Chapter 6: Sampling Distributions

McClave: Statistics, 11th ed. Chapter 6: Sampling Distributions

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Where We’ve Been

The objective of most statistical analyses is inference

Sample statistics (mean, standard deviation) can be used to make decisions

Probability distributions can be used to construct models of populations

McClave: Statistics, 11th ed. Chapter 6: Sampling Distributions

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Where We’re Going

Develop the notion that sample statistic is a random variable with a probability distribution

Define a sampling distribution for a sample statistic

Link the sampling distribution of the sample mean to the normal probability distribution

In practice, sample statistics are used to estimate population parameters. A parameter is a numerical descriptive

measure of a population. Its value is almost always unknown.

A sample statistic is a numerical descriptive measure of a sample. It can be calculated from the observations.

4McClave: Statistics, 11th ed. Chapter 6: Sampling Distributions

Parameter Statistic

Mean µ

Variance 2 s2

Standard Deviation s

Binomial proportion p p̂

5McClave: Statistics, 11th ed. Chapter 6: Sampling Distributions

6.1: The Concept of a Sampling Distribution

6McClave: Statistics, 11th ed. Chapter 6: Sampling Distributions

Since we could draw many different samples from a population, the sample statistic used to estimate the population parameter is itself a random variable.

The sampling distribution of a sample statistic calculated from a sample of n measurements is the probability distribution of the statistic.

6.1: The Concept of a Sampling Distribution

7McClave: Statistics, 11th ed. Chapter 6: Sampling Distributions

n = 1 1 12 23 3

n = 2 1, 2 1.51, 3 22, 3 2.5

n = 3 ( = N)

1, 2, 3 2

Imagine a very small population consisting of the elements 1, 2 and 3.Below are the possible samples that could be drawn, along with the

means of the samples and the mean of the means.

23

x2

3 x

21

x

6.1: The Concept of a Sampling Distribution

8McClave: Statistics, 11th ed. Chapter 6: Sampling Distributions

µTo estimate

…should we use …

… or … the median ?

6.1: The Concept of a Sampling Distribution

9McClave: Statistics, 11th ed. Chapter 6: Sampling Distributions

µTo estimate

…should we use …

… or … the median ?

Yes!(Depending on the distribution of the random variable.)

6.2: Properties of Sampling Distributions: Unbiasedness and Minimum Variance

10McClave: Statistics, 11th ed. Chapter 6: Sampling Distributions

A point estimator is a single number based on sample data that can be used as an estimator of the population parameter

µ

p

s2 2

p̂

6.2: Properties of Sampling Distributions: Unbiasedness and Minimum Variance

If the sampling distribution of a sample statistic has a mean equal to the population parameter the statistic is intended to estimate, the statistic is said to be an unbiased estimate of the parameter.

McClave: Statistics, 11th ed. Chapter 6: Sampling Distributions

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6.2: Properties of Sampling Distributions: Unbiasedness and Minimum Variance

If two alternative sample statistics are both unbiased, the one with the smaller standard deviation is preferred.

Here, A = B, but A < B, so A is preferred.

McClave: Statistics, 11th ed. Chapter 6: Sampling Distributions

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6.3: The Sampling Distribution of and the Central Limit TheoremProperties of the Sampling Distribution of

The mean of the sampling distribution equals the mean of the population

The standard deviation of the sampling distribution [the standard error (of the mean)] equals the population standard deviation divided by the square root of n

McClave: Statistics, 11th ed. Chapter 6: Sampling Distributions

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)( xEx

nx

14McClave: Statistics, 11th ed. Chapter 6: Sampling Distributions

n = 1 1 12 23 3

n = 2 1, 2 1.51, 3 22, 3 2.5

n = 3 ( = N)

1, 2, 3 2

82.

23

x

x

41.

23

x

x

0

21

x

x

Here’s our small population again, this time with the standard deviations of the sample means. Notice the mean of the sample means in each case equals the

population mean and the standard error falls as n increases.

6.3: The Sampling Distribution of and the Central Limit Theorem

6.3: The Sampling Distribution of and the Central Limit Theorem

15McClave: Statistics, 11th ed. Chapter 6: Sampling Distributions

If a random sample of n observations is drawn from a normally distributed population, the sampling distribution of will be normally distributed

6.3: The Sampling Distribution of and the Central Limit Theorem

16McClave: Statistics, 11th ed. Chapter 6: Sampling Distributions

The Central Limit TheoremThe sampling distribution of , based on a

random sample of n observations, will be approximately normal with

µ = µ and = /n .

The larger the sample size, the better the sampling distribution will approximate the normal distribution.

Suppose existing houses for sale average 2200 square feet in size, with a standard deviation of 250 ft2.

What is the probability that a randomly selected house will have at least 2300 ft2 ?

McClave: Statistics, 11th ed. Chapter 6: Sampling Distributions

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6.3: The Sampling Distribution of and the Central Limit Theorem

Suppose existing houses for sale average 2200 square feet in size, with a standard deviation of 250 ft2.

What is the probability that a randomly selected house will have at least 2300 ft2 ?

McClave: Statistics, 11th ed. Chapter 6: Sampling Distributions

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3446.)40.0(

250

22002300

)2300(

zP

zP

xP

6.3: The Sampling Distribution of and the Central Limit Theorem

Suppose existing houses for sale average 2200 square feet in size, with a standard deviation of 250 ft2.

What is the probability that a randomly selected sample of 16 houses will average at least 2300 ft2 ?

McClave: Statistics, 11th ed. Chapter 6: Sampling Distributions

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6.3: The Sampling Distribution of and the Central Limit Theorem

Suppose existing houses for sale average 2200 square feet in size, with a standard deviation of 250 ft2.

What is the probability that a randomly selected sample of 16 houses will average at least 2300 ft2 ?

McClave: Statistics, 11th ed. Chapter 6: Sampling Distributions

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0548.)60.1(

16250

22002300

)2300(

zP

zP

xP

6.3: The Sampling Distribution of and the Central Limit Theorem