Question

Like father, like son: In 1906, the statistician Karl Pearson measured the heights of 1078 pairs of fathers and sons. The following table presents a sample of 6pairs, with height measured in inches, simulated from the distribution specified by Pearson.

Father's height			Son's height
73.6			76.5
69.0			69.1
73.6			74.9
66.7			68.8
70.1			73.3
72.3			71.9
		Send data to Excel

The regression equation is Y = .9834x + 2.7077

Construct a 95% confidence interval for the slope. Round the answers to at least four decimal places.

_____ < B1 < _____

P-Value ______

Reject or Do not Reject

Evidence?

Answer 1

	ΣX	ΣY	Σ(x-x̅)²	Σ(y-ȳ)²	Σ(x-x̅)(y-ȳ)
total sum	425.30	434.50	38.43	47.97	37.79
mean	70.88	72.42	SSxx	SSyy	SSxy

Sample size,   n =   6      
here, x̅ = Σx / n=   70.883          
ȳ = Σy/n =   72.417          
SSxx =    Σ(x-x̅)² =    38.4283      
SSxy=   Σ(x-x̅)(y-ȳ) =   37.8      
              
estimated slope , ß1 = SSxy/SSxx =   37.7917/38.4283=   0.9834  

  
intercept,ß0 = y̅-ß1* x̄ =   72.4167- (0.9834 )*70.8833=   2.7077      
              
Regression line is, Ŷ=   2.7077   + (   0.9834   )*x

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

SSE=   (SSxx * SSyy - SS²xy)/SSxx =    10.8028
std error ,Se =    √(SSE/(n-2)) =    1.6434
................

confidence interval for slope                  
α=   0.05              
t critical value=   t α/2 =    2.776   [excel function: =t.inv.2t(α/2,df) ]      
estimated std error of slope = Se/√Sxx =    1.6434/√38.4283=   0.265          
                  
margin of error ,E= t*std error =    2.776   *   0.265   =   0.736040
estimated slope , ß^ =    0.9834              
                  
lower confidence limit = estimated slope - margin of error =   0.9834   -   0.736   =   0.2474
upper confidence limit=estimated slope + margin of error =   0.9834   +   0.736   =   1.7195

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

Ho:   β1=   0
H1:   β1╪   0
n=   6  
alpha =   0.05  
estimated std error of slope =Se(ß1) = Se/√Sxx =    1.6434/√38.4283=   0.2651

t stat = estimated slope/std error =ß1 /Se(ß1) =    (0.9834-0)/0.2651=   3.71
Degree of freedom ,df = n-2=   4  

      
p-value =    0.0207  
decison :    p-value<α , reject Ho  
Conclusion:   Reject Ho and conclude that slope is significantly different from zero  

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

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