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# Practice 3: Single Proportion

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?

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?

left-tailed

### Exercise 4

What symbol represents the Random Variable for this test?

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'=

• a. 7
• b. 100
• c. 0.07

### Exercise 7

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

0.0293

### Exercise 8

State the distribution to use for the hypothesis test.

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.

Find the p-value

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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##### What is in a lens?

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