Homework #8, Math 320, Spring 2001
Name:____________________________
Section:____

## Math 320 Homework #8 --- Due 3/23

Include your name, section number, and homework number on every page that
you hand in. Enter ``Section 1'' for the morning class (10-11AM) and
``Section 2'' for Professor Sawyer's class (12-1PM).

Begin the exposition of your work on this page. If more room is needed,
continue on sheets of paper of exactly the same size (8.5 x 11 inches),
lined or not as you wish, but not torn from a spiral notebook. You should
do your initial work and calculations on a separate sheet of paper before
you write up the results to hand in.

1. (Similar to exercise 7.28 on page 313.) A company is concerned about
a machine that fills cans with ground coffee. The machine is tested each
day by weighing all of the cans filled by the machine during the first
hour of production, which is always n=29 cans. The machine is assumed to
be working properly if the sample standard deviation of the weights of
the cans is not too large. The company will repair the machine only if
there is convincing evidence that the standard deviation is greater than
2.6. Otherwise, it is assumed that the standard deviation equals 2.6 and
that the machine is working properly. Assume that the weights of the
coffee cans are normally distributed with the same mean.

(a) State the hypotheses H_{0} and H_{1}.
(b) Suppose that the production manager measures T=3.08 for the
sample standard deviation of the n=29 coffee cans on one day. Does he or
she accept H_{0} or reject H_{0}? What is the P-value?
(*Hint:* It may be easier to work with sample variances rather than
sample standard deviations.)
(c) Find the critical value lambda_{0} for the test at level
of significance alpha=0.10 based on the sample standard
deviation T. Is the observed value T=3.08 below or above that
value? In this way, if the manager decides to become concerned about the
machine only when the P-value is 0.10 or less, then he or she just has
to check whether the observed sample standard deviation T is greater or
less than lambda_{0} without having to calculate a new P-value
each day.
2. Do exercise 7.44 on page 325. (This asks you to find the level of
significance and the power for a test based on a binomial distribution.
*Warning:*``Greater than'' means ``strictly greater than'' and not
``greater than or equal to''.)

3. Do exercise 7.48 on page 335. What test statistic are you using? What
is its distribution given H_{0}? Is this a one-sided or a
two-sided test? How does that affect the P-value?

4. Do exercise 7.72 on page 346.

5. Do exercise 8.6 on page 360. Is this a one-sided or a two-sided
P-value?

6. Do exercise 8.14 on page 375.