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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 Fall 2011

  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 Fall 2011

  3. Examples Fall 2011

  4. More Examples Fall 2011

  5. Basic Statistics (review) Sample Mean: A class project involved the formation of three 10-person teams (Team Q, Team R and Team S). At the end of the project, team members were asked to give themselves and each other a grade on their contribution to the group. A random sample from two of the teams yielded the following results: = 87.5 = 85.0 Fall 2011

  6. Basic Statistics (review) • Sample variance equation: • For our example: • Calculate the sample standard deviation (s) for each group. • SQgroup = 7.59386 and SSgroup = 7.25718 Fall 2011

  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: Fall 2011

  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) Note that this equation for Z applies when dealing with sample data. Compare to the Z equation for the population. Fall 2011

  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. Fall 2011

  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, Fall 2011

  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 Fall 2011

  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 Fall 2011

  13. Example • Look at example 8.7, pg. 245: 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. 739-740) Χ20.025 =11.143 Χ20.975 = 0.484 3.26 Fall 2011

  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 Fall 2011

  15. t- Distribution • Recall, by Central Limit Theorem: is n(z; 0,1) • Assumption: _____________________ (Generally, if an engineer is concerned with a familiar process or system, this is reasonable, but …) Fall 2011

  16. What if we don’t know σ? • New statistic: Where, and follows a t-distribution with ν = n – 1 degrees of freedom. Fall 2011

  17. Characteristics of the t-Distribution • Look at Figure 8.8, pg. 248 • Note: • Shape: _________________________ • Effect of ν: __________________________ • See table A.4, pp. 737-738 Note that the table yields the right tail of the distribution. Fall 2011

  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) Fall 2011

  19. 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 Fall 2011

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