Dependent (response) Variable Independent (control) Variable Random Error XY x1x1 y1y1 x2x2 y2y2 …… xnxn ynyn Raw data: Assumption:  i ‘s are independent.

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Presentation transcript:

Dependent (response) Variable Independent (control) Variable Random Error XY x1x1 y1y1 x2x2 y2y2 …… xnxn ynyn Raw data: Assumption:  i ‘s are independent normally distributed random variables with mean 0 and (common) variance  2.

Estimator for  : Estimator  : Estimator for the Y value for given x: Estimators for parameters unbiased? Confidence intervals for the true parameters? Confidence interval for the Y(x) value: Estimate E(Y(x))? Hypothesis Testing? Quality of the model? Better model? (Sec. 11.2)

Air Velocity x (cm/s) Evaporation Coefficient y (mm 2 /s) (Linear) relationship exists? Linear relationship? Prediction for x = 190 cm/s Error?

xyx^2y^2xy Sum Sxx Slope b Syy Intercept a Sxy505.4

Residuals: Residual sum of squares (or Error Sum of Squares): Estimator for  2 : Standard Error of the Estimate:

Distribution of b (as a statistic): Confidence interval for the true population slope  is To test the null hypothesis H 0 :  =  0, use the statistic (with df = n – 2.) Only when the null hypothesis H 0 :  = 0 is rejected, the independent variable x can be included in the model.

Distribution of a (as a statistic): Confidence interval for the true population intercept  is To test the null hypothesis H 0 :  =  0, use the statistic (with df = n – 2.) Only when the null hypothesis H 0 :  = 0 is rejected, the nonzero intercept can be included in the model.

Inference about E[Y(x)] =  +  x, the mean of the response value for given x. Confidence interval for E[Y(x)] is where t-distribution has df = n – 2.

To predict the “future” Y(x), we use the interval where t-distribution has df = n – 2. Note: Confidence interval for Y(x) is wider than the confidence interval for E[Y(x)]; Width of the confidence intervals for Y(x) and E[Y(x)] depend on the x value. (Limit of prediction.)

Excel Outputs: Estimators a and b T-score in Hypothesis Tests P-values in Hypothesis Tests 95% Confidence Intervals for  and  Compare with the critical t-value(s); Compare with the  value; Check if the interval contains zero.