HS 67Sampling Distributions1 Chapter 11 Sampling Distributions

HS 67Sampling Distributions2 Parameters and Statistics Parameter ≡ a constant that describes a population or probability model, e.g., μ 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 µ = 23.5 (parameter), but the average age in any sample x-bar (statistic) is going to differ from µ

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 μ = 25 and σ = 7 (µg/L)

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)

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.

HS 67Sampling Distributions6 Simulation of a Sampling Distribution of xbar

HS 67Sampling Distributions7 μ and σ of x-bar Square root law x-bar is an unbiased estimator of μ

HS 67Sampling Distributions8 Sampling Distribution of Mean Wine tasting example Population X~N(25,7) Sample n = 10 By sq. root law, σ xbar = 7 / √10 = 2.21 By unbiased property, center of distribution = μ Thus x-bar~N(25, 2.21)

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 = (C)Sketch and shade (D)Table A: Pr(Z < –2.26) =.0119

HS 67Sampling Distributions10 Central Limit Theorem No matter the shape of the population, the distribution of x-bars tends toward Normality

HS 67Sampling Distributions11 Central Limit Theorem Time to Complete Activity Example Let X ≡ time to perform an activity. X has µ = 1 & σ = 1 but is NOT Normal:

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

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 μ x-bar = 1 and σ 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) = xbar z -1.42