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Practice 1: Hypothesis Testing for Two Proportions

Module by: Susan Dean, Barbara Illowsky, Ph.D.. E-mail the authors

Summary: This module provides a practice of Two Population Means and Two Population Proportions as a part of Collaborative Statistics collection (col10522) by Barbara Illowsky and Susan Dean.

Student Learning Outcomes

  • The student will conduct a hypothesis test of two proportions.

Given

In the recent Census, 3 percent of the U.S. population reported being two or more races. However, the percent varies tremendously from state to state. (Source: http://www.census.gov/prod/cen2010/briefs/c2010br-02.pdf) Suppose that two random surveys are conducted. In the first random survey, out of 1000 North Dakotans, only 9 people reported being of two or more races. In the second random survey, out of 500 Nevadans, 17 people reported being of two or more races. Conduct a hypothesis test to determine if the population percents are the same for the two states or if the percent for Nevada is statistically higher than for North Dakota.

Hypothesis Testing: Two Proportions

Exercise 1

Is this a test of means or proportions?

Solution

Proportions

Exercise 2

State the null and alternative hypotheses.

  • a. H0:H0: size 12{H rSub { size 8{0} } :} {}
  • b. Ha:Ha: size 12{H rSub { size 8{a} } :} {}

Solution

  • a. H 0 : PN = PND H 0 : PN = PND size 12{H rSub { size 8{O} } : rSub { size 8{ ital "PN"} } = rSub { size 8{ ital "PND"} } } {}
  • a. H a : PN > PND H a : PN > PND size 12{H rSub { size 8{a} } : rSub { size 8{ ital "PN"} } > rSub { size 8{ ital "PND"} } } {}

Exercise 3

Is this a right-tailed, left-tailed, or two-tailed test? How do you know?

Solution

right-tailed

Exercise 4

What is the Random Variable of interest for this test?

Exercise 5

In words, define the Random Variable for this test.

Exercise 6

Which distribution (Normal or student's-t) would you use for this hypothesis test?

Solution

Normal

Exercise 7

Explain why you chose the distribution you did for the above question.

Exercise 8

Calculate the test statistic.

Solution

3.50

Exercise 9

Sketch a graph of the situation. Mark the hypothesized difference and the sample difference. Shade the area corresponding to the pp size 12{p - {}} {}value.

Figure 1
Figure 1 (9-1.png)

Exercise 10

Find the pp size 12{p - {}} {}value:

Solution

0.0002

Exercise 11

At a pre-conceived α=0.05α=0.05 size 12{α=0 "." "05"} {}, what is your:

  • a. Decision:
  • b. Reason for the decision:
  • c. Conclusion (write out in a complete sentence):

Solution

  • a. Reject the null hypothesis

Discussion Question

Exercise 12

Does it appear that the proportion of Nevadans who are two or more races is higher than the proportion of North Dakotans? Why or why not?

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