Today’s lesson

1. Today’s lesson • Case study involving a quality control (QC) application using the material that we have covered to date. • In class demonstration of how to use SPSS to get the numbers out. • Discussion of meaning of results.

2. Today’s lesson • Confidence interval for the mean of a normal distribution using standard normal and t-distribution. • On Thursday, we finish Chapter Eleven with one sample t and z tests and a review (yet again) of the structure of tests of statistical hypotheses.

3. Case Study • QC application in which the objective is to specify a probability model for the “manufacturing” process. • That is, specify the A part in an A vs. B comparison. • See Finch et al., Statistics in Medicine, 1999, pages 1279-1289.

4. Context • Application is the filtering of red blood to remove white blood cells. • Your white blood cells in your blood are good. • My white blood cells in the unit of blood that I donated to be transfused to you are bad for you. • Transfusion reaction • Possible infection vector

5. Practical Context • Health regulations are that a unit of filtered blood cannot contain more than so many “residual white blood cells” (RWBC). • US standard is 5x106 rwbc in a unit. • European standard is 1x106 rwbc in a unit.

6. Practical Context • Measured RWBC is product of three numbers. • Constant scaling factor • volume of blood donated in the unit • Nageotte count, which is the number of white blood cells observed in a small volume of sampled filtered blood.

7. First Question • What should be the dependent variable monitored in the QC application? • Answer is to use the Nageotte count rather than the RWBC. • American standard is then a Nageotte count of 167. • European standard is then a Nageotte count of 33.

8. Components of Variance • Here used Fisher’s fundamental idea of finding “components of variance.” • Specifically, variance of RWBC has a non-quality related component of variance from the variation of the volume of blood in the donated unit.

9. Second Question • What actually happened in the QC process when the manufacturer’s staff did the work? • That is, use the descriptive options in a statistical package to describe the data.

10. Getting numbers out • Enter SPSS • Access correct file (CAREFUL, CAREFUL, CAREFUL!!!) • Statistics menu • Descriptive submenu • Frequencies option

11. Third Question • Then, specify a probabilistic model that fits the data reasonably well so that predictions can be made. • Nageotte count variable is a ratio scale of measurement. • Nageotte count is discrete, not continuous. • Variance is very much larger than the mean. • Hence focused on the negative binomial distribution (NBD).

12. Fourth Question • ASS-U-ME a negative binomial distribution. • How well does it fit the data observed? • Use a goodness of fit test (we won’t cover this in detail until after your first exam).

13. Observed and Expected Nageotte Counts

14. Interpretation • One observation violated American, and two violated European rule. • NBD model does not predict maximum observed value of 205. • NBD model does not fit well but captures the rough order of variation (up to a Nageotte count of 33). • Choose a nonparametric test procedure because null distribution is not obvious.

15. My Most Common Three Mistakes in Making Predictions • Eliminate “outliers” from the historical data that I am using to make my prediction. • Predict a ratio scale variable without anticipating that the variance of the variable will increase when the mean increases. • ASS-U-ME independence of observations when predicting a time series with autocorrelation.

16. Fifth Question • How big of a sample is necessary to determine whether a user of this product is “in control.” • Simulation study suggested that under optimistic conditions 20 is a minimal sample size but that 80 may be required. • Client’s practice has evolved to use about 50.

17. Chapter Eleven: Testing a Hypothesis about a Single Mean • Definition of Student’s t Distribution • Using tables of Student’s t distribution. • Using the observed significance level from Student’s t. • Using the confidence limit from Student’s t.

18. Historical Background of Student’s t • Origin is quality control in the brewing industry (Guinness). • How can statistical procedures be applied with very small samples? • Nature of the advance is to describe the null distribution of the statistic that is actually used.

19. Definition of the Student’s t distribution • The pdf is also a bell-shaped curve • Continuous distribution, unimodal, symmetric, less rapid fall-off of probability for values far from mean. • Appendix C table (546-547) gives two sided tail probability by degrees of freedom. • Most statistics texts give percentile points.

20. Basic numeric facts of Student’s t percentiles • Student t 95-th percentiles are larger than the 95-th standard normal percentile. • Same holds for any percentile greater than 50. • The difference becomes larger as the “degrees of freedom” is smaller.

21. Example One-Sample Z test Problem • Test the null hypothesis H0: E(Y)=500 with level of significance 0.01 against the alternative hypothesis H1: E(Y)<500. ASS-U-ME Y is normal with known standard deviation 100 using the sample mean of a random sample of four observations. This statistic has value 360.

22. Solution • Determine the side of the test, here left-sided. • Determine the standard error of the statistic, here 100/40.5=50 • Determine the critical value of the test statistic. • In original form, 500-2.326(50)=383.7 • In standard unit form, -2.326.

23. Solution Continued • Compare the statistic to the critical value: • In original units, the observed mean of 360 is to the left of the critical value of 383.7. • In standard-score form, the z value of the mean is (statistic-hypothesized expected value)/se of statistic=(460-500)/50=-2.8, to the left of the critical value -2.326. • Make decision: reject H0 at the 0.01 level of significance.

24. Example One-Sample T test Problem • Test the null hypothesis H0: E(Y)=500 with level of significance 0.01 against the alternative hypothesis H1: E(Y)<500. ASS-U-ME Y is normal with unknown standard deviation. The mean of a random sample of four observations has value 360, and the unbiased estimate of the variance is 6400. Note that the corresponding estimate of the standard deviation is 80.

25. Solution • Determine the side of the test, here left-sided. • Determine the estimated standard error of the statistic, here 80/40.5=40. • Student’s contribution: determine the degrees of freedom. For a one-sample t-test, it is number of observations minus one, here 4-1=3.

26. Solution Continued • Determine the critical value of the test statistic. Don’t forget Student’s stretch of the critical value • In original form, 500-t3,2.326(40)=500-4.541(40) =318.36 • In standard unit form, -4.541.

27. Solution Continued • Compare the statistic to the critical value: • In original units, the observed mean of 360 is to the right of the critical value of 318.36. • In standard-score form, the z value of the mean is (statistic-hypothesized expected value)/se of statistic=(360-500)/40=-3.5, to the right of the critical value -4.541. • Make decision: accept H0 at the 0.01 level of significance

28. Major points covered • Review of material using a case study that applies descriptive statistics. • Introduction (review) of Student’s t. • The one sample standard normal test. • The one sample Student’s t test.

29. To come • Finish Chapter 11 with one sample confidence intervals. • Begin Chapter 12, the paired t-test.