, plot the residuals against the fitted values.

(d) Critically assess the usefulness of this model. What are its strengths? What are its weaknesses? Does the model raise any questions? How might the model be improved?

[Remark. These data were used on the 1992 PhD Preliminary Examination in Statistics. We shall be returning to these data from time to time during the course.]

2. ALR Exercise 1.5.

3. (a) ALR Exercise 1.2, parts 1.2.1-1.2.4.

(b) Which model (the linear model of 1.2.1 or the transformed model of 1.2.3) is better? How would you describe the relationship between pressure and temperature using your selected model? What do the coefficients mean?

(c) ALR Exercise 1.2, part 1.2.6.