1 Estimation of Standard Deviation & Percentiles Dr. Jerrell T. Stracener, SAE Fellow Leadership in Engineering EMIS 7370/5370 STAT 5340 : PROBABILITY AND STATISTICS FOR SCIENTISTS AND ENGINEERS Systems Engineering Program Department of Engineering Management, Information and Systems

2 Estimation of Standard Deviation - Normal Distribution Estimation of Ratio of Standard Deviations - Normal Distributions Estimation of Percentiles - Tolerance Intervals Estimation of Standard Deviation & Percentiles

3 Estimation of Standard Deviations

4 Point Estimate of  (1 - )  100% Confidence Interval for  is, where and Estimation of Standard Deviation - Normal Distribution

5 The following are the weights, in decagrams, of 10 packages of grass seed: 46.4, 46.1, 45.8, 47.0, 46.1, 45.9, 48.8, 46.9, 45.2, 46.0 Estimate  in terms of a point estimate and a 95% confidence interval for the standard deviation of all such packages of grass seed, assuming a normal population. Example: Estimation of 

6 First we find Example - Solution

7 Then a point estimate of  A (1 - )  100% Confidence Interval for  is, Example - Solution

8 Where Example - Solution

9 where and and is the value of X ~ x 2 n-1 such that, and Example - Solution and

10 Let X 11, X 12, …,, and X 21, X 22, …, be random samples from N(  1,  1 ) and N(  2,  2 ), respectively Point estimation of where for i = 1, 2 Estimation of Ratio of Two Standard Deviations Normal Distribution

11 (1 - )  100% Confidence Interval for where and Estimation of Ratio of Two Standard Deviations Normal Distribution

12 where is the value of the F-Distribution with anddegrees of freedom for which Estimation of Ratio of Two Standard Deviations Normal Distribution

13 A standardized placement test in mathematics was given to 25 boys and 16 girls. The boys made an average grade of 82 with a standard deviation of 8, while the girls made an average grade of 78 with a standard deviation of 7. Find a 98% confidence interval for, where and are the variances of the populations of grades for all boys and girls, respectively, who at some time have taken or will take this test. Example - Estimation of 1/2

14 We have n 1 = 25, n 2 = 16, s 1 = 8, s 2 = 7. For a 98% confidence interval,  = So that F 0.01 (24, 15) = 3.29, and F 0.01 (15, 24) = Then a 98% confidence interval for is where and Example - solution

15 Estimation of Percentiles - Tolerance Intervals

16 If X 1, X 2, …, X n is a random sample of size n from a normal distribution with unknown mean  and unknown standard deviation , tolerance limits are given by, where k is determined so that one can assert with (  )100% confidence that the given limits contain at least the proportion p of the measurements, i.e., the tolerance interval is (LTL, UTL), where and where k may be obtained from a table of tolerance factors which is located on the Information and Resources page of the website. Tolerance Limits

17 A machine is producing metal pieces that are cylindrical in shape. A sample of these pieces is taken and the diameters are found to be: 1.01, 0.97, 1.03, 1.04, 0.99, 0.98, 0.99, 1.01, and 1.03 centimeters. Find tolerance limits that will contain at least 95%, with a 99% confidence, of the metal pieces produced by this machine, assuming a normal distribution. Example - Tolerance Interval

18 The sample mean and standard deviation for the given data are x = and s = From the Tolerance Factors Table for n = 9, 1 -  = 0.99, and 1 -  = 0.95 we find k = for two-sided limits. Hence the tolerance limits are and Solution

19 That is, we are 99% confident that the tolerance interval from to will contain at least 95% of the metal pieces produced by this machine. Solution