Question

Can hair length be used to predict the height of a person? To try to answer this, the following random subset of data were collected from our class’s data:

Hair length:       8        12        18        24        1        4          24        2                   

                       

Height:            68       60        61        67        67        70        68        70

1. Is there a significant correlation between hair length and height? Use α=.05.
2. Calculate the coefficient of determination. What does this value say about the relationship between hair length and height?
3. What would you expect the height to be of a person with 4 inches of hair?
4. Create a 90% prediction interval for the height of a person with 4 inches of hair.

Answer 1

X :- Hair length

Y :- Height

ΣX = 93    ΣY =531    ΣX * Y = 6089    ΣX2 = 1705 ΣY2 = 35347

Sxx =Σ (Xi - X̅ )2 = 623.875
Syy = Σ( Yi - Y̅ )2 = 101.875
Sxy = Σ (Xi - X̅ ) * (Yi - Y̅) = -83.875

r = -0.3327

To Test :-

H0 :- ρ = 0
H1 :- ρ ≠ 0

Test Statistic :-
t = (r * √(n - 2) / (√(1 - r2))
t = ( -0.3327 * √(8 - 2) ) / (√(1 - 0.1107) )
t = -0.8642
Test Criteria :-
Reject null hypothesis if t > t(α,n-2) OR t < -t(α,n-2)
t(α/2,n-2) = t(0.05/2 , 8 - 2 ) = ± 2.4469
-2.4469 < -0.8642 < 2.4469
Result :- We fail to Reject null hypothesis

Decision based on P value
P - value = P ( t > 0.8642 ) = 0.4207
Reject null hypothesis if P value < α = 0.05 level of significance
P - value = 0.4207 > 0.05 ,hence we fail to reject null hypothesis
Conclusion :- We Accept H0

There is statistically no correlation between variables.

R2 = r2 = 0.1107

11.07% of variation in height is explained by hair length.

Ŷ = 67.9379 + -0.1344X
Ŷ = 67.4

Predictive Confidence Interval of
Ŷ = 67.9379 + -0.1344X
Ŷ = 67.4

t(30.645/2) = t(0.1/2) = 1.943
X̅ = (Xi / n ) = 93/8 = 11.625


90% Predictive confidence interval is (59.066 < < 75.734)