12 Testing Two Population Means 12 Testing Two Population Means 12.2 Paired Samples

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12.1 The Setup

Goals:

\bullet

Learn how to state the competing hypotheses;

\bullet

Learn how to identify the direction of extreme.

Suppose a physical therapist has developed a new treatment that she believes will improve flexibility for people suffering from rotator cuff injuries. What would be a reasonable way to test whether the treatment is effective? Or, suppose a drug manufacturer has developed a new medicine that is suppose to be an improvement, say in time to providing relief to patients, over a currently used medicine. What would be a reasonable process to test whether the new drug is more effective? Questions such as these can be assessed by way of studying a difference of means. In the first example, the researcher is interested in the difference between mean flexibility after treatment and mean flexibility before. Likewise, the second example looks at the difference between mean time to relief from the new drug and the mean time to relief for the current drug. Comparing means is common in the data-driven disciplines, and this chapter introduces tests for comparing two means.

12.1.1 Possible Hypotheses

Suppose that $X$ and $Y$ are two populations of numbers. Suppose that $X$ has mean $\mu_{X}$ and standard deviation $\sigma_{X},$ and likewise, population $Y$ has mean $\mu_{Y}$ and standard deviation $\sigma_{Y}$ . In mathematics, there are two fundamental ways to compare two numbers. One is to take their ratio, the other is to take the difference. In comparing means, we do the latter, i.e, we look at $\mu_{X}-\mu_{Y}.$ In a hypothesis test on the difference, the possible competing hypotheses and corresponding direction of extremes are:

$H_{0}:\mu_{X}-\mu_{Y}\geq 0$	$H_{0}:\mu_{X}-\mu_{Y}\leq 0$	$H_{0}:\mu_{X}-\mu_{Y}=0$
$H_{1}:\mu_{X}-\mu_{Y}<0$	$H_{1}:\mu_{X}-\mu_{Y}>0$	$H_{1}:\mu_{X}-\mu_{Y}\neq 0$
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 $H_{0}:\mu_{X}-\mu_{Y}\geq 0,$ why is the direction of extreme to the left? As was the case in testing a single mean, evidence against $H_{0}$ will come from samples, but in this case a sample from each population. From population $X$ , a sample is drawn and sample mean $\bar{x}$ computed. Similarly, a sample is drawn from $Y$ and a sample mean $\bar{y}$ computed. It is the difference $\bar{x}-\bar{y}$ that is used to make a decision regarding $H_{0}.$ The smaller the difference, i.e., the farther to the left of 0, the stronger the evidence against $H_{0}.$ Hence, the direction of extreme is to the left. Similar explanations can be made for the other two cases.

Since $\mu_{X}-\mu_{Y}=0$ is equivalent to $\mu_{X}=\mu_{Y},$ the hypotheses can also be rewritten as:

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

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

Example 12.1.1.

A researcher wished to test whether filling a football with air or helium would change the average distance of a punt. One set of footballs were filled with air and the other were filled with helium. Each football was punted by the researcher, and the distance recorded in yards. State the competing hypotheses and direction of extreme.

Solution. Let $\mu_{a}$ and $\mu_{h}$ denote the true averages of the researcher punting distances of air-filled and helium-filled footballs, respectively. Then the competing hypotheses are:

\begin{array}[]{ll}H_{0}:&\mu_{a}-\mu_{h}=0\\ H_{1}:&\mu_{a}-\mu_{h}\neq 0\\ \end{array}

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

Example 12.1.2.

A researcher wished to test whether consuming caffeine shortens the average time it takes to run a mile. To test the claim, she timed participants running a mile. One week later, allowing participants time to recuperate, she gave each participant a 29 mg caffeine tablet, and then ten minutes after consuming the tablet, she timed participants running another mile. State the competing hypotheses and direction of extreme.

Solution. Let $\mu_{b}$ and $\mu_{a}$ denote the true averages of the time to run a mile before and after consuming caffeine, respectively. Then the competing hypotheses are:

\begin{array}[]{ll}H_{0}:&\mu_{a}-\mu_{b}\geq 0\\ H_{1}:&\mu_{a}-\mu_{b}<0\\ \end{array}

The direction of extreme is to the left. Note that if one used the difference $\mu_{b}-\mu_{a}$ instead, then the competing hypotheses would be:

\begin{array}[]{ll}H_{0}:&\mu_{a}-\mu_{b}\leq 0\\ H_{1}:&\mu_{a}-\mu_{b}>0\\ \end{array}

With this choice the direction of extreme is to the right. $\clubsuit$

Concepts Check: 1. A study on iron deficiency among infants compared samples of infants following two different feeding regiments. One group was breastfed while the other received a standard baby formula without iron supplements. Is the average blood hemoglobin levels significantly higher in breastfed babies than in formula-fed babies? State the competing hypotheses and direction of extreme. Answer: If

\mu_{b}

and

\mu_{f}

denote the true averages blood hemoglobin levels of breast-fed and formula-fed babies, respectively, then the competing hypotheses are:

\begin{array}[]{ll}H_{0}:&\mu_{b}-\mu_{f}\leq 0\\ H_{1}:&\mu_{b}-\mu_{f}>0\\ \end{array}

The direction of extreme is to the right. 2. A researcher thinks that a summer program for high school students would improve average interest in engineering as a profession. To test the claim, she measures participant interest in engineering before and after the summer program. Measurements are rankings of interest from 0 to 100, with larger values denoting greater interest. State the competing hypotheses and direction of extreme. Answer: If

\mu_{b}

and

\mu_{a}

denote the true averages of interest in engineering before and after the summer experience, respectively, then the competing hypotheses are:

\begin{array}[]{ll}H_{0}:&\mu_{b}-\mu_{a}\leq 0\\ H_{1}:&\mu_{b}-\mu_{a}>0\\ \end{array}

The direction of extreme is to the right.

12.1.2 Exercises

1.
Answer the following as True or False.
1. (a)
  
  A null hypothesis of $H_{0}:\mu_{X}\geq\mu_{Y}$ is the same as $\mu_{Y}-\mu_{X}\geq 0$ .
2. (b)
  
  Hypothesis involving testing two population means still has the same decision outcomes: Reject $H_{0}$ and Fail to Reject $H_{0}$ .
3. (c)
  
  If $H_{0}:\mu_{X}\geq\mu_{Y}$ , then the direction of extreme is to the right.
4. (d)
  
  It is possible to have a null hypothesis read $H_{0}:\mu_{X}>\mu_{Y}$ .
5. (e)
  
  If $H_{1}:\mu_{X}\geq\mu_{Y}$ , then the direction of extreme is to the right.
2.
For the following problems, write the hypothesis test and determine the direction of extreme.
1. (a)
  
  An education professor wants to determine if the curriculum he/she has designed is improving students’ average final exam scores. Scores for the same final exam are collected over two semesters.
2. (b)
  
  Hypothyroidism can be an issue with English Bulldogs. A researcher wants to find out if a new drug has an impact on the average thyroxine (T4) levels. T4 levels are collected for a control group of 40 English Bulldogs and a treatment group of 40 English Bulldog. The treatment group has been given the new drug.
3.

It is possible to change the direction of extreme for one-sided tests. In the previous problem, rewrite the one-sided tests so that the direction of extreme is in the other direction.