HS 67 Sampling Distributions 1

Chapter 11

Sampling Distributions

HS 67 Sampling Distributions 2

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

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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)

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Law of Large Numbers

This figure shows results from an experiment that demonstrates the law of large numbers (will be discussed in class)

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Sampling Distributions of Statistics The sampling distribution of a statistic predicts

the behavior of the statistic in the long run 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.

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Simulation of a Sampling Distribution of xbar

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μ and σ of x-bar

Square root law

nx

x-bar is an unbiased estimator of μ

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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 = μThusx-bar~N(25, 2.21)

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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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Central Limit Theorem

No matter the shape of the population, the distribution of x-bars tends toward Normality

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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:

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Central Limit Theorem Time to Complete Activity ExampleThese 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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Central Limit Theorem Time to Complete Activity ExampleThe 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

0

xbar

z -1.42