AP Statistics / Mr. Hansen
1/31/2001

Name: _________________________

Test #5 (Exp. Design, §7.2, §8.1, §8.2)

 

Useful formulas that will not be included on your formula sheet for the actual AP exam:
z = (x - m)/s
For a geometric random variable X, mX = 1/p, sX = (Öq)/p, and P(X = k) = qk - 1p.

Part I.

Short Answer / Multiple Choice
You need not show your work in this section, except for the very last part of #3.

1.

In problems 2 and 3, you will be asked to categorize distributions as normal, binomial, geometric, or none of the above. For those that are normal or binomial, indicate the parameters of the distribution using (fill in the blanks here) N( __ , __ ) or B( __ , __ ) notation. For a geometric distribution, since we have no corresponding standard notation, simply state the value of the parameter.

2.

Freida Friendly is playing Whack-A-Mole at her local arcade. The moles appear randomly and independently, one at a time, from any of 5 possible holes. The object of the game is to whack the mole immediately after it appears in order to score points. Each mole appears for the same length of time before it disappears back into its hole. Moles appear at the rate of 2 per second during the game, which lasts 45 seconds. Freida wishes to know how many times she can expect a mole to appear from hole #4 during the game.

(a)

Let X = ___________________________________________________

(b)

This distribution is ___________ with parameter(s) ____________________ . (Use your proper parameter shorthand as described in the instructions for #1.)

(c)

The distribution is (circle one) exactly / approximately of the form described.

(d)

Give notation for, and compute the probability of, having exactly 20 moles appear from hole #4 during the game. WARNING: An answer without notation will not earn full credit. However, work is not required. (See instructions for Part I above.)

(e)

Give notation for, and compute the probability of, having more than 20 moles appear from hole #4 during the game. Same warning applies from above.

(f)

Give notation for and compute the expected number of moles that appear from hole #4 during the game. Same warning applies from above.
3.

Larry the Loser is trying to sell encyclopedias door-to-door. He knows from bitter experience that only 1% of the houses he visits will result in a sale. Larry chooses an SRS of houses from a large city and decides to visit these houses, instead of merely picking a street and hitting all the houses on that street. The question of interest is how many houses Larry must visit in order to achieve a sale.

(a)

Let X = ___________________________________________________

(b)

This distribution is ___________ with parameter(s) ____________________ (Use your proper parameter shorthand as described in the instructions for #1.)

(c)

The distribution is (circle one) exactly / approximately of the form described.

(d)

Sketch the probability histogram for X below. Mark numbers on your axes.

 

 

 

 

 

 

(e)

X has a (circle one) skew left / skew right / symmetric shape.

(f)

Give notation for, and compute the probability of, the event that Larry achieves a sale with the third house he visits. WARNING: An answer without notation will not earn full credit. However, as above, work is not required.

(g)

Give notation for, and compute the probability of, the event that Larry must visit more than 120 houses before making a sale. Same warning applies from above.

(h)

Give notation for and compute the expected number of houses that Larry must visit in order to achieve his first sale. Same warning applies from above.

(i)

Would the assumptions required for the distribution you indicated at the beginning of this question be satisfied if Larry were to visit all the houses on a street before moving on? ____ Why or why not? __________________________

(j)

With the problem as stated (i.e., with Larry using an SRS of houses), are the requirements for the distribution you selected met in full? ____ Why or why not? _____________________________________ What word(s) in the setup of the problem allow you to safely use the distribution you indicated above? _____________________

(k)

Draw a tree diagram to illustrate how a person could compute P(X = 4) manually, and compute the answer. P(X = 4) = ___________________________ = __________. (Show what numbers you are combining to get the answer.)

 

 

 

 

 

 

 

 

Part II.

Free Response

4.(a)

In experimental design, the technical (statistical) reason for blocking is in order to reduce the ______________ among the ________________________________ .

(b)

A difficult test is given to two groups of students, one of which has received specialized test preparation services. In the area below, sketch the N(50, 7.2) distribution of the control group and the N(53, 9) distribution of the experimental group on the same set of axes. 

 

 

 

 

 

 

 

 

 

 

(c)

After blocking on Form, we notice that the students in the Form IV control group follow the N(50.5, 0.5) distribution, while the Form IV experimental students follow the N(54, 0.4) distribution. Plot these two distributions on the axes above, using dotted lines.

 

Write a few coherent sentences explaining (d) what your diagram above says about the results from this experiment and (e) what your diagram illustrates about the value of blocking in general, i.e., in all experiments where blocking is a good strategy. Since the topics are closely related, you may write a single response paragraph if you prefer. Be sure to answer both questions in the process.

(d)

______________________________________________________________

______________________________________________________________

______________________________________________________________

(e)

______________________________________________________________

______________________________________________________________

______________________________________________________________

 

 

5.

We wish to investigate the hypothesis that math teachers at STA score tests higher, on average, if a student uses pencil instead of pen. It is well known (or, at least, there is a wealth of anecdotal evidence to suggest) that different teachers have different grading profiles, i.e., distributions of scores that they customarily compute.

(a)

Would it be desirable or practical to incorporate blinding of the graders to the type of writing instrument used? _____ Explain your answer. ________________________
________________________________________________________________

(b)

Is it desirable to incorporate blinding in this experiment? ____ Explain your answer, and explain what you mean by blinding in this context.
_______________________________________________________________

(c)

Design an experiment to address the hypothesis stated in the introduction above. Along the way, be sure to state the three principles of good experimental design and indicate explicitly what you are doing to implement these. If appropriate, please incorporate blocking and matched pairs into your design. You may use a diagram, if you wish, to indicate the skeleton outline of the design, but you will need to write some sentences as well.