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simple linear regressio

⇒ set up times are not significantly different in

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Unformatted text preview: ⇒ Set up times are not significantly different in the two systems while the per byte times (slopes) are different. 14-32 ©2008 Raj Jain CSE567M Washington University in St. Louis Confidence Intervals for Predictions Confidence Intervals for Predictions ! This is only the mean value of the predicted response. Standard deviation of the mean of a future sample of m observations is: ! m =1 ⇒ Standard deviation of a single future observation: 14-33 ©2008 Raj Jain CSE567M Washington University in St. Louis CI for Predictions (Cont) CI for Predictions (Cont) ! m = ∞ ⇒ Standard deviation of the mean of a large number of future observations at x p : ! 100(1- α )% confidence interval for the mean can be constructed using a t quantile read at n-2 degrees of freedom. 14-34 ©2008 Raj Jain CSE567M Washington University in St. Louis CI for Predictions (Cont) CI for Predictions (Cont) ! Goodness of the prediction decreases as we move away from the center. 14-35 ©2008 Raj Jain CSE567M Washington University in St. Louis Example 14.5 Example 14.5 ! Using the disk I/O and CPU time data of Example 14.1, let us estimate the CPU time for a program with 100 disk I/O's. ! For a program with 100 disk I/O's, the mean CPU time is: 14-36 ©2008 Raj Jain CSE567M Washington University in St. Louis Example 14.5 (Cont) Example 14.5 (Cont) ! The standard deviation of the predicted mean of a large number of observations is: ! From Table A.4, the 0.95-quantile of the t-variate with 5 degrees of freedom is 2.015. ⇒ 90% CI for the predicted mean 14-37 ©2008 Raj Jain CSE567M Washington University in St. Louis Example 14.5 (Cont) Example 14.5 (Cont) ! CPU time of a single future program with 100 disk I/O's: ! 90% CI for a single prediction: 14-38 ©2008 Raj Jain CSE567M Washington University in St. Louis Visual Tests for Regression Assumptions Visual Tests for Regression Assumptions Regression assumptions: 1. The true relationship between the response variable y and the predictor variable x is linear. 2. The predictor variable x is non-stochastic and it is measured without any error. 3. The model errors are statistically independent. 4. The errors are normally distributed with zero mean and a constant standard deviation. 14-39 ©2008 Raj Jain CSE567M Washington University in St. Louis 1. Linear Relationship: Visual Test 1. Linear Relationship: Visual Test ! Scatter plot of y versus x ⇒ Linear or nonlinear relationship 14-40 ©2008 Raj Jain CSE567M Washington University in St. Louis 2. Independent Errors: Visual Test 2. Independent Errors: Visual Test 1. Scatter plot of ε i versus the predicted response ! All tests for independence simply try to find dependence. 14-41 ©2008 Raj Jain CSE567M Washington University in St. Louis Independent Errors (Cont) Independent Errors (Cont) 2. Plot the residuals as a function of the experiment number 14-42 ©2008 Raj Jain CSE567M Washington University in St. Louis 3. Normally Distributed Errors: Test 3. Normally Distributed Errors: Test ! Prepare a normal quantile-quantile plot of errors....
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