Question

Consider the following data: (If you want to check data entry, the sample covariance is 80) Can we reject the hypothesis that the coefficient for Hours is zero with 90% confidence?

Sales	Hours
12	68
9	71
18	120
21	110
18	110
11	90
16	120

Answer 1

	ΣX	ΣY	Σ(x-x̅)²	Σ(y-ȳ)²	Σ(x-x̅)(y-ȳ)
total sum	689	105	2947.714286	116.0	480.00
mean	98.43	15.00	SSxx	SSyy	SSxy

sample size ,   n =   7          
here, x̅ = Σx / n=   98.43   ,     ȳ = Σy/n =   15.00  
                  
SSxx =    Σ(x-x̅)² =    2947.7143          
SSxy=   Σ(x-x̅)(y-ȳ) =   480.0          
                  
estimated slope , ß1 = SSxy/SSxx =   480.0   /   2947.714   =   0.1628
                  
intercept,   ß0 = y̅-ß1* x̄ =   -1.0279          
                  
so, regression line is   Ŷ =   -1.0279   +   0.1628   *x

Ho:   ß1=   0          
H1:   ß1╪   0          
n=   7              
alpha =   0.1              
estimated std error of slope =Se(ß1) = Se/√Sxx =    2.751   /√   2947.71   =   0.0507
                  
t stat = estimated slope/std error =ß1 /Se(ß1) =    0.1628   /   0.0507   =   3.2138
                  
t-critical value=    2.015   [excel function: =T.INV.2T(α,df) ]          
Degree of freedom ,df = n-2=   5              
p-value =    0.0236              
decison :    p-value<α , reject Ho              
Conclusion:   Reject Ho and conclude that slope is significantly different from zero        

   

Thanks in advance!

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