HS 67Sampling Distributions1 Chapter 11 Sampling Distributions

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Transcript of HS 67Sampling Distributions1 Chapter 11 Sampling Distributions

  • Slide 1
  • HS 67Sampling Distributions1 Chapter 11 Sampling Distributions
  • Slide 2
  • HS 67Sampling Distributions2 Parameters and Statistics Parameter a constant that describes a population or probability model, e.g., from a Normal distribution Statistic a random variable calculated from a sample e.g., x-bar These are related but are not the same! For example, the average age of the SJSU student population = 23.5 (parameter), but the average age in any sample x-bar (statistic) is going to differ from
  • Slide 3
  • HS 67Sampling Distributions3 Example: Does This Wine Smell Bad? Dimethyl sulfide (DMS) is present in wine causing off-odors Let X represent the threshold at which a person can smell DMS X varies according to a Normal distribution with = 25 and = 7 (g/L)
  • Slide 4
  • HS 67Sampling Distributions4 Law of Large Numbers This figure shows results from an experiment that demonstrates the law of large numbers (will be discussed in class)
  • Slide 5
  • HS 67Sampling Distributions5 Sampling Distributions of Statistics u The sampling distribution of a statistic predicts the behavior of the statistic in the long run u The next slide show a simulated sampling distribution of mean from a population that has X~N(25, 7). We take 1,000 samples, each of n =10, from population, calculate x-bar in each sample and plot.
  • Slide 6
  • HS 67Sampling Distributions6 Simulation of a Sampling Distribution of xbar
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  • HS 67Sampling Distributions7 and of x-bar Square root law x-bar is an unbiased estimator of
  • Slide 8
  • HS 67Sampling Distributions8 Sampling Distribution of Mean Wine tasting example Population X~N(25,7) Sample n = 10 By sq. root law, xbar = 7 / 10 = 2.21 By unbiased property, center of distribution = Thus x-bar~N(25, 2.21)
  • Slide 9
  • HS 67Sampling Distributions9 Illustration of Sampling Distribution: Does this wine taste bad? What proportion of samples based on n = 10 will have a mean less than 20? (A)State: Pr(x-bar 20) = ? Recall x-bar~N(25, 2.21) when n = 10 (B)Standardize: z = (20 25) / 2.21 = -2.26 (C)Sketch and shade (D)Table A: Pr(Z < 2.26) =.0119
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  • HS 67Sampling Distributions10 Central Limit Theorem No matter the shape of the population, the distribution of x-bars tends toward Normality
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  • HS 67Sampling Distributions11 Central Limit Theorem Time to Complete Activity Example Let X time to perform an activity. X has = 1 & = 1 but is NOT Normal:
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  • HS 67Sampling Distributions12 Central Limit Theorem Time to Complete Activity Example These figures illustrate the sampling distributions of x-bars based on (a) n = 1 (b) n = 10 (c) n = 20 (d) n = 70
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  • HS 67Sampling Distributions13 Central Limit Theorem Time to Complete Activity Example The variable X is NOT Normal, but the sampling distribution of x-bar based on n = 70 is Normal with x-bar = 1 and x-bar = 1 / sqrt(70) = 0.12, i.e., xbar~N(1,0.12) What proportion of x-bars will be less than 0.83 hours? (A) State: Pr(xbar < 0.83) (B) Standardize: z = (0.83 1) / 0.12 = 1.42 (C) Sketch: right (D) Pr(Z < 1.42) = 0.0778 0 xbar z -1.42