# CU-Boulder MCEN 3037 - 13 (11 pages)

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Mcen 3037 - Data Analysis
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A measure of the goodness of fit is given by the Coefficient of Determination r2 n r2 n 2 y y y y i i i 2 i 1 i 1 n 2 y y i i 1 n 1 2 y y i i i 1 n y i 1 y 2 i Square of the correlation coefficient Let s say we do the spring experiment How confident are we in our results We need a few assumptions Errors are random and independent Errors have a zero mean Errors all have same variance Errors normally distributed We can estimate the spread in the data variance through n S2 2 y y i i i 1 n 2 li 0 The standard error or uncertainty in the regression coefficients is 1 S S 0 n x2 S n 2 x x i 1 i 1 S n 2 x x i i 1 Now we are in a position to find the confidence interval 0 0 tn 2 P S 0 1 1 tn 2 P S 1 We can Find the best fit line slope and intercept Estimate the variance Estimate the standard error Report Confidence intervals for the slope and intercept y i 0 1 xi Best estimate of the mean value of yi The confidence interval on the mean value of yi yi 0 1 x tn 2 P SY Where 1 SY S n x x n 2 x x i 2 i 1 I m 95 sure that the next spring tested will fall within yi 0 1 xi tn 2 P S Pr ed S Pr ed 1 S 1 n x x 2 n 2 xi x i 1 Prediction interval Confidence interval rxy 0 413 rt 0 811 We need to accept or reject a statement about some parameter of a population Statement is called the Hypothesis Random variable Example We are interested in the burn rate of a solid rocket propellant We may be interested in determining if the mean burn rate is 50 cm s Null hypothesis we want to test H 0 50 We say H1 50 Alternative hypothesis rejecting null hypothesis leads to accepting alternative hypothesis

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