
Concept explainers
a.
Find the
Check whether the correlation coefficient is greater than zero.
a.

Answer to Problem 56CE
The
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
The hypotheses are given below:
Null hypothesis:
That is, the correlation in the population is less than or equal to zero.
Alternative hypothesis:
That is, the correlation in the population is positive.
Test statistic:
The test statistic is as follows:
Here, the
The test statistic is as follows:
Degrees of freedom:
The level of significance is 0.05. Therefore,
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
Otherwise, fail to reject H0.
Conclusion:
The value of test statistic is –0.415 and the critical value is 1.943.
Here,
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.
Find the regression equation and check whether it can be concluded that the slope of the regression line is negative.
b.

Answer to Problem 56CE
The regression equation is
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,
The test hypotheses are:
Define
Null hypothesis:
That is, the slope of the regression line is not less than zero.
Alternate hypothesis:
That is, the slope of the regression line is less than zero.
Consider the level of significance as 0.05.
The standard error of
Test statistic:
The t-test statistic is:
Where,
Thus,
Here, the sample size is
Step-by-step software procedure to obtain the critical value,
• 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,
Otherwise fail to reject H0.
Conclusion:
The t-calculated value is –0.144 and the critical value is 1.943.
That is,
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.
Find the residual for each observation and find the company that has the largest residual.
c.

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.
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Chapter 13 Solutions
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