Question

. An experiment was performed on a certain metal to determine if the strength is a function of heating time (hours). Results based on 20 metal sheets are given below. Use the simple linear regression model.
∑X = 40
∑X² = 200
∑Y =   80
∑Y² = 1120
∑XY = 460

Find the estimated y intercept and slope and write the equation of the least squares regression line. Estimate Y when X is equal to 3 hours. Also determine the standard error, the Mean Square Error, the coefficient of determination and the coefficient of correlation.

Answer 1

Σn	ΣX	ΣY	ΣXY	ΣX²	ΣY²
20	40	80	460	200	1120

sample size ,   n =   20      
here, x̅ =   2.0000 , ȳ =   4.0000

SSxx =    Σx² - (Σx)²/n =   120

SSxy=   Σxy - (Σx*Σy)/n =   300

SSyy =    Σy²-(Σy)²/n =   800

slope ,    ß1 = SSxy/SSxx =   2.5

intercept,   ß0 = y̅-ß1* x̄ =   -1

so, regression line is   Ŷ =   -1 +   2.5 *x

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when X=3

then Ŷ =   -1 +   2.5 *3 = 6.5

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SSE=   (Sx*Sy - S²xy)/Sx =    50
      
std error ,Se =    √(SSE/(n-2)) =    1.667

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Mean Square Error = SSE/(N-2)=50/18=2.778
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SSR = SST-SSE = SSyy - SSE = 800-50=750

R² = SSR/SST=750/800 = 0.9375
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the coefficient of correlation=√R² = √0.9375 = 0.9682