Chapter 9.1, Problem 1AC

Home Runs

For a sports radio talk show, you are asked to research the question whether more home runs are hit by players in the National League or by players in the American League. You decide to use the home run leaders from each league for a 40-year period as your data. The numbers are shown.

Using the data given, answer the following questions.

1. Define a population.

2. What kind of sample was used?

3. Do you feel that the samples are representative?

4. What are your hypotheses?

5. What significance level will you use?

6. What statistical test will you use?

7. What are the test results? (Assume σ1 = 8.8 and σ2 = 7.8.)

8. What is your decision?

9. What can you conclude?

10. Do you feel that using the data given really answers the original question asked?

11. What other data might be used to answer the question?

To determine

Find the population.

Answer to Problem 1AC

The population is “all home runs hit by players in the National league and or by players in the American league”.

Explanation of Solution

Population:

Population consists of the complete subjects that are included in the study for making inferences.

Thus, all home runs hit by players in the National league and or by players in the American league.

To determine

Find the type of sample.

Answer to Problem 1AC

The type of sample is “Cluster sample”.

Explanation of Solution

Cluster sampling:

In cluster sampling, the whole population is divided into different sections or clusters. From these sections some are randomly selected and all the subjects from the selected sections are taken into the sample.

Here, the home run leader is divided into two leagues (National League and American League). Hence, it involves cluster sampling.

To determine

Explain whether the samples are representative or not.

Answer to Problem 1AC

The samples are not representative of the population.

Explanation of Solution

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

Here, the sample is not generalized to the overall behavior of the general population. Thus, the samples are not representative of the population.

To determine

Find the hypotheses.

Answer to Problem 1AC

Hypotheses:

$H_0:{\mu }_1={\mu }_2$

$H_1:{\mu }_1\ne {\mu }_2$

Explanation of Solution

Here, the claim is that “there is a difference in the number of home runs hit by National League and American League baseball players”. This can be written as ${\mu }_1={\mu }_2$. The complement of the claim is ${\mu }_1\ne {\mu }_2$.

Thus, the null and alternative hypotheses are:

Null hypothesis:

$H_0:{\mu }_1={\mu }_2$

Alternative hypothesis:

$H_1:{\mu }_1\ne {\mu }_2$

To determine

Find the level of significance.

Answer to Problem 1AC

The possible significance levels are 0.01, 0.05 and 0.10.

Explanation of Solution

The level of significance that can be used for testing are 0.01, 0.05 and 0.10.

To determine

Find the appropriate statistical test.

Answer to Problem 1AC

The “z test for the difference in means” can be used.

Explanation of Solution

Here, both sample sizes are greater than 30. That is, $n_1\left(=40\right)>30$ and $n_2\left(=40\right)>30$. Thus, the z test for the difference in means is appropriate.

To determine

Explain about the test result.

Answer to Problem 1AC

The test statistic is 1.01.

Explanation of Solution

Calculation:

The z test statistic is defined as,

$z=\frac{\left({\overline{X}}_1-{\overline{X}}_2\right)-\left({\mu }_1-{\mu }_2\right)}{\sqrt{\frac{{\sigma }_1^2}{n_1}+\frac{{\sigma }_2^2}{n_1}}}$ Where ${\mu }_1$ and ${\mu }_2$ are the population means for National League and American League, respectively, ${\sigma }_1$ and ${\sigma }_2$ are the population standard deviations for National League and American League, respectively and $n_1$ and $n_2$ are the population means for National League and American League, respectively,

Software procedure:

Step by step procedure to obtain the mean using the MINITAB software:

• Choose Stat > Basic Statistics > Display Descriptive Statistics.
• In Variables enter the columns National League and American League.
• Click OK.

Output using the MINITAB software is given below:

From the output, the mean for National league is 44.75 and the mean for American league is 42.88.

Test statistic:

$\begin{array}{c}z=\frac{\left({\overline{X}}_1-{\overline{X}}_2\right)-\left({\mu }_1-{\mu }_2\right)}{\sqrt{\frac{{\sigma }_1^2}{n_1}+\frac{{\sigma }_2^2}{n_1}}}\\ =\frac{\left(44.75-42.88\right)}{\sqrt{\frac{{\left(\text{8}\text{.8}\right)}^2}{40}+\frac{{\left(7.8\right)}^2}{40}}}\\ =\frac{\text{1}\text{.87}}{1.86}\\ =1.01\end{array}$

Thus, the test statistic is 1.01.

To determine

Explain the decision.

Answer to Problem 1AC

The decision is “fail to reject the null hypothesis”.

Explanation of Solution

Critical value at $\alpha =0.05$:

From Standard normal table, the critical value for two tail test is ±1.96.

The rejection region for a two-tailed test with $\alpha =0.05$ is, $z<-z_0\left(=-1.96\right)$$z>+z_0\left(=+1.96\right)$.

Decision rule:

• If $z>z_0\left(=+1.96\right)$, then reject the null hypothesis.
• $z<-z_0\left(=-1.96\right)$, then reject the null hypothesis.

Here, the test statistic is less than the critical value. That is, $z\left(=1.01\right)<z_0\left(=+1.96\right)$.

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

To determine

Draw the conclusion.

Answer to Problem 1AC

The conclusion is that, there is no enough evidence to support the claim that there is a difference in the number of home runs hit by National League and American League baseball players.

Explanation of Solution

From the result obtained in part (8), the null hypothesis is not rejected at 5% level of significance.

Thus, it can be concluded that there is no difference in the number of home runs hit by National League and American League baseball players.

To determine

Explain whether the data provides the answer of original question.

Explanation of Solution

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

No, the given data cannot answer the original question asked because the samples are not randomly selected from National League and American League.

To determine

Find the other data that might be used to answer the question.

Explanation of Solution

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

The other data that might be used to answer the question is “the samples are randomly selected from National League and American League from a recent season”.