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Practice 3: Single Proportion

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

Summary: This module provides a practice of Hypothesis Testing of Single Mean and Single Proportion 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 a single population proportion.

Given

The National Institute of Mental Health published an article stating that in any one-year period, approximately 9.5 percent of American adults suffer from depression or a depressive illness. (http://www.nimh.nih.gov/publicat/depression.cfm) Suppose that in a survey of 100 people in a certain town, seven of them suffered from depression or a depressive illness. Conduct a hypothesis test to determine if the true proportion of people in that town suffering from depression or a depressive illness is lower than the percent in the general adult American population.

Hypothesis Testing: Single Proportion

Exercise 1

Is this a test of means or proportions?

Solution

Proportions

Exercise 2

State the null and alternative hypotheses.

  • a. Ho:Ho: size 12{H rSub { size 8{O} } :} {}
  • b. Ha:Ha: size 12{H rSub { size 8{a} } :} {}

Solution

  • a. H o : p = 0 . 095 H o : p = 0 . 095 size 12{H rSub { size 8{O} } :p=0 "." "095"} {}
  • b. H a : p < 0 . 095 H a : p < 0 . 095 size 12{H rSub { size 8{a} } :P<0 "." "095"} {}

Exercise 3

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

Solution

left-tailed

Exercise 4

What symbol represents the Random Variable for this test?

Solution

P'P'

Exercise 5

In words, define the Random Variable for this test.

Solution

the proportion of people in that town surveyed suffering from depression or a depressive illness

Exercise 6

Calculate the following:

  • a: x=x= size 12{x={}} {}
  • b: n=n= size 12{n={}} {}
  • c: p'=p'=

Solution

  • a. 7
  • b. 100
  • c. 0.07

Exercise 7

Calculate σp'σp'. Make sure to show how you set up the formula.

Solution

0.0293

Exercise 8

State the distribution to use for the hypothesis test.

Solution

Normal

Exercise 9

Sketch a graph of the situation. Label the horizontal axis. Mark the hypothesized mean and the sample proportion, p-hat. Shade the area corresponding to the p-value.

Blank horizontal axis of P'.

Exercise 10

Find the p-value

Solution

0.1969

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. Do not reject the null hypothesis

Discusion Question

Exercise 12

Does it appear that the proportion of people in that town with depression or a depressive illness is lower than general adult American population? Why or why not?

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