Name:____________________________ Section:____

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.

**Answers:** (a) H_{0} is that the machine is working
properly and that T has the distribution stated. That is, the population
standard deviation of the coffee can weights is sigma=2.6. H_{1}
is that sigma>2.6.

`X2cdf(`

. The syntax
for `X2cdf(`

is `X2cdf(Lower,Upper,df)`

where
`df`

stands for degrees of freedom. To calculate
P(X(28)>39.29), enter 39.29 then COMMA, then 1E99 (1 then
2nd COMMA then 99) then COMMA then 28 for degrees of freedom then
`)`

(right parenthesis) then press Enter. The number 0.07637
should appear on the TI-83 screen, which is the P-value. Note that
0.05<0.7637<0.10, consistent with before.
2. Do exercise 7.44 on page 325. Here a random sample of size n=30 is
taken from a population, and H_{0}:p=0.70, H_{1}:p=0.80
for the population proportion of some property. A decision rule is to
reject H_{0} if the number T in the sample with that property
satisfies T>23. What is the significance level? What is the power?

**Answer:** Here alpha=P(T>23)=1-P(T<=23) where T has a binomial
distribution with n=30 and p=0.70, and the power is P(T>23)=1-P(T<=23)
for n=30 and p=0.80.

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?

**Answer:** This is a two-tailed test for H_{0}:mu=25.0g. The
easiest test statistic to use is T=(Xbar-25)/(Sx/root(n)), which under
the assumption of normality, has a Student's t-distribution with n-1
degrees of freedom. Here n=6, Xbar=23.9, Sx=0.885, T=-3.043, so that the
two-sided P-value is P = P(|T(5)|>=3.043) = 2P(T(5)>=3.043), where
T(5) denotes a random variable with a Student's t distribution with 5
degrees of freedom.

4. Do exercise 7.72 on page 346.

**Answer:** Some summary statistics for the n=16 numbers are listed
in a table at the bottom of page 346. From this table, n=16,
Xbar=1497.50, and Sx=10.708. (Otherwise, you could calculate Xbar and Sx
yourself from the 16 numbers.) Assume that the observations are normal.
The Z-scores that are listed in Problem 7.70 show no significant
outliers, so that this assumption is probably safe.

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

**Answer:** The setup of Problem 8.6 has a pair of values (X,Y)
for each of n=20 claims. We are to test H_{0}:muX=muY versus
H_{1}:muX(notequal)muY, which means a two-sided test. The
problem structure suggests the use of paired differences, so that we
define the differences D_{i}=X_{i}-Y_{i} and
test H_{0}:muD=0 versus H_{1}:muD(notequal)0.

6. Do exercise 8.14 on page 375.

**Answer:** Here we are told explicitly to use the Wilcoxon Signed
Rank test for n=10 paired differences for married couples.