1. MATH408: Probability & Statistics Summer 1999 WEEK 6 Dr. Srinivas R. Chakravarthy Professor of Mathematics and Statistics Kettering University (GMI Engineering & Management Institute) Flint, MI 48504-4898 Phone: 810.762.7906 Email: schakrav@kettering.edu Homepage: www.kettering.edu/~schakrav

2. Sample Size Determination for a given 

3. Examples

4. Confidence Interval Recall point estimate for the parameter under study. For example, suppose that µ= mean tensile strength of a piece of wire. If a random sample of size 36 yielded a mean of 242.4psi. Can we attach any confidence to this value? Answer: No! What do we do?

5. Confidence Interval (cont’d) Given a parameter, say,, let denote its UMV estimator. Given , 100(1-  )% CI for is constructed using the sampling (probability) distribution of as follows. Find L and U such that P(L < < U) = 1- . Note that L and U are functions of.

6. Interpretation of CI With 100(1-  )% confidence, we can say that the true value of will lie between L and U; or equivalently, if 100 samples of size n were taken, then we would expect at least 100(1-  ) of the 100 values of will be between L and U. We will illustrate this in the laboratory.

8. Confidence Interval for the population mean

9. Horsepower Example (Revisited)

10. Choice of Sample Size

11. Examples

12. Student’s t-distribution Referring to HP example, we assumed that the population standard deviation  was known (to be 10). However, in practice, it is usually unknown. Hence, we need to estimate it first. If the sample size is reasonably large (n  30), we can still use the normal distribution for inferential part (as justified by the CLT).

13. Student’s t-distribution What happens if the sample is small (n < 30)? In this case we cannot use normal since the sample size is small and by using the sample standard deviation to estimate s, we bring in more variability into the picture and the appropriate distribution to use is the student's t-distribution. In 1908, William S.Gosset, a chemist working for a brewery company, under the pseudonym Student, first deduced this distribution.

14. Student’s t-distribution Suppose that X 1, X 2, …, X n are n random samples from a normal distribution with mean  and standard deviation . Then the PDF of is given by

16. Student’s t-distribution (cont’d) Student’s t-distribution, like normal, –is bell-shaped. It depends on the sample size. –It is more spread than normal and approaches normal as n approaches infinity. So in the case when n is small,  is unknown and with the assumption that the population is approximately normal, 100(1-  )% C.I for  is given by

17. HP Example (cont’d):

18. VERIFYING THE NORMALITY ASSUMPTION Note that in constructing the above confidence interval, we assumed that the populations (for 4500 RPM and 5500 RPM) are normal. How do we verify that the assumptions are not grossly violated? Recall normal probability plot? If there is a reason to believe that this assumption is not valid in any given problem, then one has to transform the data or to rely on nonparametric methods.

19. Examples

20. One-sided Confidence Intervals

21. Testing on the mean using T-distribution The test statistics is given by: Examples

22. HOMEWORK PROBLEMS Sections 4.1 through 4.5 1-4, 11-13, 15-18, 21, 22, 24-27, 30,31, 33-38, 40

23. INFERENCE ON THE VARIANCE Recall that sample variance is an UMV estimator for the population variance. Here, we will see how to construct CI and test hypotheses on  2. Assuming that the population is normal, the statistic: follows a chi-square distribution with n-1 degrees of freedom

24. Chi-square Distribution