AP Statistics / Mr. Hansen 12/16/2004

Name: _________________________

Test on Chapters 5, 6, and 7
(and classroom discussions)

1.	Define what is meant by a random variable (r.v.): ______________________________ _________________________________________________________________
2.	There are two basic types of random variables, namely __________________ and __________________ . Without re-using any of the examples given in class, give an example of each type, in which you assign an appropriate letter to the r.v. It would be best if you chose something from your own personal experience. First example (a _______ r.v.): Let ____ = _____________________________                                                              _________________________________ Second example (a _______ r.v.): Let ____ = _____________________________                                                               _________________________________
3.	Another term that means exactly the same thing as “expected value of a random variable” is _______________________________________________ .
4.	Let X = the numeric value of a single card randomly drawn from a 36-card deck of playing cards in which all the aces and face cards have been removed.
(a)	Compute the expected value of X. Show correct notation, formula, plug-ins, and answer.
(b)	Convince me that the variance of X equals 20/3. Use correct notation.
(c)	Let Y = the outcome of throwing a single fair die, and let S = X + Y. What is the lowest possible value for S? _____ The highest? _____ (No work required.) Now use the result of part (b) to compute the s.d. of S, showing your work. Use correct notation. Hint: We learned in yesterday’s quiz that the variance of Y equals 35/12, and you can use that result without re-deriving it.
(d)	Let Y be defined as in part (c) and let T = X – Y. What is the lowest possible value for T? _____ The highest? _____ (No work required.) Now find the s.d. of T. Either show your work or provide a reason for your answer.

5.

Explain why it may not be a good idea to conduct a screening test for a rare disease, even if early detection saves lives.

6.

The disease of gloppiosis has incidence of 1% in the asymptomatic population. The standard screening test for gloppiosis is 97% accurate in producing a positive reading if someone has the disease, and the standard test is 98% accurate in producing a negative reading if someone does not have the disease. In other words, P(positive reading | disease) = .97, which is what we call the ________________ of the test, and P(negative reading | ~disease) = .98, which is what we call the ________________ of the test. Compute the PPV (positive predictive value) of the test, namely P(disease | positive reading).

7.	We will define the term “face card” to mean jack, queen, or king only. Are the events “a king is drawn” and “a face card is drawn” independent events for the situation involving a single card drawn from a standard deck? Justify your answer.
8.	State the LOLN (proportion/probability version) using words only.
9.	State the LOLN (proportion/probability version) using mathematical symbols only.
10.	State the LOLN (mean version, sometimes called the law of averages) using words only.
11.	State the LOLN (mean version, sometimes called the law of averages) using mathematical symbols only.
12.	Give a concise definition (approximately 4 words) of probability from the “frequentist” point of view. This is the definition that your textbook uses and the one that I emphasized in class as being the conventional version.
13.	The Bayesian point of view holds that probability is really a state of mind or a degree of belief. The followers of Bayes believe that probability should be continually readjusted in order to _______________________________________________________ .
14.	Below are several arguments allegedly based upon the law of large numbers (or the law of averages, which is essentially the same thing). For each, identify whether the reasoning is essentially correct or deeply flawed, and if the latter, write a sentence of explanation.
(a)	In Las Vegas, the odds at roulette are such that 94.7% of the amount wagered, on average, is returned to the players in the form of prizes. (The rest, of course, is the casino’s profit margin.) Over the long run, a player will lose, before taxes, 5.3% of the money that he or she wagers.
(b)	A fair coin is flipped 100,000,000,000 times. By the law of large numbers, the probability is greater than .5 that an exact 50/50 split (i.e., exactly 50 billion heads, 50 billion tails) will occur.
(c)	A fair die (with faces numbered 1 through 6) is rolled 100 times. The number 3 appears only 10 times, which is significantly less than the expected number of 3’s, namely ___________ . By the law of large numbers, the probability of a 3 on the 101st roll equals 0.1.
(d)	A professional baseball player is in a slump; in his last 17 at-bats, he has had only 1 hit, even though his lifetime batting average (consistently for the last 10 seasons) has been approximately .200. By the law of averages, he is quite likely to go 3-for-3 in his next 3 at-bats, so that the average of the most recent 20 at-bats will be .200. (Here we will define the phrase “quite likely” to mean that the probability is greater than .5.)
(e)	A fair die (with faces numbered 1 through 6) is rolled 100 times. The number 3 appears only 10 times, which is significantly less than the expected number of 3’s. Therefore, in the next 6000 rolls, the law of large numbers implies that I will have slightly more than 1000 rolls of 3. This is necessary so that the overall probability of 3 will approach closer to 1/6.
15.	Repeat #14(d), except this time evaluate the following claim: By the law of averages, the player is quite likely to make at least one hit in his next 3 at-bats. (Assume that at-bats are independent trials, in much the same way that coin flips are independent trials. Although at-bats are not independent, strictly speaking, data analysis has supported the claim that this assumption is realistic.) Show your work.
16.	What is “blocking” in experimental design, and why do we do it? Be very specific about the benefit that is produced. Use reverse side if necessary.