13 Testing Two Population Proportions 13.1 Introduction 13.3 Testing a Difference of Two Population Proportions

13.2 Possible Hypotheses for $p_{1}-p_{2}$

Goals:

\bullet

Learn how to state the competing hypotheses;

\bullet

Learn how to identify the direction of extreme.

Suppose that we have two populations, Population 1 and Population 2. Each population can be characterized as containing two types of things: things of Type A, and things that are not of Type A. Let $p_{1}$ denote the proportion of things of Type A in Population 1, and similarly, let $p_{2}$ be the proportion of things in Population 2 of Type A. As in the case of comparing two means, to compare two proportions we take their difference, $p_{1}-p_{2}.$ In a hypothesis test on the difference, the possible competing hypotheses and corresponding direction of extremes are:

$H_{0}:p_{1}-p_{2}\geq 0$	$H_{0}:p_{1}-p_{2}\leq 0$	$H_{0}:p_{1}-p_{2}=0$
$H_{1}:p_{1}-p_{2}<0$	$H_{1}:p_{1}-p_{2}>0$	$H_{1}:p_{1}-p_{2}\neq 0$
to the left	to the right	two-sided

Regardless of choice of $H_{0},$ the goal is to make a decision just as in the tests covered in prior chapters: reject $H_{0},$ or fail to reject $H_{0}.$

In the case of $H_{0}:p_{1}-p_{2}\leq 0,$ why is the direction of extreme to the right? As was the case in testing two means mean, evidence against $H_{0}$ will come from a sample from each population. From Population 1, a sample is drawn and the sample proportion $\hat{p}_{1}$ of things of Type A is computed. Similarly, a sample is drawn from Population 2 and the sample proportion $\hat{p}_{1}$ of things of Type A is also computed. It is the difference $\hat{p}_{1}-\hat{p}_{2}$ that is used to make a decision regarding $H_{0}.$ The larger the difference, i.e., the farther to the right of 0, the stronger the evidence against $H_{0}.$ Hence, the direction of extreme is to the right. Similar explanations can be made for the other two cases.

Since $p_{1}-p_{2}=0$ is equivalent to $p_{1}=p_{2},$ the hypotheses can also be rewritten as:

$H_{0}:p_{1}\geq p_{2}$	$H_{0}:p_{1}\leq p_{2}$	$H_{0}:p_{1}=p_{2}$
$H_{1}:p_{1}<p_{2}$	$H_{1}:p_{1}>p_{2}$	$H_{1}:p_{1}\neq p_{2}$
to the left	to the right	two-sided

For purposes of consistency, we’ll use the former notation throughout this chapter.

Example 13.2.1.

Consider the opening scenario in this chapter: suppose that a physical therapist has developed a new treatment that she believes will reduce injuries of a certain type. State the competing hypotheses and direction of extreme.

Solution: Let $p_{1}$ and $p_{2}$ denote the population proportion of injuries for those who receive the treatment and the population proportion of those who do not receive the treatment, respectively. Then the competing hypotheses are:

\begin{array}[]{ll}H_{0}:&p_{1}-p_{2}\geq 0\\ H_{1}:&p_{1}-p_{2}<0\\ \end{array}

The direction of extreme is to the left.⁵²⁵²Note that choosing $p_{2}-p_{1}$ instead changes the direction of extreme. The choice doesn’t matter, as long as once you’ve made the choice, you stick with it through the entire analysis. $\clubsuit$

Example 13.2.2.

A researcher wished to test whether a newly developed mathematics lesson would increase the percentage of students passing a standard first-year algebra course. State the competing hypotheses and direction of extreme.

Solution: Let $p_{1}$ and $p_{2}$ denote the population proportions of those who would pass having received the lesson and those who would pass without receiving the lesson. Then the competing hypotheses are:

\begin{array}[]{ll}H_{0}:&p_{1}-p_{2}\leq 0\\ H_{1}:&p_{1}-p_{2}>0\\ \end{array}

The direction of extreme is to the right. $\clubsuit$

Example 13.2.3.

A researcher wishes to test whether there is a difference in the percent of voters that favor Proposition A in Districts 1 and 2. State the competing hypotheses and direction of extreme.

Solution: Let $p_{1}$ and $p_{2}$ denote the population proportions that favor Proposition A in Districts 1 and 2, respectively. Then the competing hypotheses are:

\begin{array}[]{ll}H_{0}:&p_{1}-p_{2}=0\\ H_{1}:&p_{1}-p_{2}\neq 0\\ \end{array}

The direction of extreme is two-sided. $\clubsuit$

Concepts Check: 1. A study on college students was conducted to test whether student-athletes were more likely to vote than those who do not participate in a sport. State the competing hypotheses and direction of extreme. Answer: If

p_{1}

and

p_{2}

denote the true proportions of student-athletes and non-athletes that intend to vote, then the competing hypotheses are:

\begin{array}[]{ll}H_{0}:&p_{1}-p_{2}\leq 0\\ H_{1}:&p_{1}-p_{2}>0\\ \end{array}

The direction of extreme is to the right. 2. A researcher thinks that women are more likely to accept science than men. To test the claim, he randomly selects 100 men and 100 women, and asks each if he/she “accepts science.” State the competing hypotheses and direction of extreme. Answer: If

p_{1}

and

p_{2}

denote the population proportions of men and women, respectively, that accept science, then the competing hypotheses are:

\begin{array}[]{ll}H_{0}:&p_{1}-p_{2}\geq 0\\ H_{1}:&p_{1}-p_{2}<0\\ \end{array}

The direction of extreme is to the left.

13.2.1 Exercises

1.

A college provost wishes to test whether freshmen and sophomores take more than 16 credits more often than juniors and seniors take more than 16 credits. State the competing hypotheses and direction of extreme.
2.

A basketball player wishes to test whether she is more accurate when shooting three pointers from the top of the key versus shooting from the corner. State the competing hypotheses and direction of extreme.

13.2 Possible Hypotheses for p1-p2

Example 13.2.1.

Example 13.2.2.

Example 13.2.3.

13.2.1 Exercises

13.2 Possible Hypotheses for $p_{1}-p_{2}$