1. Interval Estimates A point estimate gives a plausible single number estimate for a parameter. We may also be interested in a range of plausible values for a parameter or even future measurements Called an interval estimate Confidence intervals Prediction intervals Tolerance intervals Credibility intervals (Bayesian. No time unfortunately…)

2. Confidence Intervals  is a parameter we are interested in. Usually: a mean a variance/standard deviation a proportion Consider an experiment that will collect n data points. Then BEFORE we collect the data, we can devise procedure such that:

3. Confidence Intervals The PROCEDURE produces intervals: that cover  with probability 1-  The interval is “random”. We don’t know the upper-estimate or lower-estimate yet. Since it is random and it is the result of a repeatable experiment, it can be associated with a probability under the frequentist definition

4. Confidence Intervals BUT, in order to get a numerical interval we perform the experiment and plug in the data Once the sample of data is measured, the interval is no longer “random”. We cannot say that  is in the specific, realized interval with probability 1-   is either in the interval or it is not, we just don’t know which. We can’t do repeatable experiments to compute the probability that  is in the specific interval we realized with the sample we measured. So let’s make up a new word: confidence

5. Confidence Intervals Any realized interval: contains  with 1-  “confidence”. We can be 1-  confident that the realized interval covers . The width of the confidence interval varies with sample size and level of confidence Confidence increases, width increases Sample size increases, width decreases

6. Confidence Intervals Caution: IT IS NOT CORRECT to say that there a (1-  α)×100% probability that the true value of a parameter is between the bounds of any given CI. true value of parameter Here 90% of the CIs contain the true value of the parameter Graphical representation of 90% CIs is for a parameter: Take a sample. Compute a CI.

7. Construction of a large sample two-sided CI for a mean depends on: Sample size: n Standard error for means: Desired level of confidence 1-α α is significance level Use α/2 to compute N(0,1) z-value (1-α)×100% CI for population mean using a sample average and standard error is: Confidence Intervals Assumes  = s

8. Compute a 95% confidence interval for the mean using this sample set: Confidence Intervals Fragment #Fragment nD 11.52005 21.52003 31.52001 41.52004 51.52000 61.52001 71.52008 81.52011 91.52008 101.52008 111.52008 Putting this together: [1.52005 – (-1.96)(0.00001), 1.52005 + (-1.96)(0.00001)] 95% CI for sample = [1.520123, 1.519980]

9. Confidence Intervals large sample two-sided CI for a mean

10. Construction of a small sample two-sided CI for a mean depends on: Sample size: n Standard error for means: Desired level of confidence 1-α α is significance level Use α/2 to compute Student-t(n) t-value (1-α)×100% CI for population mean using a sample average and standard error is: Confidence Intervals

11. small sample two-sided CI for a mean

12. Construction of a large sample two-sided CI for a proportion p, depends on: Sample size: n Standard error for proportion: Desired level of confidence 1-α α is significance level Use α/2 to compute N(0,1) z-value (1-α)×100% CI for proportion using sample standard error is: Confidence Intervals

13. Mike Neel and Major Wells of the ATF carried out a comparison of known matching (KM) striation pattern tool marks (AFTE J 39(3):176-198, 2007). Of the 914 KM comparisons they examined, it was found that 19 had 3 4-line (4X) consecutive matching striation (CMS) patches. Compute the large sample 95% confidence interval around the proportion of KM pairs with 3 4X CMS patches.

14. Construction of a small sample two-sided CI for a proportion: For desired level of confidence (1-α)×100% CI for population is: Confidence Intervals Upper Bound Lower Bound

15. Confidence Intervals Compute the small sample 95% confidence interval around the proportion of KM pairs with 3 4X CMS patches.

16. One-sided CIs Use z  or t(n)  instead of z  /2 or t(n)  /2 and only + or – depending on which bound is desired. Confidence Intervals What is the lower one-sided small sample 95% confidence limit on the proportion of KM pairs with 3 4X CMS patches?

17. Prediction Intervals What if we want to predict, a value of a future data for a given experiment? What can be said about the values produced by a future experiment based on data in hand Vardman, Morris ? We want to predict at a specified level of confidence (ehhhh… probability…) We want an interval estimate for our prediction.

18. Prediction Intervals Since we know the distribution of the data generating process, along with the distributions parameters, this is an easy question to answer. The reliability of analytical balances in a narcotics lab is set by policy. A balance’s “lifetime” is it’s reliable use time (in hr) before it needs major servicing. A vendor states that it’s balances have a Weibull distributed lifetime with shape (  ) = 15 and scale (  ) = 5000 What is the 90% prediction interval for the lifetime of the next balance shipped out of the factory?

19. Prediction Intervals Most of the units have this lifetime Predict: 90% chance next unit will have a lifetime in here 4100 hr 5400 hr

20. Prediction Intervals We pretty much never know the data’s distribution and it’s parameters precisely. Usually we are stuck with just a sample of data. What can we say about the prediction intervals if all we have is a sample? If the data’s distribution is Gaussian (normal): Prediction interval for the n th + 1 data point, having observed n data points: For the 1-  probability prediction

21. Prediction Intervals 15 RI measurements from a pane of glass were: 1.51881, 1.51874, 1.51883, 1.51865, 1.51878, 1.51882, 1.51876, 1.51877, 1.51882, 1.51882, 1.51883, 1.51882, 1.51882, 1.51882, 1.51879 Assuming these RI measurements follow a normal distribution with unknown mean and variance, compute the 95% prediction interval for the 16 th measurement.

22. Prediction Intervals 95% Prediction interval for the 16 th measurement after the first 15 RI measurements The actual 16 th RI measurement. Did we make it in? Load some glass RI data from Prof. James Curran’s (Auckland) dafs package

23. Prediction Intervals If the data’s distribution is unknown Vardman, Morris : Prediction interval for the n th + 1 data point, having observed n data points, x i : Prediction probability: Prediction probability dictated by sample size { min(x i ), max(x i ) } Also called the non-parametric prediction interval

24. Prediction Intervals 15 RI measurements from a pane of glass were: 1.51881, 1.51874, 1.51883, 1.51865, 1.51878, 1.51882, 1.51876, 1.51877, 1.51882, 1.51882, 1.51883, 1.51882, 1.51882, 1.51882, 1.51879 Compute the prediction interval for the 16 th measurement, assuming an unknown distribution for the data. How does the width compare the the Gaussian prediction interval? What is the sample size required to have prediction interval with a probability of at least 95%?

25. Prediction Intervals non-parametric prediction interval prediction probability Width of 95% Gaussian PI Width of 87.5% non-parametric PI

26. Tolerance Intervals What if we want to predict a large fraction of future data for a given experiment? A prediction interval is good for only the next, not-yet- produced value We want to give an interval where 100×p% of future values should fall with 1 -  confidence (Ehhhh,….. probability if they haven’t been produced yet)

27. Tolerance Intervals If the data’s distribution is Gaussian (normal): Two sided tolerance interval for 100×p% of the mass of future data values with 1 -  confidence (probability): Two sided tolerance scale factor for 1-  probability, p-fraction of the of the future values: Chi-squared quantile, qchisq Degrees of freedom, df

28. Tolerance Intervals If the data’s distribution is Gaussian (normal): One sided tolerance interval for 100×p% of the mass of future data values with 1 -  confidence (probability): One sided tolerance scale factor for 1-  probability, p-fraction of the of the future values: For large samples (n>30): For small samples: Degrees of freedom, df Non-centrality param, ncp

29. Tolerance Intervals If the data’s distribution is unknown Vardman, Morris : Two sided tolerance interval for 100×p% of the mass of future data values: Tolerance confidence (probability): Probability depends on proportion p and sample size, n { min(x i ), max(x i ) } Also called the non-parametric tolerance interval

30. Tolerance Intervals 15 RI measurements from a pane of glass were: 1.51881, 1.51874, 1.51883, 1.51865, 1.51878, 1.51882, 1.51876, 1.51877, 1.51882, 1.51882, 1.51883, 1.51882, 1.51882, 1.51882, 1.51879 Based on this sample, with what probability can a tolerance interval be produced which covers 99% of future RI, assuming the data’s distribution is unknown?

31. Tolerance Intervals 15 RI measurements from a pane of glass were: 1.51881, 1.51874, 1.51883, 1.51865, 1.51878, 1.51882, 1.51876, 1.51877, 1.51882, 1.51882, 1.51883, 1.51882, 1.51882, 1.51882, 1.51879 Assuming the RI measurements follow a normal distribution, compute the two-sided tolerance interval for where 83% of future measurement will lay with 95% probability. Assuming the data follows a Gaussian distribution. Reference Vardeman and Morris notes for Industrial Engineering 361, Iowa State: http://www.public.iastate.edu/~vardeman/IE361/Modules/Module%2018C.pdf

32. Tolerance Intervals Make life a little easier and use the package “tolerance” to do the tolerance interval calcs tolerance interval assuming the data is Gaussian