Question

A. The heights and weights of six men are given below:

Height (meters) 2.00 1.80 1.85

Weight (kgs) 85.0 78.0 80.0

(i) Determine the lines of regression of Height on Weight and Weight on Height.

Interpret the results as well.

(ii) Estimate weight when height is 1.70 meters.

(iii) Estimate height when weight is 70.0 kg.

(iv) Find the correlation coefficient between Height and Weight and comment on

its value. [4 Marks]

(v) Test the hypothesis that the correlation coefficient between Height and Weight

is at least 0.8. Use a = 0.05 . [5 Marks]

B. Draw scatter diagrams for the different values of correlation co-efficient ‘r’ given

below:

(i) r = 1 (ii) r = – 1 (iii) r = 0 (iv) r = 0.90 (v) r = 0.10 (vi) r = – 0.88

Explain the type of relationship you would expect between ‘x’ and ‘y’ in each of the above cases. [09 Marks]

C. Find a 95% confidence interval for the average life time of all the television sets produced by a company when a random sample of 25 television sets lasted an average of 10,000 hours with a standard deviation of 1,000 hours. Interpret your result.

Answer 1

1)

Height (x) and Weight (y):

	ΣX	ΣY	Σ(x-x̅)²	Σ(y-ȳ)²	Σ(x-x̅)(y-ȳ)
total sum	5.65	243	0.021666667	26.0	0.75
mean	1.88	81.00	SSxx	SSyy	SSxy

sample size ,   n =   3          
here, x̅ = Σx / n=   1.88   ,     ȳ = Σy/n =   81.00  
                  
SSxx =    Σ(x-x̅)² =    0.0217          
SSxy=   Σ(x-x̅)(y-ȳ) =   0.8          
                  
estimated slope , ß1 = SSxy/SSxx =   0.8   /   0.022   =   34.6154
                  
intercept,   ß0 = y̅-ß1* x̄ =   15.8077          
                  
so, regression line is   Ŷ =   15.8077   +   34.6154   *x

--------

Height (y) and Weight (x):

	ΣX	ΣY	Σ(x-x̅)²	Σ(y-ȳ)²	Σ(x-x̅)(y-ȳ)
total sum	243	5.65	26	0.0	0.75
mean	81.00	1.88	SSxx	SSyy	SSxy

sample size ,   n =   3          
here, x̅ = Σx / n=   81.00   ,     ȳ = Σy/n =   1.88  
                  
SSxx =    Σ(x-x̅)² =    26.0000          
SSxy=   Σ(x-x̅)(y-ȳ) =   0.8          
                  
estimated slope , ß1 = SSxy/SSxx =   0.8   /   26.000   =   0.0288
                  
intercept,   ß0 = y̅-ß1* x̄ =   -0.4532          
                  
so, regression line is   Ŷ =   -0.4532   +   0.0288   *x

2)

Predicted Y at X=   1.7   is                  
Ŷ =   15.80769   +   34.615385   *   1.7   =   74.654

3)

Predicted Y at X=   70   is                  
Ŷ =   -0.45321   +   0.028846   *   70   =   1.566

4)

SSE=   (SSxx * SSyy - SS²xy)/SSxx =    0.038
      
std error ,Se =    √(SSE/(n-2)) =    0.196
      
correlation coefficient ,    r = Sxy/√(Sx.Sy) =   0.9993

5)

Ho:   ρ = 0.8 tail=   1  
Ha:   ρ > 0.8
n=   3              
alpha,α =    0.05              
correlation , r=   0.1993              
t-test statistic = r*√(n-2)/√(1-r²) =        0.203          
DF=n-2 =   1              
p-value =    0.4361              
Decison:   P value > α, So, Do not reject Ho              

Thanks in advance!

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