9. 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 = 388
Find the estimated y intercept and slope. Write the equation of the least squares regression line and explain the coefficients. Estimate Y when X is equal to 5 hours. Also determine the standard error, the Mean Square Error, the coefficient of determination and the coefficient of correlation. Check the relation between correlation coefficient and Coefficient of Determination. Test the significance of the slope.
Sample size, n = | 20 |
Ʃ x = | 40 |
Ʃ y = | 80 |
Ʃ xy = | 388 |
Ʃ x² = | 200 |
Ʃ y² = | 1120 |
x̅ = Ʃx/n = | 2 |
y̅ = Ʃy/n = | 4 |
SSxx = Ʃx² - (Ʃx)²/n = | 120 |
SSyy = Ʃy² - (Ʃy)²/n = | 800 |
SSxy = Ʃxy - (Ʃx)(Ʃy)/n = | 228 |
Slope, b = SSxy/SSxx = 1.9
y-intercept, a = y̅ -b* x̅ = 0.2
Regression equation :
ŷ = 0.2 + 1.9 x
If x is increased by 1 unit then there is an increase of 1.9 units in y.
Predicted value of y at X = 5
ŷ = 0.2 + 1.9 * 5 = 9.7
Sum of Square error, SSE = SSyy -SSxy²/SSxx = 366.8
Standard error, se = √(SSE/(n-2)) =
4.51418
Mean square error = se2 = 20.37778
Correlation coefficient, r = SSxy/√(SSxx*SSyy) = 0.7359
Coefficient of determination, r² = (SSxy)²/(SSxx*SSyy) = 0.5415
Slope significance test:
Null and alternative hypothesis:
Ho: β₁ = 0 ; H1: β₁ ≠ 0
Test statistic:
t = b /(se/√SSxx) = 1.9/(4.51418/√120) = 4.611
df = n-2 =18
p-value = T.DIST.2T(4.611, 18) = 0.0002
Decision :
p-value < 0.05, Reject the null hypothesis.
9. An experiment was performed on a certain metal to determine if the strength is a...
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