# 8.14 Practice 3: single proportion

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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

Is this a test of means or proportions?

Proportions

State the null and alternative hypotheses.

• ${H}_{o}:$
• ${H}_{a}:$
• ${H}_{o}:p=0\text{.}\text{095}$
• ${H}_{a}:p<0\text{.}\text{095}$

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

left-tailed

What symbol represents the Random Variable for this test?

$\text{P}\text{'}$

In words, define the Random Variable for this test.

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

Calculate the following:

• $x=$
• $n=$
• $\text{p}\text{'}=$

• 7
• 100
• 0.07

Calculate ${\sigma }_{p\text{'}}$ . Make sure to show how you set up the formula.

0.0293

State the distribution to use for the hypothesis test.

Normal

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

At a pre-conceived $\alpha =0\text{.}\text{05}$ , what is your:

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

• Do not reject the null hypothesis

## Discusion question

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?