Chapter 9, Problem 18CQ

a.

To determine

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

Answer to Problem 18CQ

The claim is that “there is a significant difference between the two states in chemists’ salaries”.

The hypotheses are given below:

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

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

Explanation of Solution

Given info:

${\overline{x}}_1=\$39,420$, ${\overline{x}}_2=\$30,215$, $s_1=\$1,659$, $s_2=\$4,116$, $n_1=12$, and $n_2=26$

Justification:

Here, there is a significant difference between the two states in chemists’ salaries is tested. Hence, the claim is that there is a significant difference between the two states in chemists’ salaries. This can be written as ${\mu }_1\ne {\mu }_2$. 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:

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

b.

To determine

To find: The critical value $\alpha =0.02$.

Answer to Problem 18CQ

The critical value at $\alpha =0.02$ is $\pm 2.718$.

Explanation of Solution

Calculation:

Here, the test is two tailed test.

Critical value:

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

$\begin{array}{c}\text{df}=\text{Smallar of }{n}_1-1\text{ and }{n}_2-1\\ =12-1\\ =11\end{array}$

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 11.
• Click the Shaded Area tab.
• Choose Probability value and Two Tail for the region of the curve to shade.
• Enter the Probability value as 0.02.
• Click OK.

Output using the MINITAB software is given below:

From the output, the critical value is $\pm 2.718$.

c.

To determine

To find: The test value.

Answer to Problem 18CQ

The test value is 9.81.

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 as 12, Mean as 39,420, Standard deviation as 1,659.
• In second, enter Sample size as26, Mean as 30,215, Standard deviation as 4,116.
• Choose Options.
• In Confidence level, enter 98.
• In Alternative, select not equal.
• Click OK in all the dialogue boxes.

Output using the MINITAB software is given below:

From the MINITAB output, the test value is 9.81.

d.

To determine

To make: The decision.

Answer to Problem 18CQ

The decision is that, the null hypothesis isrejected.

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 11.
• Click the Shaded Area tab.
• Choose Probability value and Two Tail for the region of the curve to shade.
• Enter the Probability value as 0.02.
• Enter 9.81 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 isrejected.

e.

To determine

To summarize: The result.

To construct: The 99% confidence interval for the difference of the mean.

Answer to Problem 18CQ

The conclusion is that, there is enough evidence to support the claim that there is a significant difference between the two states in chemists’ salaries.

The 98% confidence interval for the difference of the mean is $\left(\$6,917,\$11,493\right)$.

Explanation of Solution

Justification:

From part (d), the null hypothesis is rejected. Thus, there is enough evidence to support the claim that there is a significant difference between the two states in chemists’ salaries.

Confidence interval:

From the MINITAB output in part (c), the 98% confidence interval for the difference of the mean is $\left(\$6,917,\$11,493\right)$.