Question

Question #11 – Regression Analysis

Use the data provided to:

• Perform the “Tests to Check the Validity of a Regression”
  • Show both the calculated and critical values
• Estimate Y when X = 4 (round to 2 decimal places)

Use a level of significance of 5% (α = .05).

Clearly show the null and alternate hypothesis.

Graphs are not required.

X	Y
3	14
7	26
6	23
4	17
7	28
5	20
8	29
2	11

Answer 1

x	y	(x-x̅)²	(y-ȳ)²	(x-x̅)(y-ȳ)
3	14	5.06	49.00	15.75
7	26	3.06	25.00	8.75
6	23	0.56	4.00	1.50
4	17	1.56	16.00	5.00
7	28	3.06	49.00	12.25
5	20	0.06	1.00	0.25
8	29	7.56	64.00	22.00
2	11	10.56	100.00	32.50

	ΣX	ΣY	Σ(x-x̅)²	Σ(y-ȳ)²	Σ(x-x̅)(y-ȳ)
total sum	42	168	31.500	308.0	98.00
mean	5.25	21.00	SSxx	SSyy	SSxy

sample size ,   n =   8          
here, x̅ = Σx / n=   5.25   ,     ȳ = Σy/n =   21.00  
                  
SSxx =    Σ(x-x̅)² =    31.5000          
SSxy=   Σ(x-x̅)(y-ȳ) =   98.0          
                  
estimated slope , ß1 = SSxy/SSxx =   98.0   /   31.500   =   3.1111
                  
intercept,   ß0 = y̅-ß1* x̄ =   4.6667          
                  
so, regression line is   Ŷ =   4.6667   +   3.1111   *x

SSE=   (SSxx * SSyy - SS²xy)/SSxx =    3.111
      
std error ,Se =    √(SSE/(n-2)) =    0.720

........

slope hypothesis test               tail=   2
Ho:   ß1=   0          
H1:   ß1╪   0          
n=   8              
alpha =   0.05              
estimated std error of slope =Se(ß1) = Se/√Sxx =    0.720   /√   31.50   =   0.1283
                  
t stat = estimated slope/std error =ß1 /Se(ß1) =    3.1111   /   0.1283   =   24.2487
                  
t-critical value=    2.447   [excel function: =T.INV.2T(α,df) ]          
Degree of freedom ,df = n-2=   6              
decison : test stat > critical value , reject Ho              
Conclusion:   Reject Ho and conclude that slope is significantly different from zero              

so regression is significant

...............

Predicted Y at X=   4   is                  
Ŷ =   4.66667   +   3.111111   *   4   =   17.111