AP Statistics / Mr. Hansen
2/6/2002

Name: _________________________

Part I. Fill in the blanks (3/4 pt. per blank)

1.

The ______________ distribution of a ______________ is the distribution of all possible values that that ______________ could have when ______________ of a fixed size are drawn from a fixed ______________ . The two ______________ for which we are most interested in constructing these types of distributions are (1)______[symbol] (the ______________ ______________ ) and (2)______[symbol] (the ______________ ______________ ). However, IQR, s, and median are also examples of ______________ for which we could compute ______________ distributions if we were so inclined.

2.

The ______________ ______________ ______________ states that the ______________ ______________ of xbar approaches N(_____ , _____) as ______________ grows large. This is due, in part, to the fact that ______________ is a(n) ______________ estimator of ______________ . In fact, regardless of the sample size, E(xbar) = ______________ .

3.

The expected value of X + Y, where X and Y are ______________ ______________ , always equals ______________ + ______________ . (The alternate notation for E(X) is ______________ .) However, we can compute the variance of a sum or difference a priori only if X and Y are ______________ , in which case V(X + Y) = ______________ and V(X – Y) = ______________ .

Part II. Free-response problems (SHOW STEPS, using standard notation and good "AP style," and give answers to at least 3 decimal places or more if the situation seems to warrant)

4.

A large collection of weather-related temperature data (recorded in ° F) can be considered to be the distribution of a random variable with mean 54 and s.d. 11.8. As you may recall, Fahrenheit readings can be converted to Celsius by the formula C = 5/9 (F – 32). Compute the mean, s.d., and variance of the same r.v. when readings are expressed in ° C.

5.

In the large metropolis of Buffalonia, 1.8% of the residents are HIV positive. In an SRS of 216 residents, compute

(a)

the expected number of HIV-positive subjects

(b)

the probability that at least 3, but no more than 5, are HIV positive

(c)

the probability that the first HIV-positive subject is found in the middle third of the SRS, i.e., on or after person #73 but before person #145

(d)

In part (b), what distribution did you use? ______________ Which conditions are satisfied? _____________________________________________________________________ Are any conditions violated? ______________ Explain briefly below:

(e)

In part (c), what distribution did you use? ______________ Which conditions are satisfied? _____________________________________________________________________ Are any conditions violated? ______________ Explain briefly below:

6.

A fair die is rolled 600 times.

(a)

What is the expected number of sixes? Remember to show some notation and work, as always.

(b)

What is the probability that exactly that number of sixes will actually occur?

(c)

A Las Vegas casino decides to accept bets (at 2:1 odds) from people who think that the number of sixes in 600 rolls will be either higher or lower than the answer you gave in part (a). In other words, winners hand their bet to the casino (the money is no longer in their possession) and then either win nothing or win $2 per dollar wagered. What is the casino’s mean profit per dollar wagered? Assume that equal numbers of bettors choose "higher" and "lower."

7.

An engine plant manufactures cylinders whose diameters follow N(4.3, 0.01) and pistons whose diameters follow N(4.2, 0.02), where all measurements are in cm. A piston is said to "fit" a cylinder iff the cylinder’s diameter is at least 0.08 cm larger than that of the piston, but no more than 0.13 cm larger. Compute the scrap percentage, i.e., the probability that a randomly selected piston fails to fit a randomly selected cylinder. State any assumptions that you make.