STAtistics / Mr. Hansen
12/12/2005

Name: _________________________

Test #3

Probability, symbolic logic, random variables, LOLN, Type I/Type II error, PPV, and simulations

Time limit: 50 minutes. (Extended timers: Complete everything here. Then do an additional sheet at Math Lab.)

 

Scoring

Each question is worth 5 points, and your name is worth 5 free (bonus) points. There is a 1-point bonus at the end. Therefore, you can omit any one question and still score up to 101 points out of 100. If you answer all questions, you could score up to 106 out of 100.

Other

·      Write legibly, using correct notation.

·      Illegible or ambiguous writing (e.g., uncrossed z, or a m that looks like u) will be deliberately misinterpreted in a way that does not help you.

·      Mark a single “X” through anything you wish to be ignored during grading.

·      Calculator notation (e.g., normalcdf) will result in a deduction unless “X”ed out.

·      Standard abbreviations such as r.v. and s.d. are permitted.

·      You may use sentence fragments or bulleted lists if your meaning is clear.

 

 

Part I: Translate each of the following formulas into English.

 

 

Example:

P(X = k) =

 



The probability that the binomial r.v. X registers exactly k successes in n trials equals the binomial coefficient “n choose k,” times single-trial probability of success to the k power, times single-trial probability of failure to the (n – k) power.

 

 

1.

 


The _____________________________________________________________

 

 

 

equals ___________________________________________________________ .

 

 

2.

E(X) =

 

 

 

The ___________  ___________ of ___________  ___________  ___________ , also

 

 

 

called the ___________ of X, equals ________________________________

 

 

 

_______________________________________________________________________ .

 

 

3.

var(X) =

 

 

 

The ___________ of r.v. X equals the probability-weighted ________ of ________

 

 

 

___________ from the ___________ of X.

 

 

4.

      The _______________________________________ equals

______________________________________________________________________

 

 

 

______________________________________________________________________ .

 

 

Part II: Symbolic Logic.

 

 

5.

Use algebraic transformations, not a truth table, to show that the complicated expression  can be simplified to . This is exactly like a Precal identity proof, except that instead of using a chain of = signs, you will use a chain of  signs. If you can’t do this, remember that you can leave one question blank and still score up to 101 points.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6.

Now use a truth table to prove that  is equivalent to .

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Part III: Short Answer (no work required, but you must show work if you desire partial credit).
There is no partial credit for most of these. All answers must be correct to 3 decimal places. Use the word “undefined” or the abbreviation DNE (“does not exist”) if an answer is impossible.

 

 

7-13.

Unlike Mr. Hansen, who is a terrible free-throw shooter, Mr. Hansen’s younger brother Carl is a good free-throw shooter who consistently sinks 85% of free throws. This probability is independent of the outcome of other shots. Let X denote the number of free throws Carl sinks in 12 trials, and let Y be the number of shots Carl needs in order to sink the first one.

 

 

7.

X has a __________________ distribution with n = ___________ and p = _______ .

 

 

8.

Y has a __________________ distribution with n = ___________ and p = _______ .

 

 

9.

Compute P(X < 9) and state what this means in plain English.

P(X < 9) = _______ and represents ________________________________

 

 

 

_______________________________________________________________________ .

 

 

10.

Compute the mean and s.d. of Y. Identify each answer with proper notation. The s.d. has been filled in for you to show the desired pattern (symbols on the left, numbers on the right).

 

 

 

mean: _____ = ______________

 

 

 

s.d.: _____ = ______________

 

 

11, 12.

Compute the expected value and s.d. of X. As in #10, identify each answer with proper notation. Symbols go on the left, and numbers go on the right.

 

 

 

expected value: _____ = ______________

 

 

 

s.d.: _____ = ______________

 

 

13.

Compute P(Y > 2) and state what your answer means in plain English.

P(Y > 2) = _______ and represents ________________________________

 

 

 

_______________________________________________________________________ .

 

 

14.

Let Z = height of a randomly selected adult American woman, in inches. If Z is N(65, 2.5), compute P(63 < Z < 68) and state what your answer means in plain English.

P(63 < Z < 68) = _______ and represents ____________________________

 

 

 

_______________________________________________________________________ .

 

 

15.

In #14, is Z a binomial random variable? (Mark a check next to one of the answers below, and fill in the blanks.)

Yes, Z is a binomial random variable with n = ________ and p = ________ .

No, Z is a ____________  ____________  ____________ .

 

 

16.

Airline crashes on commercial U.S. carriers are extremely improbable events. For the sake of this problem, let p = .0000005 be the probability that a scheduled commercial U.S. flight crashes. Assuming that p is invariant and independent across days and regions of the country, compute the probability that an airline that operates 4,000 flights per day will experience at least one crash in a given year. Answer: ________________

 

 

 

17-20.

A disease has an incidence of 0.8% in the general population (in other words, 8 per thousand) and has no obvious symptoms. A screening test for the disease is 97% accurate in assigning a reading of “positive” to a subject who is infected with the disease, and the test is 98% accurate in assigning a reading of “negative” to a subject who is not infected with the disease. In other words,

P(positive reading | infected) = .97
P(negative reading | ~infected) = .98

Also, assume that there are no ambiguous test results. In other words, every test is either a clear “positive” or a clear “negative.”

 

 

17.

State the null hypothesis (H0): ________________________________________

 

 

18.

We will reject H0 whenever a subject’s screening test is “positive” for the presence of infection. Of course, sometimes this happens even though the subject is healthy. Compute P(rejecting H0 | H0 is true): __________________________________________

 

 

 

What vocabulary term do we use for that number? _________________________

 

 

19.

We will fail to reject H0 whenever a subject’s screening test is “negative” for the presence of infection. Of course, sometimes this happens even though the subject is actually infected. Compute P(failing to reject H0 | H0 is false): _________________________

 

 

 

What vocabulary term do we use for that number? ________________________

 

 

Part IV: Free Response.
Show your work as you compute one final answer related to the screening test described above.

 

 

20.

Compute the PPV of the test, i.e., P(infected | positive reading). Answer: ____________

 

 

BONUS
(1 pt.)

What do the letters PPV stand for? __________________________________

 

 

 

Note: For #20, work is required. Show a tree diagram or other suitable work below.