Homework #10, Math 320, Spring 2001
Math 320 Homework #10 --- Due 4/6
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.
1. (15 points) Problem 5 of Homework 9 had burning times for two
independent samples of smudge pots:
Type 1: 612 583 629 595 653 596 624 564 576 593
Type 2: 592 607 696 686 680 669 697 729 694 662
It asked you to carry out a pooled-variance (classical) t-test for
whether there was a different in the population-mean burning times.
(i) Is it reasonable that the variances of the two samples in
the last exercise are the same, as is required for the classical
pooled-variance test? Test the hypothesis that the variances are the
same at alpha=0.05, and find the P-value.
(ii) What is the distribution under the null hypothesis of the test
statistic that you used? How many degrees of freedom does it have?
(Warning: An F distribution has TWO numbers for degrees of
freedom, not just one.)
(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. You can find an exact P-value by doing linear
interpolation in Table A5, but it is easier to use the
Fcdf( function on a TI-83 calculator.
Click here for the syntax.)
2. (15 points) An experimenter measures the weights of 12 rabbits from
one species and 8 rabbits from another. The experimenter is interested
in whether there is a difference in average weight between the two types
of rabbit. Assume that the weights of the rabbits are normally
distributed within each species. The weights in ounces are
Species 1: 81 87 82 86 90 87 86 93 88 91 87 84
Species 2: 93 77 79 69 66 89 87 72
(i) First, assume that the variances of the weights of the two
species are the same, and test for a difference in means. Is the
difference significant at alpha=0.05, using a two-sided test?
(ii) Is it reasonable to assume that the variances are the same in
part (i)? Test this at alpha=0.05. What is the P-value? What are
the numbers of degrees of freedom of the F-statistic?
(iii) Your answer to part (ii) should be no, so that the test that
you carried out in part (i) is not valid. Instead, carry out
Satterthwaite's test for an equality of means. Is the difference
significant at alpha=0.05, using a two-sided test? How many degrees of
freedom did you use?
3. (15 points) (i) Apply the Wilcoxon Rank Sum test to the data in the
previous problem. Is there a significant difference between mean weights
of the two samples using this test? What is the (two-sided) P-value?
(ii) Was it reasonable to apply the pooled-variance t-test to the
Wilcoxon ranks in part (i) of this problem? Apply the F-test
for equality of variances to the ranks. Do the variances of the
ranks appear to be the same for the two species of rabbits? What
is the P-value?
4. (15 points) Consider problem 10.2 on page 432.
(i) Construct a 2 x 2 contingency table for the data. What are the
two row sums? What are the two column sums? What is the total sum?
(Hint: Let `Vitamin C' be the first row and let `Placebo' be
the second row. Let `Caught cold' be the first column and `Didn't catch
cold' be the second column. You will have to fill in some cells by
subtraction. A `placebo' is a pill or treatment that has no physical
effect, to allow for the possibility that taking part in the study might
be more fun or curative than the treatment itself.)
(ii) Test the table for significance using the Pearson chi-square
test. (That is, use the test based on the statistic T in equation (10.1)
on page 433.) What is the hypothesis H0? Do you accept or
reject H0 at alpha=0.05? What is the P-value?
(iii) In what direction does the data point? Are the skiers who take
vitamin C more likely to catch colds or less likely? What are the
sample proportions of those who catch colds among the Vitamin-C takers
and among the placebo-takers?
(Hint: Some physicians would argue, or used to argue,
that people who take that much Vitamin C are more likely to
catch colds, since doses of vitamin C that are this massive might
interfere with vitamin B metabolism.)