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.

Month   Rates_of_return_of_the_index_-_x   Rates_of_return_of_the_company_stock_-_y
Apr-07   4.33   3.38
May-07   3.35   5.09
Jun-07   -1.78   0.54
Jul-07   -3.20   2.88
Aug-07   1.29   2.69
Sept-07   3.58   7.41
Oct-07   1.48   -4.83
Nov-07   -4.40   -2.38
Dec-07   -0.86   2.37
Jan-08   -6.12   -4.27
Feb-08   -3.48   -3.77

​(a) Treating the rate of return of the index as the explanatory​ variable, x, use technology to determine the estimates of β0 and β1.

The estimate of β0 is __?__.

​(Round to four decimal places as​ needed.)

The estimate of β1 is __?__.

​(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

α=0.10 level of significance.

Choose the correct answer below.

State the null and alternative hypotheses.

A.

H0​: β1=0

H1​: β1>0

B.

H0​: β0=0

H1​: β0≠0

C.

H0​: β0=0

H1​: β0>0

D.

H0​: β1=0

H1​: β1≠0

Determine the​ P-value for this hypothesis test.

​P-value=__?__

​(Round to three decimal places as​ needed.)

State the appropriate conclusion at the α=0.10 level of significance.

Choose the correct answer below.

A.

Do not reject H0. 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.

B.

Do not reject 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.  

C.

Reject H0. 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.

D.

Reject 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. 

​

(c) Assuming the residuals are normally​ distributed, construct a​ 90% confidence interval for the slope of the true​ least-squares regression line.

Lower​ bound: __?__

​(Round to four decimal places as​ needed.)

Upper​ bound: __?__

​(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 3.25​%?

The mean rate of return for the company stock if the rate of return of the index is 3.25​%

is __?__​%.

​(Round to three decimal places as​ needed.)

Answer 1

Sol:

The statistical software output for this problem is:

Simple linear regression results:
Dependent Variable: y
Independent Variable: x
y = 1.2321253 + 0.76478103 x
Sample size: 11
R (correlation coefficient) = 0.66485999
R-sq = 0.4420388
Estimate of error standard deviation: 3.223176

Parameter estimates:

Parameter	Estimate	Std. Err.	Alternative	DF	T-Stat	P-value
Intercept	1.2321253	0.98352766	≠ 0	9	1.2527612	0.2419
Slope	0.76478103	0.28640965	≠ 0	9	2.6702348	0.0256

Parameter estimates:

Parameter	Estimate	Std. Err.	DF	90% L. Limit	90% U. Limit
Intercept	1.2321253	0.98352766	9	-0.57079201	3.0350425
Slope	0.76478103	0.28640965	9	0.23975979	1.2898023

Analysis of variance table for regression model:

Source	DF	SS	MS	F-stat	P-value
Model	1	74.074195	74.074195	7.130154	0.0256
Error	9	93.499769	10.388863
Total	10	167.57396

Predicted values:

X value	Pred. Y	s.e.(Pred. y)	95% C.I. for mean	95% P.I. for new
3.25	3.7176636	1.4544412	(0.42748898, 7.0078382)	(-4.2816332, 11.71696)

Hence,

a) Estimate of P0 is 1.2321

Estimate of P1 is 0.7648

b) P - value is 0.0256

c) Lower bound = 0.2398

Upper bound = 1.2898

d) 3.7177

Note - I have rounded all the values to 4 decimal places. Please round off accordingly as mentioned in the question.