STAtistics / Mr. Hansen
12/12/2005 [modified 12/2/2010]

Name: _________________________

Test #3

Probability, symbolic logic, random variables, LOLN

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

 

Scoring

Each question is worth 5 points, except for #6, where each part is worth 7 points. Your name is worth 5 free points. Therefore, you can score up to 101 points out of 100.

Other

·       Write legibly, using correct notation.

·       Illegible or ambiguous writing (e.g., uncrossed z, or a  that looks like M or 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: Normal Distributions.

 

 

5.

Every normal distribution is (circle one)   discrete   continuous.

 

 

6.(a)

The 70th percentile of a normal distribution has a z-score of approximately _______ .

 

 

(b)

For a standardized test having a mean of 610 and a standard deviation of 110, find the probability that a randomly chosen test taker scores more than 1.5 standard deviations above or below the mean. Assume a normal distribution.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c)

For the distribution in part (b), characterize the most extreme 1% of scores. (In other words, what cutoff scores identify someone who is far enough above or below the mean to be in the tiny tails that add up to 1% of the total distribution?)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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 partly filled in 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: ________________