Homework #9, Math 320, Spring 2001
Name:____________________________
Section:____
Math 320 Homework #9 --- Due 3/30
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.
Note: You can use a TI-83 calculator or Excel to carry out one-
and two-sample Z- and T-tests, using either data or summary statistics.
The calculator or computer will display the exact P-value as well as the
value of the test statistic. As before, you can also find exact P-values
for Student-t, chi-square, and F distributions, even for a fractional
number of degrees of freedom for the Student t distribution. Click
here for the details on the
Math 320 calculator page.
(Note: The material for the one- and two-sample tests has just
been added.)
1. (Like exercise 8.20 on page 377.)
The data for n=10 paired differences in problem 8.19 has P=0.0145
for a one-sided Student t-test and P=0.0191 for a one-sided Wilcoxon
signed-rank test. (This is either stated in the problem or else is given
in the back of the book.) Thus the two tests give essentially the same
results, in the sense that 0.01 < P < 0.025 in both cases. (You do
not have to repeat these calculations.)
Now assume that the ``After course'' score for Student #6 is 85
instead of 45. This changes the difference for the 6th
student from 10 to 50, and the average of the 10 differences from 4.5 to
8.5.
Carry out both the Student t-test and the one-sided Wilcoxon
rank-sum test for the altered data. Are the one-sided P-values still
more-or-less the same for the two tests? In particular, are they still
both significant at the 5% level or else both nonsignificant?
A rule of thumb for deciding if data is normal:
Recall that the Student's t-test is only valid if you can assume
that the data are normally distributed.
Step (i): Calculate the standardized values (sometimes also
called Z-scores) for the data. If the data is X1,
X2, ...., Xn, then the Z-scores are
X*i = (Xi-Xbar)/sX. (See page 357 in
the text.)
Step (ii): If the sample size n<=10, reject normality if there
are any standardized values with |X*| >= 2.50.
Step (iii): If the sample size n<=15, reject normality if there
are any standardized values with |X*| >= 2.80.
Steps (ii,iii) are overly conservative and may reject normality in
some data sets that are really normal. However, the Wilcoxon rank-sum
test is valid whether the data is normal or not.
Step (iv): If the sample size n<=30, reject normality if there
are two standardized values with |X*| >= 2.50 or
else one standardized value with |X*| >= 3.00.
In general, if you are sure that the data is normal, check every
value with |X*| >= 3.00 to make sure that it is not the
result of a typographical error or some other error, such as writing
down a measurement in millimeters when it should be in inches.
2. Calculate the standardized values or Z-scores for the altered
observations in Problem 1. Do the standardized values meet
the rule for normality above? Could this be related to your conclusions
in Problem 1?
3. Also, carry out the sign test for the data in exercise 8.19 on
page 377. That is, note that only 2 of the 10 differences are negative
and the rest are positive. If H0 were true and the typing
course had no effect, then one would expect the signs of the
differences to be like 10 tosses of a fair coin.
(i) Calculate the one-sided sign-test P-value as the
probability that T<=2, where T has a binomial distribution with n=10
and p=0.50. Use Table A1 to find the exact value of this
probability.
(ii) How does the resulting P-value compare with that of the Wilcoxon
rank-sum test? Does the sign test appear to be more powerful or less
powerful than the Wilcoxon rank-sum test for this data?
(iii) How do the results of the sign test change if the difference
value of 10 for the 6th student is changed to 50, as in
Problem 1?
4. An owner of a large plant nursery wants to test the effect of a new
fertilizer on the growth of a popular shrub. After four months of growth
using the standard fertilizer, the average heights of 437 plants was
38.6 inches with a standard deviation of 3.1 inches. After four months
of growth using the new fertilizer, the average height of a
nonoverlapping group of 101 plants was 39.9 inches with standard
deviation 4.2 inches.
(i) Was there a significant improvement with the new fertilizer,
using a level of significance of alpha=0.01? What is the (one-sided)
P-value?
(ii) Find a 95% confidence interval for the improvement in height
due to the new fertilizer.
5. (Like Exercise 9.12 on page 397.) Two types of smudge pots were
tested for use to protect orchards from frost. A grapefruit grower wants
to compare them in terms of how long they burn before dying out. Two
samples of burning times for 10 smudge pots each are recorded below. It
is desired to test whether or not there is a difference between mean
burning times for the two types of smudge pot. The burning times (in
minutes) were:
Type 1: 612 583 629 595 653 596 624 564 576 593
Type 2: 592 607 696 686 680 669 697 729 694 662
(i) Assume that the data behave as if they were two independent
random samples from two different normal populations with the same
variance. State the appropriate null and alternative hypothesis, and
test the null hypothesis at alpha=0.05. What is the resulting P-value?
(Hint:: The first sample has mean 602.50 and sample
standard deviation 26.99. The second sample has mean 671.20 and sample
standard deviation 42.06. )
(ii) What test statistic did you use? What is its distribution
under the null hypothesis?
(iii) Find a 95% confidence interval for the difference in mean
burning time between the two types of pot.