Chapter 13, Problem 56CE

a.

To determine

Find the correlation coefficient.

Check whether the correlation coefficient is greater than zero.

Answer to Problem 56CE

The correlation coefficient is –0.059.

There is not enough evidence to infer that the population correlation is positive.

Explanation of Solution

Step-by-step procedure to obtain the correlation coefficient using 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 Sheet6!$H$1:$I$9.
• Click on OK.

Output obtained using MegaStat is given as follows:

The correlation coefficient is –0.059.

Denote the population correlation as $\rho$.

The hypotheses are given below:

Null hypothesis:

$H_0:\rho \le 0$

That is, the correlation in the population is less than or equal to zero.

Alternative hypothesis:

$H_1:\rho >0$

That is, the correlation in the population is positive.

Test statistic:

The test statistic is as follows:

$t=\frac{r\sqrt{n-2}}{\sqrt{1-r^2}}$

Here, the sample size is 8 and the correlation coefficient is –0.059.

The test statistic is as follows:

$\begin{array}{c}t=\frac{-0.059\sqrt{8-2}}{\sqrt{1-\left(-{0.059}^2\right)}}\\ =\frac{-0.059\times \sqrt{6}}{\sqrt{1.003481}}\\ =-0.145\end{array}$

Degrees of freedom:

$\begin{array}{c}df=n-2\\ =8-2\\ =6\end{array}$

The level of significance is 0.05. Therefore, $1-\alpha =0.95$.

Critical value:

Step-by-step software procedure to obtain the critical value using EXCEL software:

• Open an EXCEL file.
• In cell A1, enter the formula “=T.INV (0.95, 6)”.

Output obtained using the EXCEL is given as follows:

Decision rule:

Reject the null hypothesis H₀, if $t\text{-calculated}>t\text{-critical value}$.

Otherwise, fail to reject H₀.

Conclusion:

The value of test statistic is –0.415 and the critical value is 1.943.

Here, $t\text{-calculated}\left(=-0.415\right)<t\text{-critical value}\left(=1.943\right)$.

By the rejection rule, do not reject the null hypothesis.

Thus, there is not enough evidence to infer that the population correlation is positive.

b.

To determine

Find the regression equation and check whether it can be concluded that the slope of the regression line is negative.

Answer to Problem 56CE

The regression equation is $\text{Return}=17.9321-0.0322\text{Growth}$.

There is sufficient evidence to conclude that the slope of the regression line is not negative at 5% level of significance.

Explanation of Solution

Step-by-step procedure to obtain the ‘Regression equation’ using the MegaStat software:

• In an EXCEL sheet enter the data values of x and y.
• Go to Add-Ins > MegaStat > Correlation/Regression > Regression Analysis.
• Select input range as ‘Sheet6!$I$1:$I$9’ under Y/Dependent variable.
• Select input range ‘Sheet6!$H$1:$H$9’ under X/Independent variables.
• Click on OK.

Output using the Mega Stat software is given below:

From the output, the regression equation is, $\text{Return}=17.9321-0.0322\text{Growth}$

The test hypotheses are:

Define $\beta$ as the slope of the regression line.

Null hypothesis:

$H_0:\beta \ge 0$

That is, the slope of the regression line is not less than zero.

Alternate hypothesis:

$H_1:\beta <0$

That is, the slope of the regression line is less than zero.

Consider the level of significance as 0.05.

The standard error of $b$$\left(=-0.0322\right)$ is given as $s_b=0.2226$.

Test statistic:

The t-test statistic is:

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

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

Thus,

$\begin{array}{c}t=\frac{b}{s_b}\\ =\frac{-0.0322}{0.2226}\\ =-0.144\end{array}$

Here, the sample size is $n=8$. Thus, the degrees of freedom is:

$\begin{array}{c}n-2=8-2\\ =6\end{array}$

Step-by-step software procedure to obtain the critical value, $t_{\alpha }$ using EXCEL software is as follows:

•         Open an EXCEL file.

•         In cell A1, enter the formula “=T.INV(0.95,6)”.

Output using the EXCEL is given as follows:

From the EXCEL output, the critical value is 1.943.

Decision based on critical value:

Reject the null hypothesis if, $t-\text{calculated value <}-t_{\alpha }$.

Otherwise fail to reject H₀.

Conclusion:

The t-calculated value is –0.144 and the critical value is 1.943.

That is, $t-\text{calculated value}\left(=-0.144\right)\text{ >}-t_{\alpha }\left(=-1.943\right)$.

Thus, the null hypothesis is not rejected.

Hence, there is sufficient evidence to conclude that the slope of the regression line is not negative at 5% level of significance.

c.

To determine

Find the residual for each observation and find the company that has the largest residual.

Answer to Problem 56CE

Company RC has the largest residual.

Explanation of Solution

The error estimate is the difference between actual return and estimated return. The error estimates for each of the companies are as follows:

Company	y	y-cap	Error
AT	23.1	17.6745	5.4255
B	13.2	17.42978	–4.22978
GD	24.2	16.92746	7.27254
H	11.1	17.8516	–6.7516
LC	10.1	16.79222	–6.69222
NG	10.8	17.7389	–6.9389
RC	27.3	17.65196	9.64804
UT	20.1	17.82906	2.27094

From the table, it is clear that company RC has the largest residual.