Stats for Engineers Lecture 9

1. Stats for Engineers Lecture 9

2. Summary From Last Time Confidence Intervals for the mean If is known, confidence interval for is to , where is obtained from Normal tables. Ifis unknown (only know sample variance ): Assuming independent Normal data, the confidence interval for is: to . Student t-distribution Q t-tables

3. Sample size How many random samples do you need to reach desired level of precision? Suppose we want to estimate to within , where (and the degree of confidence) is given. Want Need: - Estimate of (e.g. previous experiments) - Estimate of . This depends on n, but not very strongly. e.g. take for 95% confidence. Rule of thumb: for 95% confidence, choose

4. Example A large number of steel plates will be used to build a ship. Ten are tested and found to have sample mean weight and sample variance How many need to be tested to determine the mean weight with 95% confidence to within ? Answer: Assuming plates have independent weights with a Normal distribution Take for 95% confidence. i.e. need to test about 28

5. Number of samplesIf you need 28 samples for the confidence interval to be approximately how many samples would you need to get a more accurate answer with confidence interval • 88.5 • 280 • 2800 • 28000 so need more. i.e. 2800

6. Linear regression We measure a response variable at various values of a controlled variable e.g. measure fuel efficiency at various values of an experimentally controlled external temperature Linear regression: fitting a straight line to the mean value of as a function of

7. Distribution of when Regression curve: fits the mean values of the distributions

8. From a sample of values at various , we want to fit the regression curve. e.g.

9. Straight line plotsWhich graph is of the line ? 1. • Plot • Plot2 • Plot3 • Plot4 2. 3. 4.

10. From a sample of values at various , we want to fit the regression curve. e.g.

11. Or is it What do we mean by a line being a ‘good fit’?

12. Equation of straight line is Simple model for data: Straight line Random error Simplest assumption: for all , and 's are independent - Linear regression model

13. Model is Want to estimate parametersa and b, using the data. e.g. - choose and to minimize the errors Maximum likelihood estimate = least -squares estimate Minimize Data point Straight-line prediction E is defined and can be minimized even when errors not Normal – least-squares is simple general prescription for fitting a straight line (but statistical interpretation in general less clear)

14. The line has been proposed as a line of best fit for the following four sets of data. For which data set is this line the best fit (minimum )? 1. • Pic1 • Pic2 • Pic correct • pic4 Question from Derek Bruff 2. 3. 4.

15. How to find and that minimize ? For minimum want and , see notes for derivation Solution is the least-squares estimates and : Sample means Where Equation of the fitted line is

16. Most of the things you need to use are on the formula sheet

17. Note that since i.e. is on the line

18. Example: The data y has been observed for various values of x, as follows: Fit the simple linear regression model using least squares. Answer: Want to fit n = 9 , ,

19. Answer: Want to fit n = 9 , , Now just need So the fit is approximately

20. Which of the following data are likely to be most appropriately modelled using a linear regression model? 1. • Correct • Errors change • Not straight 2. 3.

21. Quantifying the goodness of the fit Estimating : variance of y about the fitted line Estimated error is: , so the ordinary sample variance of the 's is In fact, this is biased since two parameters, a and b have been estimated. The unbiased estimate is: [derivation in notes] Residual sum of squares

22. Which of the following plots would have the greatest residual sum of squares [variance of about the fitted line]? 1. • Pic1 • Pic2 • Pic correct Question from Derek Bruff 2. 3.