AP Statistics / Mr. Hansen
3/7/2002

Name: _________________________

Ground rules: No collaboration if you are doing all or part of this in order to earn extra credit on your test. Your essays must be your own words, your own thoughts. Honor code applies. Please initial here _____ to indicate that you have read these ground rules.

Part I: Essay/short answer (5 points per lettered problem).

1.

(a) State the central limit theorem, and (b) explain why and under what conditions (roughly) we may safely conduct a z hypothesis test for a mean even if the population is not normal.

2.

You are Director of Manufacturing for a dog food company that makes kibbled food (see visual aid) whose pieces are nearly uniform, with a standard deviation of only 150 mg. The desired mean kibble mass is 1200 mg. You have instituted a quality control procedure in which an SRS of 350 pieces from each lot (randomly pulled from a number of bags) will be examined. You announce to the workers on the plant floor that they should try to keep the kibbles as close to 1200 mg as possible, since you will reject any lot that shows (at the a = 0.05 level) evidence of having a mean kibble that differs from 1200 mg.

A wealthy investor is visiting your plant and is asking you a number of somewhat naïve questions about your quality control. Answer each one using the context of the problem (i.e., dog food).

(a)

"I have heard that Type I error is bad. What do you mean by ‘Type I’ error in this situation? In other words, what would you be doing if you committed a Type I error?"

(b)

"When you examine a lot, what is your probability of making a Type I error?"

(c)

"I have heard that Type II error is also bad. What would you be doing if you committed a Type II error?"

(d)

"When you examine a lot, what is your probability of making a Type II error?"

(e)

"Your answer to part (d) really confused me. Even if there isn’t a real number, can’t you give me some idea? Make a diagram if you have to."

(f)

"Why can’t you just avoid these errors? Why is there always a probability of making Type I and Type II errors?"

(g)

"OK, I almost half understand why the errors are unavoidable. But surely we could reduce the errors by reducing the a level, right? Maybe if we just reduced a to 0.01, we’d have a lot fewer errors of both types, right?"

(h)

"Whew. I guess that wasn’t such a great idea. But surely there is something you could do that would reduce the probability of both Type I and Type II errors. Come on, you’re a smart guy—that’s what I’m paying you to do. Show me something we can do, and explain it to me with pictures so that I can understand."

Part II: Problem solving. Show adequate support for all problems. Write the full procedures that we discussed for any problems requiring a hypothesis test. In general, final answers should be correct to at least 3 decimal places, which means that you should carry all intermediate results at the full precision that your calculator gives you (approx. 14 significant digits). You may use ". . ." to indicate precision beyond the precision that you record on your paper.

3.
(18 pts.)

In the dog food problem as originally stated, what would you conclude about a lot whose mean kibble (in the sample of 350) was 1215 mg?

4.
(18 pts.)

A laboratory researcher has a low-quality balance that is quite inaccurate: The s.d. of repeated measurements of the same mass is known to be 9 g. However, she feels that her balance is not only variable but also biased, consistently returning low values. She measures a standard mass known to be precisely 2000 g and achieves the following readings over a period of several days with repeated attempts:

2012 g
1984 g
1988 g
2002 g
2008 g
1994 g
1996 g
1999 g

Is the researcher justified in her hunch?

5.
(4 pts.)

For problem #4, explain why a 95% confidence interval, though useful in some hypothesis tests, does not readily apply to this one.

6.
(10 pts.)

Compute a 95% confidence interval for problem #4 and state your conclusion in the context of the problem. Show your work, of course.