Question

1. Researchers are studying the effect of alcohol consumption on a person's short-term memory. Six volunteer subjects are assigned to consume a certain number of drinks X. A short time later, the subjects are given a memory test, and their score Y on the test is recorded. The data are shown below, together with the means and standard deviations:

Subject 1 2 3 4 5 6 mean std dev

No. of Drinks 4 2 8 3 1 6 4 2.61

Test Score 81 90 22 58 80 53 64 25.07

The correlation between number of drinks and test score is calculated to be r = 0.869.

(a) What fraction of the variation in test score can be accounted for by its regression on number of drinks consumed?

(b) Find the equation of the least squares regression line.

(c) Find the predicted test scores for a subject who consumes 5 drinks and for a subject who consumes 11 drinks.

(d) Is one of your predictions in (c) more reliable than the other? Explain.

Answer 1

a)

R² = 0.7550
Approximately    75.50%  variation in test score can be accounted for by its regression on number of drinks consumed

b)

	ΣX	ΣY	Σ(x-x̅)²	Σ(y-ȳ)²	Σ(x-x̅)(y-ȳ)
total sum	24.00	384.00	34.00	3142.00	-284.00
mean	4.00	64.00	SSxx	SSyy	SSxy

sample size ,   n =   6      
here, x̅ = Σx / n=   4.000          
ȳ = Σy/n =   64.000          
SSxx =    Σ(x-x̅)² =    34.0000      
SSxy=   Σ(x-x̅)(y-ȳ) =   -284.0      
              
estimated slope , ß1 = SSxy/SSxx =   -284/34=   -8.35294      
intercept,ß0 = y̅-ß1* x̄ =   64- (-8.3529 )*4=   97.41176      
              
Regression line is, Ŷ=   97.412   + (   -8.353   )*x

......

c)

Predicted Y at X=   5   is          
Ŷ=   97.4118   +   -8.3529   *5=   55.65

Predicted Y at X=   11   is          
Ŷ=   97.4118   +   -8.3529   *11=   5.53

...........

d)

55.65 i more relable than other

as for 11 drinks it is extrapollated valeu

........

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