Chapter 13, Problem 48CE

a.

To determine

Construct a scatter diagram.

Answer to Problem 48CE

The scatter diagram of the data is as follows:

Explanation of Solution

It is given that ‘Estimated cost’ is the dependent variable.

Step-by-step procedure to obtain the scatterplot using the MegaStat software:

• In an EXCEL sheet enter the data values of x and y.
• Go to Add-Ins > MegaStat > Correlation/Regression > Scatterplot.
• Enter horizontal axis as $B$1:$B$15 and vertical axis as $A$1:$A$15.
• Click on OK.

From the scatterplot of the data, it indicates an inverse relationship.

b.

To determine

Find the correlation coefficient.

Answer to Problem 48CE

The correlation coefficient is –0.822.

Explanation of Solution

Step-by-step procedure to obtain the correlation coefficient using the MegaStat software:

• In an EXCEL sheet enter the data values of x and y.
• Go to Add-Ins > MegaStat > Correlation/Regression > Correlation matrix.
• Enter Input Range as $A$1:$B$15.
• Click on OK.

Output obtained using the MegaStat is given as follows:

The correlation coefficient is –0.822. Since the correlation coefficient is negative and close to –1, there is a strong negative correlation between ‘Estimated cost’ and ‘age’.

c.

To determine

Interpret the slope of the regression equation.

Explanation of Solution

The estimated regression equation is $\text{Estimated cost}=18,358-\text{1,534 Age}$. Thus, the slope of the regression equation is –1,534.

The interpretation is that for each unit added to age, there is a decrease of $1,534 in cost.

d.

To determine

Estimate the cost of a five-year-old car.

Answer to Problem 48CE

The estimated value of cost of a five-year-old car is $10,688.

Explanation of Solution

Substitute x as 5 in the regression equation.

$\begin{array}{c}\text{Estimated cost}=18,358-1,534x\\ =18,358-1,534\left(5\right)\\ =18,358-7,670\\ =10,688\end{array}$

Thus, the estimated value of cost of a five-year-old car is $10,688.

e.

To determine

Explain the given portion of the software output.

Explanation of Solution

From the output p-value corresponding to the variable age is 0. That is, the p-value is less than any common level of significance. Thus, the variable age is significant.

f.

To determine

Test whether the slope is significant or not.

Interpret the result.

Check whether there is any significant relationship between the two variables.

Answer to Problem 48CE

There is sufficient evidence to conclude that the slope of the regression line is different from zero at the 10% level of significance.

Explanation of Solution

It is given that the regression equation is $\text{Estimated cost =18,358}-\text{1,534 Age}$. The sample size is 14 and the standard error of the slope is 306.3.

From the regression equation, the estimated slope of the regression line is $b=-1,534$ and the standard error of $b$ is $s_b=306.3$.

Let $\beta$ be the slope of the regression line.

The given test hypotheses are as follows:

Null hypothesis:

$H_0:\beta =0$

That is, the slope of the regression line is equal to zero.

Alternate hypothesis:

$H_1:\beta \ne 0$

That is, the slope of the regression line is not equal to zero.

It is given that the level of significance is 0.10.

The standard error of $b$ is given as $s_b=306.3$.

Test statistic:

The t-test statistic is as follows:

$t=\frac{b-0}{s_b}$,

Where, $b$ is the slope of the computed regression line and $s_b$ is the standard error of $b$.

Thus, the following is obtained:

$\begin{array}{c}t=\frac{b}{s_b}\\ =\frac{-1,534}{306.3}\\ =5.008\\ \approx 5.01\end{array}$

Here, the sample size is $n=14$. Thus, the degrees of freedom is as follows:

$\begin{array}{c}n-2=14-2\\ =12\end{array}$

Critical value:

Software procedure:

Step-by-step software procedure to obtain the critical value $t_{\frac{\alpha }{2}}$ using the EXCEL software:

• Open an EXCEL file.
• In cell A1, enter the formula “=T.INV(0.05,12)”.

Output obtained using the EXCEL is given as follows:

From the EXCEL output, the critical value is –1.782 ($-t_{\frac{\alpha }{2}}$).

Decision based on critical value:

Reject the null hypothesis, if $\left|t\right|\text{>}\left|t_{\frac{\alpha }{2}}\right|$. Otherwise, fail to reject H₀.

Conclusion:

The t-calculated value is 5.01 and the critical value is 1.782.

That is, $t\text{-calculated value}\left(5.01\right)>t_{\frac{\alpha }{2}}\left(1.782\right)$.

Thus, the null hypothesis is rejected.

Hence, there is sufficient evidence to conclude that the slope of the regression line is different from zero at the 10% level of significance.

Since the slope of the regression line is different from zero, there is a relationship between age and cost.