1. For a new type of tire, a NASCAR team found the av- erage distance a set of tires would run during a race is 168 miles, with a standard deviation of 14 miles. Assume that tire mileage is independent and follows a Normal model.
a) If the team plans to change tires twice during a 500-mile race, what is the expected value and standard deviation of miles remaining after two changes?
b) What is the probability they won't have to change tires a third time before the end of a 500-mile race?
2. In the 4 * 100 medley relay event, four swimmers swim 100 yards, each using a different stroke. A college team preparing for the conference championship looks at the times their swimmers have posted and creates a model based on the following assumptions:
- The swimmers' performances are independent.
- Each swimmer's times follow a Normal model.
- The means and standard deviations of the times (in sec- onds) are as shown here.
|
Swimmer
|
Mean
|
SD
|
|
1 (backstroke)
|
50.72
|
0.24
|
|
2 (breaststroke)
|
55.51
|
0.22
|
|
3 (butterfly)
|
49.43
|
0.25
|
|
4 (freestyle)
|
44.91
|
0.21
|
a) What are the mean and standard deviation for the relay team's total time in this event?
b) The team's best time so far this season was 3:19.48. (That's 199.48 seconds.) What is the probability that they will beat that time in the next event?