Question

A publisher of college textbooks conducted a study to relate profit per text y to cost of sales x over a 6-year period when its sales force (and sales costs) were growing rapidly.  These inflation-adjusted data (in thousands of dollars) were collected:

x  = {5.0, 5.6, 6.1, 6.8, 7.4, 8.6}                y = {16.5, 22.4, 24.9, 28.8, 31.5, 35.8}

Multiple Choice question:

Question 1. What is the correct null and alternative hypothises?

A. H_o: β₁ = 0   vs. H₁: β₁ not= 0

B. H_o: β₁ not= 0   vs. H₁: β₁ = 0

C. none of these

Question 2. What is the correct decision?

A. Reject Ha

B. Reject Ho

C. Do not Reject Ho

D. inconclusive

Question 3. What is the conclusion?

A. There is enough evidence to say that x contributes information to the prediction of y.

B. There is not enough evidence to say that x contributes information to the prediction of y

C. x explains y

D. y explains x

Question 4. Construct a 95% CI for Beta one

Answer 1

1)A. H_o: β₁ = 0   vs. H₁: β₁ not= 0

2)

X	Y	(x-x̅)²	(y-ȳ)²	(x-x̅)(y-ȳ)
5	16.5	2.5069444	103.0225	16.07083
5.6	22.4	0.9669444	18.0625	4.179167
6.1	24.9	0.2336111	3.0625	0.845833
6.8	28.8	0.0469444	4.6225	0.465833
7.4	31.5	0.6669444	23.5225	3.960833
8.6	35.8	4.0669444	83.7225	18.4525

	ΣX	ΣY	Σ(x-x̅)²	Σ(y-ȳ)²	Σ(x-x̅)(y-ȳ)
total sum	39.5	159.9	8.488333	236.015	43.975
mean	6.583333333	26.65	SSxx	SSyy	SSxy

sample size ,   n =   6      
here, x̅ =   6.583333333       ȳ =   26.65
              
SSxx =    Σ(x-x̅)² =    8.488333333      
SSxy=   Σ(x-x̅)(y-ȳ) =   43.975      
              
slope ,    ß1 = SSxy/SSxx =   5.18064      
SSE=   (Sx*Sy - S²xy)/Sx =    8.20
      
std error ,Se =    √(SSE/(n-2)) =    1.4315
n=   6              
alpha=   0.05              
estimated std error of slope =Se(ß1) =                s/√Sxx =    0.4913
                  
t stat =    ß1 /Se(ß1) =        10.54421697      
                  
t-critical value=        2.776445105          
                  
p-value =    0.0005              
decision :    p-value<α , reject Ho              
answer: B. Reject Ho

3)

A. There is enough evidence to say that x contributes information to the prediction of y.

4)

confidence interval for slope              
t critical value=   t α/2 =    2.776445105      
margin of error ,E=       t*std error =        1.3641
lower confidence limit =        ß̂1-E =   3.8165
upper confidence limit=       ß̂1+E =   6.5448