Chapter 9.3, Problem 1AC

Air Quality

As a researcher for the EPA, you have been asked to determine if the air quality in the United States has changed over the past 2 years. You select a random sample of 10 metropolitan areas and find the number of days each year that the areas failed to meet acceptable air quality standards. The data are shown.

Source: The World Almanac and Book of Facts.

Based on the data, answer the following questions.

1. What is the purpose of the study?

2. Are the samples independent or dependent?

3. What hypotheses would you use?

4. What is (are) the critical value(s) that you would use?

5. What statistical test would you use?

6. How many degrees of freedom are there?

7. What is your conclusion?

8. Could an independent means test have been used?

9. Do you think this was a good way to answer the original question?

To determine

To find: The purpose of the study.

Explanation of Solution

The purpose of the givens study is “to determine if the air quality in the United States has changed over the past 2 years”.

To determine

To classify: The samples as independent or dependent.

Answer to Problem 1AC

The samples are dependent.

Explanation of Solution

Independent samples:

If the sample values from one population do not associate with the sample values from other population, then the two samples are said to be independent samples.

Dependent samples:

If the sample values from one population associated or matched with the sample values from other population, then the two samples are said to be dependent samples.

Matched pair design occurred at two situations, which are listed below:

• Subjects are matched with pairs and each treatment is given to one subject in each pair.
• Before and after observations on the same subjects.

Here, the samples are dependent because same metropolitan areas are taken and the samples that are related. Thus, it can be concluded that the samples are dependent.

To determine

To find: The hypotheses of the study.

Answer to Problem 1AC

Hypotheses:

Null hypothesis: $H_0:{\mu }_D=0$

Alternative hypothesis: $H_1:{\mu }_D\ne 0$

Explanation of Solution

Null hypothesis:

Null hypothesis is a statement about population parameter, its value is equal to the claim value, which is denoted by $H_0$. The population parameter can be mean, standard deviation, proportion, variance, and so forth. The possible symbols used in the null hypothesis would be $\le ,\ge$, or =.

Alternative hypothesis:

It is complementary to the null hypothesis. That is, it differs from the null hypothesis. The possible symbols used in the alternative hypothesis would be <,>, or ≠. It is denoted by $H_1$.

State the null and alternative hypotheses:

Null hypothesis:

$H_0:{\mu }_D=0$

Alternative hypothesis:

$H_1:{\mu }_D\ne 0$

To determine

To find: The critical value.

Answer to Problem 1AC

The critical value is ±2.262.

Explanation of Solution

Calculation:

Degrees of freedom:

$\begin{array}{c}n-1=10-1\\ =9\end{array}$

Software Procedure:

Step-by-step procedure to obtain the critical value using the MINITAB software:

• Choose Graph > Probability Distribution Plot choose View Probability> OK.
• From Distribution, choose ‘t’ distribution.
• In Degrees of freedom, enter 9.
• Click the Shaded Area tab.
• Choose Probability value and Both Tail for the region of the curve to shade.
• Enter the Probability value as 0.05.
• Click OK.

Output using the MINITAB software is given below:

From the output, the critical value is ±2.262.

To determine

To find: The statistical test.

Answer to Problem 1AC

The t test for dependent samples can be used.

Explanation of Solution

Here, the samples are dependent because same metropolitan areas are taken and the samples that are related.

Thus, the t test for dependent samples can be used.

To determine

To find: The degrees of freedom.

Answer to Problem 1AC

The degrees of freedom is 9.

Explanation of Solution

Calculation:

Degrees of freedom:

$\begin{array}{c}n-1=10-1\\ =9\end{array}$

Thus, the degrees of freedom is 9.

To determine

To describe: The conclusion.

Answer to Problem 1AC

The conclusion is that there is no enough evidence to support the claim that the air quality in the United States has changed over the past 2 years.

Explanation of Solution

Calculation:

Software Procedure:

Step-by-step procedure to obtain the test value using the MINITAB software:

• Choose Stat > Basic Statistics > 1-Sample t.
• In Samples in Column, enter the column of Difference.
• In Perform hypothesis test, enter the test mean as 0.
• Check Options; enter Confidence level as 95%.
• Choose not equal in alternative.
• Click OK.

Output using the MINITAB software is given below:

From the output, the test value is –1.88.

Decision:

Decision rule:

If $t>t_0\left(=+2.262\right)$, then reject the null hypothesis.

If $t<-t_0\left(=-2.262\right)$, then reject the null hypothesis.

Here, the value of test statistic is greater than the critical value.

That is, $t\left(=-1.88\right)>-t_0\left(=-2.262\right)$.

Therefore, the null hypothesis is not rejected,

Thus, the decision is “fail to reject the null hypothesis”.

Hence, there is no enough evidence to support the claim that the air quality in the United States has changed over the past 2 years.

To determine

To check: Whether the independent test can be used.

Answer to Problem 1AC

The independent means test cannot be used.

Explanation of Solution

Here, each metropolitan area had two readings. That is, the samples are related. Thus, the independent means test cannot be used.

To determine

To describe: The result.

Explanation of Solution

Answer will vary. One of the possible answers is given below:

The answer is that there are other measures of air quality in the U.S that could have examined to answer the original question.