Random Variables and their PDFs

 Solutions to Practice Problems.

1) Swimming Medley (from DeVeaux & Velleman p 315 #37)
In the 4x100 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 seconds) are shown;
Swimmer          Mean     Standard deviation
Backstroke          50.72          0.24
Breaststroke          55.51          0.22
Butterfly               49.43          0.25
Freestyle               44.91          0.21

a) What are the mean and standard deviation for the relay team’s total time in this event?

200.57 and 0.46

b) The team’s best time so far this season was 3:19.48 or 199.48 seconds. Do you think that the team is likely to swim faster than this at the conference championship? Explain.

No, the results would be about 2.36 SDs below the mean.

2) A commuter airline flies planes between San Luis Obispo and San Francisco. For small planes, the baggage weight is a concern, particularly on foggy mornings, because weight has an effect on how quickly the plane can ascend. Suppose we know that the weight of baggage, checked by a random passenger has a mean and standard deviation of 42 and 16 pounds, respectively. Consider a flight on which ten passengers, all traveling alone, are flying.

a) Determine the mean and standard deviation for the total weight of the checked baggage.
(From POD, p. 332, ex 7.16)
420 pounds and 50.6 pounds.

These last few problems make use of the binomial distribution. However, they also illustrate some intuitive lessons about the law of large numbers. The law of large numbers says that sample average is more likely to be close to the expected value (a.k.a. the mean) for a large sample size than for a small sample size.

Sometimes Multiple Choice is a good teaching tool! Particularly if you add the "Explain."

3) A die will be rolled some number of times, and you win $1 if it shows an ace more than 20% of the time. Which is better? Circle one.

a) 60 rolls is correct. Share your explanation on the discussion board.
b) 600 rolls

4) A die will be rolled some number of times and you win $1 if the percentage of aces is exactly 16 and 2/3 %. Which is best? Circle one:

a) roll the die 60 times
b) Roll the die 600 times is correct.

The SD for the percent of aces is smaller for 600 tosses. Therefore, the percent of aces is likely to be closer to the mean (which 16 2/3 %).