Question

An experiment was performed on a certain metal to determine if its strenath is a function of heating time. Partial results based on a sample of 10 metal sheets are given below. The simple linear regression equation is ý1+1X. Time is in minutes, strength is measured in pounds per square inch, MSE Ex5, Ex 30, and Ex 104. The distance value has been found to be equal to 17143. Determine the 95 percent prediction interval for the strength ofa metal sheet when the average heating time is 4 minutes

Answer 1

MSE = 0.5

Standard error of regression, s = $\sqrt{MSE} = \sqrt{0.5}$ = 0.7071

Predicted y at x = 4,  $\hat{y}$ = 1 + 4 = 5

Degree of freedom = n - 2 = 10 - 2 = 8

Critical value of t at 95% interval and df = 8 is 2.31

$S_{xx} = \sum x^2 - (\sum x)^2 / n = 104 - 30^2 / 10 = 14$

$\bar{x} = \sum x / n = 30 / 10 = 3$

95% prediction interval for the strength of the metal sheet at average heating time of 4 minutes is,

$\hat{y} \pm t * s \sqrt{1 + (1/n) + (x - \bar{x})^2 / S_{xx}}$

$= 5 \pm 2.31 * 0.7071 \sqrt{1 + (1/10) + (4 - 3)^2 / 14}$

$= 5 \pm 1.767872$

= (5 - 1.767872, 5 + 1.767872)

= (3.232128, 6.767872)