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 3.28
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​: β1=0

H1​:β1>0

C. H0​: β0=0

H1​:β0>0

D. H0​:β0=0

H1​:β0≠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.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.  

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.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.

D.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.

​(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 3.45​%?

The mean rate of return for the company stock if the rate of return of the index is 3.45​% is ____​%. ​(Round to three decimal places as​ needed.)

Answer 1

X	Y	(x-x̅)²	(y-ȳ)²	(x-x̅)(y-ȳ)
4.33	3.28	23.602	6.056	11.956
3.35	5.09	15.040	18.241	16.563
-1.78	0.54	1.567	0.078	0.349
-3.2	2.88	7.139	4.247	-5.506
1.29	2.69	3.306	3.500	3.402
3.58	7.41	16.877	43.440	27.077
1.48	-4.83	4.033	31.912	-11.344
-4.4	-2.38	14.991	10.234	12.386
-0.86	2.37	0.110	2.405	-0.515
-6.12	-4.27	31.268	25.899	28.457
-3.48	-3.77	8.713	21.060	13.546

	ΣX	ΣY	Σ(x-x̅)²	Σ(y-ȳ)²	Σ(x-x̅)(y-ȳ)
total sum	-5.81	9.01	126.646	167.1	96.3709182
mean	-0.53	0.82	SSxx	SSyy	SSxy

sample size ,n    =11

here,   x̅   =-0.53,ȳ =0.82

SSxx = Σ(x-x̅)² = 126.65

SSxy=Σ(x-x̅)(y-ȳ) =96.4

a)

intercept,ß0 = y̅-ß1* x̄ =1.2210

slope , ß1 = SSxy/SSxx =0.7610

b)

Ho:ß1=0

H1:ß1╪0

SSE=(Sx*Sy - S²xy)/Sx = 93.74

std error ,Se = √(SSE/(n-2)) = 3.22731

estimated std error of slope =Se(ß1) = s/√Sxx = 3.22731/ 93.74    =    0.2868

t stat = ß1 /Se(ß1) = 2.65344

p-value = 0.0263

decision : p-value<α =0.10, reject Ho

.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.

c)

confidence interval for slope

t critical value=t α/2 = 1.833    [excel function:    =t.inv.2t(0.10,9) ]

margin of error ,E=t*std error = 1.833* 0.2868    = 0.526

lower confidence limit = ß̂1-E =0.2353

upper confidence limit=ß̂1+E =1.2866

d)

Ŷ    =1.22+0.7609*x

x=3.45%

Ŷ    =1.22+0.7609*3.45 = 3.846%