1. EGR 252 - Ch. 8 Part 1 and 2 Spring 2009 Slide 1 Fundamental Sampling Distributions  Introduction to random sampling and statistical inference  Populations and samples  Sampling distribution of means  Central Limit Theorem  Other distributions  S 2  t-distribution  F-distribution  Data displays / Graphical methods

2. EGR 252 - Ch. 8 Part 1 and 2 Spring 2009 Slide 2 Populations and Samples  Population: “a group of individual persons, objects, or items from which samples are taken for statistical measurement” 1  Sample: “a finite part of a statistical population whose properties are studied to gain information about the whole” 1  Population – the totality of the observations with which we are concerned 2  Sample – a subset of the population 2 1 (Merriam-Webster Online Dictionary, http://www.m-w.com/, October 5, 2004)http://www.m-w.com/ 2 Walpole, Myers, Myers, and Ye (2007) Probability and Statistics for Engineers and Scientists

3. EGR 252 - Ch. 8 Part 1 and 2 Spring 2009 Slide 3 Examples PopulationSample Students pursuing undergraduate engineering degrees 1000 engineering students selected at random from all engineering programs in the US Cars capable of speeds in excess of 160 mph. 50 cars selected at random from among those certified as having achieved 160 mph or more during 2003

4. EGR 252 - Ch. 8 Part 1 and 2 Spring 2009 Slide 4 More Examples PopulationSample Potato chips produced at the Frito-Lay plant in Kathleen 10 chips selected at random every 5 minutes as the conveyor passes the inspector Freshwater lakes and rivers 4 samples taken from randomly selected locations in randomly selected and representative freshwater lakes and rivers

5. EGR 252 - Ch. 8 Part 1 and 2 Spring 2009 Slide 5 Basic Statistics (review) Sample Mean:  At the end of a team project, team members were asked to give themselves and each other a grade on their contribution to the group. The results for two team members were as follows: = ___________________ 87.5 = ___________________ 85.0 QS 9285 9588 8575 7892

6. EGR 252 - Ch. 8 Part 1 and 2 Spring 2009 Slide 6 Basic Statistics (review) 1. Sample Variance:  For our example: S Q 2 = ___________________ S S 2 = ___________________  S 2 Q = 7.59386  S 2 S = 7.25718 QS 9285 9588 8575 7892

7. EGR 252 - Ch. 8 Part 1 and 2 Spring 2009 Slide 7 Sampling Distributions  If we conduct the same experiment several times with the same sample size, the probability distribution of the resulting statistic is called a sampling distribution  Sampling distribution of the mean: if n observations are taken from a normal population with mean μ and variance σ 2, then:

8. EGR 252 - Ch. 8 Part 1 and 2 Spring 2009 Slide 8 Central Limit Theorem  Given:  X : the mean of a random sample of size n taken from a population with mean μ and finite variance σ 2,  Then,  the limiting form of the distribution of is the standard normal distribution n(z;0,1)

9. EGR 252 - Ch. 8 Part 1 and 2 Spring 2009 Slide 9 Central Limit Theorem-Distribution of X  If the population is known to be normal, the sampling distribution of X will follow a normal distribution.  Even when the distribution of the population is not normal, the sampling distribution of X is normal when n is large.  NOTE: when n is not large, we cannot assume the distribution of X is normal.

10. EGR 252 - Ch. 8 Part 1 and 2 Spring 2009 Slide 10 Sampling Distribution of the Difference Between Two Averages  Given:  Two samples of size n 1 and n 2 are taken from two populations with means μ 1 and μ 2 and variances σ 1 2 and σ 2 2  Then,

11. EGR 252 - Ch. 8 Part 1 and 2 Spring 2009 Slide 11 Sampling Distribution of S 2  Given:  If S 2 is the variance of of a random sample of size n taken from a population with mean μ and finite variance σ 2,  Then, has a χ 2 distribution with ν = n - 1

12. EGR 252 - Ch. 8 Part 1 and 2 Spring 2009 Slide 12 Chi-squared ( χ 2 ) Distribution  χ α 2 represents the χ 2 value above which we find an area of α, that is, for which P(χ 2 > χ α 2 ) = α. α

13. EGR 252 - Ch. 8 Part 1 and 2 Spring 2009 Slide 13 Example  Look at example 8.10, pg. 256: A manufacturer of car batteries guarantees that his batteries will last, on average, 3 years with a standard deviation of 1 year. A sample of five of the batteries yielded a sample variance of 0.815. Does the manufacturer have reason to suspect the standard deviation is no longer 1 year? μ = 3 σ = 1n = 5 s 2 = 0.815 If the χ 2 value fits within an interval that covers 95% of the χ 2 values with 4 degrees of freedom, then the estimate for σ is reasonable. (See Table A.5, pp. 755-756) Χ 2 0.025 =11.143 Χ 2 0.975 = 0.484

14. EGR 252 - Ch. 8 Part 1 and 2 Spring 2009 Slide 14 Your turn …  If a sample of size 7 is taken from a normal population (i.e., n = 7), what value of χ 2 corresponds to P(χ 2 < χ α 2 ) = 0.95? (Hint: first determine α.) 12.592

15. EGR 252 - Ch. 8 Part 1 and 2 Spring 2009 Slide 15 t- Distribution  Recall, by CLT: is n(z; 0,1)  Assumption: _____________________ (Generally, if an engineer is concerned with a familiar process or system, this is reasonable, but …)

16. EGR 252 - Ch. 8 Part 1 and 2 Spring 2009 Slide 16 What if we don’t know σ?  New statistic: Where, and follows a t-distribution with ν = n – 1 degrees of freedom.

17. EGR 252 - Ch. 8 Part 1 and 2 Spring 2009 Slide 17 Characteristics of the t-Distribution  Look at fig. 8.11, pg. 221***  Note:  Shape:_________________________  Effect of ν: __________________________  See table A.4, pp. 753-754

18. EGR 252 - Ch. 8 Part 1 and 2 Spring 2009 Slide 18 Comparing Variances of 2 Samples  Given two samples of size n 1 and n 2, with sample means X 1 and X 2, and variances, s 1 2 and s 2 2 … Are the differences we see in the means due to the means or due to the variances (that is, are the differences due to real differences between the samples or variability within each samples)? See figure 8.16, pg. 226

19. EGR 252 - Ch. 8 Part 1 and 2 Spring 2009 Slide 19 F-Distribution  Given:  S 1 2 and S 2 2, the variances of independent random samples of size n 1 and n 2 taken from normal populations with variances σ 1 2 and σ 2 2, respectively,  Then, has an F-distribution with ν 1 = n 1 - 1 and ν 2 = n 2 – 1 degrees of freedom. (See table A.6, pp. 757-760)

20. EGR 252 - Ch. 8 Part 1 and 2 Spring 2009 Slide 20 Data Displays/Graphical Methods  Box and Whisker Plot  Page 236  Min Max values  Q1 Q2 Q3  Interquartile range  Quantile-Quantile Plot  Normal Probability Plot  Minitab