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Chapter 6: Sampling Distributions
McClave: Statistics, 11th ed. Chapter 6: Sampling Distributions
2
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
3
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
11
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
12
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
13
)( 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
17
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
18
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
19
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
20
0548.)60.1(
16250
22002300
)2300(
zP
zP
xP
6.3: The Sampling Distribution of and the Central Limit Theorem