Recitation 13

Comparing Two Populations or Treatments

1.

An experiment for comparing the protein content in Sundance winter wheat and Manitou spring wheat used nine plots. Each plot was planted with the Suncance wheat and then the Manitou wheat. The results are given in the following table. Protein content is measured in grams per kilogram of wheat.

	Location
	1	2	3	4	5	6	7	8	9
Sundance	77	84	116	145	100	154	114	149	137
Manitou	125	111	144	171	133	166	143	175	170
Difference	48	27	28	26	33	12	29	26	33

The sample mean difference is 29.11 and the sample standard deviation of the differences is 9.387.

(a)

Is there evidence that the mean protein levels for the two wheat varieties are different?

(b)

Give a 90% confidence interval for the difference in the population mean protein concentrations for the two varieties of wheat.

2.

A researcher studies a sample of 60 male soccer players to examine the relation between IQ and frequency of ``heading'' the ball. The soccer players were divided into two groups, based on whether or not they averaged 10 or more headers per game. The sample sizes, sample average IQs, and sample standard deviation of the IQs are

		Sample	Sample
	n	mean IQ	sd of IQs
Fewer than 10 headers	35	112	10
10 or more headers	25	103	8

(a)

Is there evidence that the average IQ of male soccer players who head the ball 10 or fewer times is lower than for players who head the ball more often?

(b)

Find a 95% confidence interval for the difference of the average IQs of male soccer players who head the ball 10 or fewer times and those who head the ball more often.

(c)

A report on this research claimed that it had shown that frequently heading the ball in soccer lowers players' IQs. Comment on this claim.

Comparing Two Populations or Treatments Using Rweb

Go to

http://www.superior.stat.umn.edu/Rweb

on your web browser. Click on the first option for Rweb under ``Which Version Do You Want?''. Type the following in the box to enter the data into Rweb and calculate the mean and standard deviation of the dfferences within plots.

sundance <- c(77,84,116,145,100,154,114,149,137)
manitou <- c(125,111,144,171,133,166,143,175,170)
mean(sundance-manitou)
sd(sundance-manitou)

Then press the Submit button. The resulting output should tell you that the mean difference is -29.11111 and the standard deviation of the differences is 9.386752. If you do not get these answers use the Back option on your browser and check that you have entered the data in correctly. If you have not then edit your data and click Submit again.

When your data is correct add the line

t.test(sundance,manitou,paired=T)

to the bottom of your other commands. This will give a 95% confidence interval and a p-value for the paired t-test for this data. The p-value should be $1.451 \times 10^{-5}$ and the confidence interval -36.3 to -21.9. Rweb used the default settings of a two-sided alternative and a 95% confidence interval. The same answer is given by:

t.test(sundance,manitou,paired=T,alternative="two.sided",mu=0,paired=T,conf.level=0.95)

Type

help("t.test")

to see other options. Try them. Also try

stem(sundance-manitou)
boxplot(sundance-manitou)

to plot the data and examine whether the normal assumption is appropriate. Is it? You can use the function pt to calculate t-probabilities. For example

pt(-9.3039,8)

gives the probability that a T random variable with 8 degrees of freedom is less than -9.3039. The answer is $7.253775\times 10^{-6}$ and the p-value for the two-sided test above is twice this or

2*pt(-9.3039,8)

If you omit the paired=T in the t.test line and just type

t.test(sundance,manitou)

you will get a two sample test for this data set - which is not appropriate for this data. You can use Rweb to check your calculations, do your homework, and calculate p-values more precisely than from Table C. (But as Rweb will not be available for the final and you will only have Table C, please make sure you know how to use Table C also!) You can also try going to

http://www.stat.umn.edu/CLASSES/faculty/luke/classes/3011/rwebnotes

and examine the examples given there.