Homework #10, Math 320, Spring 2001

Name:____________________________      Section:____

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.)