Populations and Samples

1. Populations and Samples • Population: “a group of individual persons, objects, or items from which samples are taken for statistical measurement” • Sample: “a finite part of a statistical population whose properties are studied to gain information about the whole” (Merriam-Webster Online Dictionary, http://www.m-w.com/, October 5, 2004) Spring 2008

2. Population Students pursuing undergraduate engineering degrees Cars capable of speeds in excess of 160 mph. Samples Examples • 1000 engineering students selected at random from all engineering programs in the US • 50 cars selected at random from among those certified as having achieved 160 mph or more during 2003 Spring 2008

3. Population Potato chips produced at the Frito-Lay plant in Kathleen Freshwater lakes and rivers Samples 10 chips selected at random every 5 minutes as the conveyor passes the inspector 4 samples taken from randomly selected locations in randomly selected and representative freshwater lakes and rivers Examples (cont.) Spring 2008

4. 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 Spring 2008

5. Basic Statistics (review) 1. Sample Variance: • For our example: SQ2 = ___________________ SS2= ___________________ • S2Q = 7.593857 • S2S = 7.25718 Spring 2008

6. 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 σ2, then: Spring 2008

7. Central Limit Theorem • Given: • X :the mean of a random sample of size n taken from a population with mean μ and finite variance σ2, • Then, • the limiting form of the distribution of is the standard normal distribution n(z;0,1) Spring 2008

8. Central Limit Theorem • 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. Spring 2008

9. Sampling Distribution of the Difference Between Two Averages • Given: • Two samples of size n1 and n2 are taken from two populations with means μ1 and μ2 and variances σ12 and σ22 • Then, Spring 2008

10. Sampling Distribution of S2 • Given: • If S2 is the variance of of a random sample of size n taken from a population with mean μ and finite variance σ2, • Then, has a χ2 distribution with ν = n - 1 Spring 2008

11. χ2 Distribution • χα2represents the χ2value above which we find an area of α, that is, for which P(χ2> χα2) = α. Spring 2008

12. 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 = 1 n = 5 s2 = 0.815 If the χ2 value fits within an interval that covers 95% of the χ2 values with 4 degrees of freedom, then the estimate for σ is reasonable. (See Table A.5, pp. 755-756) Χ20.025 =11.143 Χ20.975 = 0.484 Spring 2008

13. Your turn … • If a sample of size 7 is taken from a normal population (i.e., n = 7), what value of χ2 corresponds to P(χ2 < χα2) = 0.95? (Hint: first determine α.) 12.592 Spring 2008

14. 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 …) Spring 2008

15. What if we don’t know σ? • New statistic: Where, and follows a t-distribution with ν = n – 1 degrees of freedom. Spring 2008

16. Characteristics of the t-Distribution • Look at fig. 8.11, pg. 221*** • Note: • Shape: _________________________ • Effect of ν: __________________________ • See table A.4, pp. 753-754 Spring 2008

17. Comparing Variances of 2 Samples • Given two samples of size n1and n2, with sample means X1 and X2, and variances, s12and s22 … 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 Spring 2008

18. F-Distribution • Given: • S12 and S22, the variances of independent random samples of size n1 and n2taken from normal populations with variances σ12 and σ22, respectively, • Then, has an F-distribution with ν1 = n1 - 1 and ν2 = n2 – 1 degrees of freedom. (See table A.6, pp. 757-760) Spring 2008