STA 291Summer 2010

Lecture 11Dustin Lueker

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Reduce Sampling Variability The larger the sample size, the smaller the

sampling variability Increasing the sample size to 25…

10 samplesof size n=25

100 samplesof size n=25

1000 samplesof size n=25

STA 291 Summer 2010 Lecture 11

X

Population with mean m and standard deviation s

X

X

XXXXX

X

• If you repeatedly take random samples andcalculate the sample mean each time, thedistribution of the sample mean follows apattern• This pattern is the sampling distribution

Sampling Distribution

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Example of Sampling Distribution of the Mean

As n increases, the variability decreases and

the normality (bell-shapedness) increases.

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Effect of Sample Size The larger the sample size n, the

smaller the standard deviation of the sampling distribution for the sample mean◦ Larger sample size = better precision

As the sample size grows, the sampling distribution of the sample mean approaches a normal distribution◦ Usually, for about n=30, the sampling

distribution is close to normal◦ This is called the “Central Limit Theorem”

xn

STA 291 Summer 2010 Lecture 11

If X is a random variable from a normal population with a mean of 20, which of these would we expect to be greater? Why?◦ P(15<X<25)◦ P(15< <25)

What about these two?◦ P(X<10)◦ P( <10)

Examples

STA 291 Summer 2010 Lecture 11 6

x

x

7

Sampling Distribution of the Sample Mean When we calculate the sample mean, ,

we do not know how close it is to the population mean ◦ Because is unknown, in most cases.

On the other hand, if n is large, ought to be close to

x

x

STA 291 Summer 2010 Lecture 11

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Parameters of the Sampling Distribution If we take random samples of size n from a

population with population mean and population standard deviation , then the sampling distribution of

◦ has mean

◦ and standard error

The standard deviation of the sampling distribution of the mean is called “standard error” to distinguish it from the population standard deviation

x

nxSD x

)(

xxE )(

STA 291 Summer 2010 Lecture 11

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Standard Error The example regarding students in STA 291 For a sample of size n=4, the standard

error of is

For a sample of size n=25,

0.50.25

4X n

0.50.1

25X n

x

STA 291 Summer 2010 Lecture 11

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Central Limit Theorem

For random sampling, as the sample size n grows, the sampling distribution of the sample mean, , approaches a normal distribution◦ Amazing: This is the case even if the population

distribution is discrete or highly skewed Central Limit Theorem can be proved

mathematically◦ Usually, the sampling distribution of is

approximately normal for n≥30◦ We know the parameters of the sampling

distribution

x

x

xxE )(n

xSD x

)(STA 291 Summer 2010 Lecture

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Example Household size in the United States (1995)

has a mean of 2.6 and a standard deviation of 1.5

For a sample of 225 homes, find the probability that the sample mean household size falls within 0.1 of the population mean

Also find

)7.25.2( xP

)1.32(. xP

STA 291 Summer 2010 Lecture 11

Binomial Population

with proportion p of successes

• If you repeatedly take random samples andcalculate the sample proportion each time, thedistribution of the sample proportion follows apattern

p̂p̂p̂p̂p̂p̂p̂p̂

p̂

Sampling Distribution

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Example of Sampling Distributionof the Sample Proportion

As n increases, the variability decreases and

the normality (bell-shapedness) increases.

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For random sampling, as the sample size n grows, the sampling distribution of the sample proportion, , approaches a normal distribution◦ Usually, the sampling distribution of is

approximately normal for np≥5, nq≥5◦ We know the parameters of the sampling

distribution

Central Limit Theorem (Binomial Version)

14

p̂

p̂

ppE p ˆ)ˆ(

n

qp

n

pppSD p

)()1()ˆ( ˆ

STA 291 Summer 2010 Lecture 11

Take a SRS with n=100 from a binomial population with p=.7, let X = number of successes in the sample

Find

Does this answer make sense?

Also Find

Does this answer make sense?

Example

15

)8.ˆ( pP

)65( XP

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Mean of sampling distribution

Mean/center of the sampling distribution for sample mean/sample proportion is always the same for all n, and is equal to the population mean/proportion.

ppE

xE

p

x

ˆ)ˆ(

)(

STA 291 Summer 2010 Lecture 11

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Reduce Sampling Variability The larger the sample size n, the smaller

the variability of the sampling distribution

Standard Error◦ Standard deviation of the sample mean or

sample proportion◦ Standard deviation of the population divided by n

nxSD x

)(n

qp

n

pppSD p

)()1()ˆ( ˆ

STA 291 Summer 2010 Lecture 11