Question

The data in the accompanying table represent the rate of return of a certain company stock for 11​ months, compared with the rate of return of a certain index of 500 stocks. Both are in percent. Complete parts ​(a) through ​(d) below. LOADING... Click the icon to view the data table. ​(a) Treating the rate of return of the index as the explanatory​ variable, x, use technology to determine the estimates of beta 0 and beta 1. The estimate of beta 0 is nothing. ​(Round to four decimal places as​ needed.) The estimate of beta 1 is nothing. ​(Round to four decimal places as​ needed.) ​(b) Assuming the residuals are normally​ distributed, test whether a linear relation exists between the rate of return of the​ index, x, and the rate of return for the company​ stock, y, at the alphaequals0.10 level of significance. Choose the correct answer below. State the null and alternative hypotheses. A. Upper H 0​: beta 1equals0 Upper H 1​: beta 1not equals0 B. Upper H 0​: beta 0equals0 Upper H 1​: beta 0greater than0 C. Upper H 0​: beta 1equals0 Upper H 1​: beta 1greater than0 D. Upper H 0​: beta 0equals0 Upper H 1​: beta 0not equals0 Determine the​ P-value for this hypothesis test. ​P-valueequals nothing ​(Round to three decimal places as​ needed.) State the appropriate conclusion at the alphaequals0.10 level of significance. Choose the correct answer below. A. Reject Upper H 0. There is not sufficient evidence to conclude that a linear relation exists between the rate of return of the index and the rate of return of the company stock.   B. Do not reject Upper H 0. There is sufficient evidence to conclude that a linear relation exists between the rate of return of the index and the rate of return of the company stock. C. Do not reject Upper H 0. There is not sufficient evidence to conclude that a linear relation exists between the rate of return of the index and the rate of return of the company stock.   D. Reject Upper H 0. There is sufficient evidence to conclude that a linear relation exists between the rate of return of the index and the rate of return of the company stock. ​(c) Assuming the residuals are normally​ distributed, construct a​ 90% confidence interval for the slope of the true​ least-squares regression line. Lower​ bound: nothing ​(Round to four decimal places as​ needed.) Upper​ bound: nothing ​(Round to four decimal places as​ needed.) ​(d) What is the mean rate of return for the company stock if the rate of return of the index is 3.15​%? The mean rate of return for the company stock if the rate of return of the index is 3.15​% is nothing​%. ​(Round to three decimal places as​ needed.) Month Rates of return of the​ index, x Rates of return of the company​ stock, y ​Apr-07 4.23             3.28             ​May-07 3.25             5.09             ​Jun-07 negative 1.78             0.54             ​Jul-07 negative 3.20             2.88             ​Aug-07 1.29             2.69             ​Sept-07 3.58             7.41             ​Oct-07 1.48             negative 4.83             ​Nov-07 negative 4.40             negative 2.38             ​Dec-07 negative 0.86             2.37             ​Jan-08 negative 6.12             negative 4.27             ​Feb-08 negative 3.48             negative 3.77            

Answer 1

The regression equation is defined as,

The regression analysis is done in excel by following steps

Step 1: Write the data values in excel. The screenshot is shown below,

Step 2: DATA > Data Analysis > Regression > OK. The screenshot is shown below,

Step 3: Select Input Y Range: 'y' column, Input X Range: 'x' column then OK. The screenshot is shown below,

The result is obtained. The screenshot is shown below,

The regression equation is,

a)

The estimate of

The estimate of

b)

The null and alternative hypotheses.

Correct Answer: A)

Explanation: The hypothesis test for the linear relation is determine by the estimate of the slope of independent variable. If the slope is significant, we can conclude that there is significant linear relation between the dependent and independent variable.

P-value

Answer: P-value = 0.026

Explanation: From the regression output summary, the P-value for the estimates slope coefficient of independent variable is,

	Coefficients	P-value
rates of the return of the index, X	0.764814665	0.026071433

Conclusion

Correct Answer: C. Do not reject Upper H₀. There is not sufficient evidence to conclude that a linear relation exists between the rate of return of the index and the rate of return of the company stock.

Explanation: P-value = 0.026 > 0.01

Since the p-value is greater than 0.01 at 1% significance level. The null hypothesis is not rejected. Hence it is concluded that there is no significant linear relation between dependent and independent variable.

90% confidence interval for the slope

Lower Bound: 0.1142

Upper Bound 1.4154

Explanation: From the regression output summary,

	Coefficients	Lower 95%	Upper 95%
rates of the return of the index, X	0.764814665	0.114242663	1.415386667

Mean rate of return at x=3.15%

Answer: The mean rate of return for the company stock if the rate of return of the index is 3.15% is 3.639%

Explanation:

The regression equation is,

For X = 3.15%