Chapter 9.2, Problem 4E

a.

To determine

To identify: The claim and state $H_0$ and $H_1$.

Answer to Problem 4E

The claim is that “the mean age of those playing the slot machines is less than those playing roulette”.

The hypotheses are given below:

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

Alternative hypothesis: $H_1:{\mu }_1<{\mu }_2$ (claim)

Explanation of Solution

Given info:

${\overline{x}}_1=48.7\text{ years}$, ${\overline{x}}_2=55.3\text{ years}$, $s_1=6.8\text{ years}$, $s_2=3.2\text{ years}$, $n_1=25$, and $n_2=35$.

Justification:

Here, the mean age of those playing the slot machines is less than those playing roulette is tested. Hence, the claim is that the mean age of those playing the slot machines is less than those playing roulette. This can be written as ${\mu }_1<{\mu }_2$. The complement of the claim is ${\mu }_1={\mu }_2$. In the given experiment, the alternative hypothesis indicates the claim.

The hypotheses are given below:

Null hypothesis:

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

Alternative hypothesis:

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

b.

To determine

To find: The critical value $\alpha =0.05$.

Answer to Problem 4E

The critical value at $\alpha =0.05$ is $-1.711$.

Explanation of Solution

Calculation:

Here, the test is left tailed test.

Critical value:

Here, variances are not equal. Hence, the degrees of freedom is,

$\begin{array}{c}\text{df}=\text{Smallar of }{n}_1-1\text{ and }{n}_2-1\\ =25-1\\ =24\end{array}$

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 24.
• Click the Shaded Area tab.
• Choose Probability value and Left 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 $-1.711$.

c.

To determine

To find: The test value.

Answer to Problem 4E

The test value is –4.51.

Explanation of Solution

Calculation:

Test statistic:

Software Procedure:

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

• Choose Stat > Basic Statistics > 2-Sample t.
• Choose Summarized data.
• In first, enter Sample size as25, Mean as 48.7, Standard deviation as 6.8.
• In second, enter Sample size as35, Mean as 55.3, Standard deviation as 3.2.
• Choose Options.
• In Confidence level, enter 95.
• In Alternative, select less than.
• Click OK in all the dialogue boxes.

Output using the MINITAB software is given below:

From the MINITAB output, the test value is –4.51.

d.

To determine

To make: The decision.

Answer to Problem 4E

The decision is that, the null hypothesis is rejected.

Explanation of Solution

Calculation:

Software Procedure:

Step-by-step procedure to indicate the appropriate area and 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 24.
• Click the Shaded Area tab.
• Choose Probability value and Left Tail for the region of the curve to shade.
• Enter the Probability value as 0.05.
• Enter –4.51 under show reference lines at X values
• Click OK.

Output using the MINITAB software is given below:

From the output, it can be observed that the test statistic value falls in the critical region. Therefore, the null hypothesis is rejected.

e.

To determine

To summarize: The result.

Answer to Problem 4E

The conclusion is that, there is enough evidence to support the claim that the mean age of those playing the slot machines is less than those playing roulette.

Explanation of Solution

Justification:

From part (d), the null hypothesis is rejected. Thus, there is enough evidence to support the claim that the mean age of those playing the slot machines is less than those playing roulette.