Chapter 11 Sampling Distributions HS 67 Sampling Distributions

Chapter 11 Sampling Distributions HS 67 Sampling Distributions 1

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 of the SJSU student population µ = 23. 5 (parameter), but the average in any sample x-bar (statistic) is going to differ from µ HS 67 Sampling Distributions 2

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 67 Sampling Distributions 3

Law of Large Numbers This figure shows results from an experiment that demonstrates the law of large numbers (will be discussed in class) HS 67 Sampling Distributions 4

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 67 Sampling Distributions 5

Simulation of a Sampling Distribution of xbar HS 67 Sampling Distributions 6

μ and σ of x-bar is an unbiased estimator of μ HS 67 Sampling Distributions Square root law 7

Sampling Distribution of Mean Wine tasting example Population X~N(25, 7) Sample n = 10 By sq. root law, σxbar = 7 / √ 10 = 2. 21 By unbiased property, center of distribution = μ Thus x-bar~N(25, 2. 21) HS 67 Sampling Distributions 8

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 HS 67 Sampling Distributions 9

Central Limit Theorem No matter the shape of the population, the distribution of x-bars tends toward Normality HS 67 Sampling Distributions 10

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 67 Sampling Distributions 11

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 67 Sampling Distributions 12

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 μx-bar = 1 and σx-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 xbar z HS 67 Sampling Distributions -1. 42 0 13