OBJECTIVES: This lab covers the following topics: comparing two proportions, sample size, and power calculations.

DIRECTIONS: Follow the instructions below, answering all questions. Your answers should be in the form of a brief report to be handed in to the instructor before you leave.
__________________________________________________________________________________________________________
 

1.  Data Description:

In a Christmas tree survey, respondents who had a tree during the holiday season were asked whether the tree was natural or artificial. Respondents were also asked if they lived in an urban area or in a rural area. Of the 421 households displaying a Christmas tree, 160 lived in rural areas and 261 were urban residents. The tree growers want to know if there is a difference in preference for natural trees versus artificial trees between urban and rural households. Here are the data:

Population       n       X(natural)

1 (rural)        160        64

2 (urban)      261        89
 

a. What are the null and alternative hypotheses in this study?

b. What is the proportion of natural trees displayed by urban residents who had a tree during the holiday season? What is the proportion of natural trees displayed by rural residents who had a tree during Christmas?

c. What test statistic are you using to test the hypotheses? What will the approximate distribution be (for large samples)?

d. You can use Minitab to do the test.

Stat->Basic Statistics->2 proportions , select Summarized data ; and input 160 in the first entry, 64 in the first row second column, 261 in the second row first column and 89 in the second row and second column. Click Options ; make sure Alternative is "not equal". OK.

e. Based on the results from Minitab, what conclusion do you draw?

f. Suppose we want to see if the rural residents have a higher preference for natural trees than urban residents. What should the alternative hypothesis be? Make your conclusion based on your minitab output.

2.   Sample size and power calculations for proportions comparison.

The Women's Institutional Studies group at a Midwestern university wanted to compare the proportions of female faculty memebers in the various units on campus in Spring 2000. Of particular interest were the proportions of female faculty in the College of Education and the College of Bussiness. In a random sample of 80 College of Education faculty, there were 36 females, and in a random sample of 80 College of Business faculty, there were 17 females. Based on the two samples, can we conclude that there are differences in the proportion of female faculty in these colleges?

a.  Take a glance at the data, which College tends to have more female faculty?

b.  Based on persistence of gender roles, we are interested in the question of whether there are more females in Education than in Business.  What are your null hypothesis and research hypothesis? Perform the test of these hypotheses using Minitab.

c.  Given the sample sizes (80 for each group) and assuming the proportion of female faculty in College of Bussiness is 0.2 (20%), what is the power to detect a 0.1 (10%) difference in the proportion of female faculty in these colleges?

Hint: "Stat-> Power and Sample size-> 2 proportions", select Calculate power for each sample size , Sample size: 80, Proportion 1: 0.3 , Proportion 2: 0.2, click Options, check Greater than in the Alternative hypothesis , O.K.  (This is if you have selected the Education Faculty as population 1 and the Business Faculty as population 2.)

d.  Based on the same assumptions as in part c, what are the powers to detect 0.15, 0.2, 0.25, 0.3 differences in the proportions of female faculty in these colleges? When the differences between the two proportions increases, will the power increase or decrease? Why? Try to give a reasonable explanation.

Hint: "Stat-> Power and Sample size-> 2 proportions", select Calculate power for each value of proportion 1, Sample size: 80, Proportion 1: 0.35 0.4 0.45 0.5 , Proportion 2: 0.2,click Options, check Greater than in the Alternative hypothesis , O.K.

e. Assume the proportion of female faculty in College of Education is 0.45 ( group 1) and the proportion of female faculty in College of Bussiness is 0.2 ( group 2), what the sample sizes should be to achieve powers of 0.75, 0.8, 0.85, 0.9?

Hint: "Stat-> Power and Sample size-> 2 proportions", select Calculate sample size for each power value : 0.75, 0.8, 0.85, 0.9,  for Proportion 1 equal to 0.45, and Proportion 2 equal to 0.2, click Options, check Greater than in the Alternative hypothesis, O.K.

f. Assume the proportion of female faculty in College of Education is 0.4 (group 1) and the proportion of female faculty in College of Bussiness is 0.25 (group 2), what the sample sizes should be to achieve powers of 0.75, 0.8, 0.85, 0.9?

Hint: "Stat-> Power and Sample size-> 2 proportions", select Calculate sample size for each power value 1, Power value: 0.75 0.8 0.85 0.9, Proportion 1: 0.4 , Proportion 2: 0.25,click Options, check Greater than in the Alternative hypothesis , O.K.

g. Based on the results in parts e and f, if we want to increase the power of a test, should we increase the sample size or decrease it?