Question

For the data set (−3,−1),(2,2),(5,4),(8,6),(10,11),

carry out the hypothesis test H0 β1=0 HA β1≠0

Determine the value of the test statistic and the associated p-value.

Answer 1

x	y	(x-x̅)²	(y-ȳ)²	(x-x̅)(y-ȳ)
-3	-1	54.7600	29.1600	39.960
2	2	5.7600	5.7600	5.760
5	4	0.3600	0.1600	-0.240
8	6	12.9600	2.5600	5.760
10	11	31.3600	43.5600	36.960

	ΣX	ΣY	Σ(x-x̅)²	Σ(y-ȳ)²	Σ(x-x̅)(y-ȳ)
total sum	22.00	22.00	105.20	81.20	88.20
mean	4.40	4.40	SSxx	SSyy	SSxy

Sample size,   n =   5      
here, x̅ = Σx / n=   4.400          
ȳ = Σy/n =   4.400          
SSxx =    Σ(x-x̅)² =    105.2000      
SSxy=   Σ(x-x̅)(y-ȳ) =   88.2      
              
estimated slope , ß1 = SSxy/SSxx =   88.2/105.2=   0.8384      

slope hypothesis test      
Ho:   β1=   0
H1:   β1╪   0
n=   5  
alpha =   0.1  
estimated std error of slope =Se(ß1) = Se/√Sxx =    1.5549/√105.2=   0.1516
t stat = estimated slope/std error =ß1 /Se(ß1) =    (0.8384-0)/0.1516=   5.53
Degree of freedom ,df = n-2=   3  
      
p-value =    0.0116  
decison :    p-value<α , reject Ho  
Conclusion:   Reject Ho and conclude that slope is significantly different from zero