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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. Fundamental Sampling Distributions. Populations and Samples.

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Fundamental Sampling Distributions

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  1. Introduction to random sampling and statistical inference Populations and samples Sampling distribution of means Central Limit Theorem Other distributions S2 t-distribution F-distribution Data displays / Graphical methods Fundamental Sampling Distributions Spring 2009

  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) 2 Walpole, Myers, Myers, and Ye (2007) Probability and Statistics for Engineers and Scientists Spring 2009

  3. Examples Spring 2009

  4. More Examples Spring 2009

  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 Spring 2009

  6. Basic Statistics (review) 1. Sample Variance: • For our example: SQ2 = ___________________ SS2= ___________________ • S2Q = 7.59386 • S2S = 7.25718 Spring 2009

  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 σ2, then: Spring 2009

  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 σ2, • Then, • the limiting form of the distribution of is the standard normal distribution n(z;0,1) Spring 2009

  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. Spring 2009

  10. 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 2009

  11. 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 2009

  12. Chi-squared (χ2) Distribution • χα2represents the χ2value above which we find an area of α, that is, for which P(χ2> χα2) = α. α Spring 2009

  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 = 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 2009

  14. 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 2009

  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 …) Spring 2009

  16. What if we don’t know σ? • New statistic: Where, and follows a t-distribution with ν = n – 1 degrees of freedom. Spring 2009

  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 Spring 2009

  18. 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 2009

  19. 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 2009

  20. Box and Whisker Plot Page 236 Min Max values Q1 Q2 Q3 Interquartile range Quantile-Quantile Plot Normal Probability Plot Minitab Data Displays/Graphical Methods Spring 2009

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