Chapter 7.1, Problem 40E

(a)

Section 1:

To determine

To find: The difference of estimates for each subject.

Answer to Problem 40E

Solution: The table for difference of estimates is as follows,

Jocko	Other	Difference
1410	1250	160
1550	1300	250
1250	1250	0
1300	1200	100
900	950	-50
1520	1575	-55
1750	1600	150
3600	3380	220
2250	2125	125
2840	2600	240

Explanation of Solution

Calculation: To calculate Difference follow the below mentioned steps in Minitab;

Step 1: Enter the variable name as ‘Jocko’ in column C1 and enter the provided data of ‘Jocko’s’. Enter the variable name as ‘Other’ in column C2 and enter the provided data of ‘Other’.

Step 2: Enter variable name as ‘Difference’ in column C3.

Step 3: Go to $\text{"Calc"}\to "\text{Calculater"}$

Step 4: In the dialog box that appears, select ‘Difference’ in ‘Store result in variable’.

Step 5: Enter 'Jocko-Other' in ‘Expression’ and click ‘OK’.

From Minitab result, the table for weight change is as follows,

Jocko	Other	Difference
1410	1250	160
1550	1300	250
1250	1250	0
1300	1200	100
900	950	-50
1520	1575	-55
1750	1600	150
3600	3380	220
2250	2125	125
2840	2600	240

Section 2:

To determine

To find: The mean and standard deviation of difference of estimates.

Answer to Problem 40E

Solution: The required mean is 114.0 and the standard deviation is 114.4

Explanation of Solution

Calculation: Follow the below mentioned steps in Minitab,

Step 1: Follow the step 1 to 5 performed in part (a).

Step 2: Go to $\text{"Stat"}\to "\text{Basic}\text{statistics"}\to "\text{Display Discriptive statistics"}$. Then click on ‘OK’.

Step 3: In dialog box that appears select ‘Difference’ under the field marked as ‘Variables’. Then click on ‘Statistics’.

Step 4: In dialog box that appears select ‘None’ and then ‘Mean’. Finally, click ‘OK’ on both dialog boxes.

From Minitab result, the mean is 114.0 and the standard deviation is 114.4.

(b)

To determine

To test: The hypothesis that the estimates of the two garages do not differ significantly.

Answer to Problem 40E

Solution: The mean estimates for Jocko and Other differ significantly at 5% significance level.

Explanation of Solution

Calculation:

The null hypothesis is the hypothesis of no difference. If ${\mu }_1\text{and }{\mu }_2$ are mean estimate for ‘jocko’ and ‘Other’ respectively and the null and alternative hypotheses are as follows,

The null hypothesis assumes that there is no significant difference and is formulated as,

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

And the alternative hypothesis assumes a significant change in the average estimates and is formulated as,

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

The formula of test statistic for paired t-test is as follows,

$t=\frac{\overline{d}}{\left(\frac{s}{\sqrt{n}}\right)}$

In the above formula,

$\begin{array}{c}\overline{d}=\text{mean}\text{difference}\\ s=\text{sample standard deviation}\\ n=\text{ sample size}\end{array}$

The test statistic is calculated as,

$\begin{array}{c}t=\frac{114}{\left(\frac{114.4}{\sqrt{10}}\right)}\\ =3.15\end{array}$

The formula for degrees of freedom is as follows,

$\text{Degrees}\text{of}\text{freedom = }n-1$

And it is calculated as,

$\begin{array}{c}\text{Degrees}\text{of}\text{freedom = }n-1\\ =10-1\\ =9\end{array}$

To calculate the P-value follow the below mentioned steps in Minitab,

Step 1: Follow the step 1 to 5 performed in part (a).

Step 2: Go to $\text{Stat}\to \text{Basic}\text{statistics}\to \text{Paired}t$

Step 3: In the dialogue box that appears, select samples in columns.

Step 4: Enter ‘Jocko’ under the field marked as ‘First sample’ and ‘Other’ under the field marked as ‘Second sample’.

Step 5: In the dialogue box that appears, Click on ‘Option’. Enter 95.0 in the field of Confidence interval, 0.0 in ‘Test mean’ and ‘not equal’ in ‘Alternative’.

From Minitab results, the P-value is 0.012.

Conclusion: Since, the P-value is 0.012 which is less than 0.05, the significance level. So the null hypothesis is rejected and it is concluded that the mean estimate for ‘Jocko’ is different from mean estimate for ‘Other’.

(c)

To determine

To find: The confidence interval.

Answer to Problem 40E

Solution: The required confidence interval is $\left(32.2,195.8\right)$.

Explanation of Solution

Calculation:

The confidence interval is an interval for which there is 95% chances that it contains the population parameter (population mean).

To calculate the confidence interval, follow the below mentioned steps in Minitab,

Step 1: Follow the step 1 to 5 performed in part (a).

Step 2: Go to $\text{Stat}\to \text{Basic}\text{statistics}\to \text{Paired}t$.

Step 3: In the dialogue box that appears, select samples in columns.

Step 4: Enter ‘Jocko’ under the field marked as ‘First sample’ and ‘Other’ under the field marked as ‘Second sample’.

Step 5: In the dialogue box that appears, Click on ‘Option’. Enter 95.0 in the field of Confidence interval, 0.0 in ‘Test mean’ and ‘not equal’ in ‘Alternative’.

From Minitab result, the confidence interval is $\left(32.2,195.8\right)$.

(d)

To determine

To explain: The repayment that is recommended to the insurance company seeks.

Answer to Problem 40E

Solution: The seeking repayment is recommended between $32.2 and $195.8.

Explanation of Solution

The confidence interval of the mean difference is $\left(32.2,195.8\right)$ which is calculated in part (c). So the seeking repayment is recommended between $32.2 and $195.8. The conclusion is made with 95% confidence.