Chapter 9 One and Two Sample Estimation q

Chapter 9: One- and Two- Sample Estimation q Statistical Inference Estimation Ø Tests of hypotheses Ø q Interval estimation: (1 – α) 100% confidence interval for the unknown parameter. Example: if α = 0. 01, we develop a 99% confidence interval. Ø Example: if α = 0. 05, we develop a 95% confidence interval. Ø EGR 252 Ch. 9 Lecture 1 JMB 2014 9 th edition Slide 1

Single Sample: Estimating the Mean q Given: Ø σ is known and X is the mean of a random sample of size n, q Then, Ø the (1 – α)100% confidence interval for μ is 1 -a a/2 EGR 252 Ch. 9 Lecture 1 JMB 2014 9 th edition Slide 2

Example A traffic engineer is concerned about the delays at an intersection near a local school. The intersection is equipped with a fully actuated (“demand”) traffic light and there have been complaints that traffic on the main street is subject to unacceptable delays. To develop a benchmark, the traffic engineer randomly samples 25 stop times (in seconds) on a weekend day. The average of these times is found to be 13. 2 seconds, and the variance is known to be 4 seconds 2. Based on this data, what is the 95% confidence interval (C. I. ) around the mean stop time during a weekend day? EGR 252 Ch. 9 Lecture 1 JMB 2014 9 th edition Slide 3

Example (cont. ) σ = ________ X = _______ α = ________ Z 0. 025 = _______ α/2 = _______ Z 0. 975 = ______ Z 0. 025 = -1. 96 13. 2 -(1. 96)(2/sqrt(25)) = 12. 416 Solution: Z 0. 975 = 1. 96 13. 2+(1. 96)(2/sqrt(25)) = 13. 984 12. 416 < μSTOP TIME < 13. 984 EGR 252 Ch. 9 Lecture 1 JMB 2014 9 th edition Slide 4

Your turn … q What is the 90% C. I. ? What does it mean? 5% 90% 5% Z(. 05) = + 1. 645 All other values remain the same. The 90 % CI for μ = (12. 542, 13. 858) Note that the 95% CI is wider than the 90% CI. EGR 252 Ch. 9 Lecture 1 JMB 2014 9 th edition Slide 5

What if σ 2 is unknown? q For example, what if the traffic engineer doesn’t know the variance of this population? 1. If n is sufficiently large (n > 30), then the large sample confidence interval is calculated by using the sample standard deviation in place of sigma: 2. If σ 2 is unknown and n is not “large”, we must use the t-statistic. EGR 252 Ch. 9 Lecture 1 JMB 2014 9 th edition Slide 6

Single Sample: Estimating the Mean (σ unknown, n not large) q Given: Ø σ is unknown and X is the mean of a random sample of size n (where n is not large), q Then, Ø the (1 – α)100% confidence interval for μ is: EGR 252 Ch. 9 Lecture 1 JMB 2014 9 th edition Slide 7

Recall Our Example A traffic engineer is concerned about the delays at an intersection near a local school. The intersection is equipped with a fully actuated (“demand”) traffic light and there have been complaints that traffic on the main street is subject to unacceptable delays. To develop a benchmark, the traffic engineer randomly samples 25 stop times (in seconds) on a weekend day. The average of these times is found to be 13. 2 seconds, and the sample variance, s 2, is found to be 4 seconds 2. Based on this data, what is the 95% confidence interval (C. I. ) around the mean stop time during a weekend day? EGR 252 Ch. 9 Lecture 1 JMB 2014 9 th edition Slide 8

Small Sample Example (cont. ) n = _______ df = _______ X = ______ s = _______ α/2 = ____ 13. 2 - (2. 064)(2/sqrt(25)) = 13. 374 t 0. 025, 24 = _______ 13. 2 + (2. 064)(2/sqrt(25)) = 14. 026 ________ < μ < ________ EGR 252 Ch. 9 Lecture 1 JMB 2014 9 th edition Slide 9

Your turn A thermodynamics professor gave a physics pretest to a random sample of 15 students who enrolled in his course at a large state university. The sample mean was found to be 59. 81 and the sample standard deviation was 4. 94. Find a 99% confidence interval for the mean on this pretest. EGR 252 Ch. 9 Lecture 1 JMB 2014 9 th edition Slide 10

Solution X = _______ s = ________ α/2 = _______ (draw the picture) t___ , ____ = __________________ < μ < __________ X = 59. 81 s = 4. 94 α =. 01 α/2 =. 005 t (. 005, 14) = 2. 977 Lower Bound 59. 81 - (2. 977)(4. 94/sqrt(15)) = 56. 01 Upper Bound 59. 81 + (2. 977)(4. 94/sqrt(15)) = 63. 61 EGR 252 Ch. 9 Lecture 1 JMB 2014 9 th edition Slide 11

Standard Error of a Point Estimate q Case 1: σ known Ø The standard deviation, or standard error of X is q Case 2: σ unknown, sampling from a normal distribution Ø The standard deviation, or (usually) estimated standard error of X is EGR 252 Ch. 9 Lecture 1 JMB 2014 9 th edition Slide 12

9. 6: Prediction Interval q For a normal distribution of unknown mean μ, and standard deviation σ, a 100(1 -α)% prediction interval of a future observation, x 0 is if σ is known, and if σ is unknown EGR 252 Ch. 9 Lecture 1 JMB 2014 9 th edition Slide 13

9. 7: Tolerance Limits q For a normal distribution of unknown mean μ, and unknown standard deviation σ, tolerance limits are given by x + ks where k is determined so that one can assert with 100(1 -γ)% confidence that the given limits contain at least the proportion 1 -α of the measurements. q Table A. 7 (page 745) gives values of k for (1 -α) = 0. 9, 0. 95, or 0. 99 and γ = 0. 05 or 0. 01 for selected values of n. EGR 252 Ch. 9 Lecture 1 JMB 2014 9 th edition Slide 14

Case Study 9. 1 c (Page 281) q Find the 99% tolerance limits that will contain 95% of the metal pieces produced by the machine, given a sample mean diameter of 1. 0056 cm and a sample standard deviation of 0. 0246. q Table A. 7 (page 745) Ø Ø Ø (1 - α ) = 0. 95 (1 – Ƴ ) = 0. 99 n=9 k = 4. 550 x ± ks = 1. 0056 ± (4. 550) (0. 0246) q We can assert with 99% confidence that the tolerance interval from 0. 894 to 1. 117 cm will contain 95% of the metal pieces produced by the machine. EGR 252 Ch. 9 Lecture 1 JMB 2014 9 th edition Slide 15

Summary Confidence interval population mean μ Prediction interval a new observation x 0 Tolerance interval a (1 -α) proportion of the measurements can be estimated with 100( 1 -Ƴ )% confidence EGR 252 Ch. 9 Lecture 1 JMB 2014 9 th edition Slide 16