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.