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Business Statistics: A Decision-Making Approach

8th Edition

Chapter 7Introduction to Sampling Distributions

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Chapter Goals

After completing this chapter, you should be able to:

Define the concept of sampling error

Determine the mean and standard deviation for the sampling distribution of the sample mean, x

Determine the mean and standard deviation for the sampling distribution of the sample proportion, p

Describe the Central Limit Theorem and its importance

Apply sampling distributions for both x and p

_

_ _

_

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Sampling Error

Sample Statistics are used to estimate Population Parameters

ex: X is an estimate of the population mean, μ

Problems:

Different samples provide different estimates of the population parameter

Sample results have potential variability, thus sampling error exits

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Calculating Sampling Error

Sampling Error:

The difference between a value (a statistic) computed from a sample and the corresponding value (a parameter) computed from a population

Example: (for the mean)

where:

μ - xError Sampling

mean population μmean samplex

Always present

just because you sample!

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Example

If the population mean is μ = 98.6 degrees and a sample of n = 5 temperatures yields a sample mean of = 99.2 degrees, then the sampling error is

degrees0.698.699.2μx

x

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Sampling Errors

The sampling error may be positive or negative ( may be greater than or less than μ)

The size of the error depends on the sample selected i.e., a larger sample does not necessarily produce a smaller

error if it is not a representative sample Download “Sampling Distributions using Excel”

First sheet: “Sampling Error”

x

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Sampling Distribution

A sampling distribution is a distribution of all possible values of a statistic for a given sample size that has been randomly selected from a population.

Download “Sampling Distributions using Excel” Second sheet: “Sampling Dist”

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For any population, The average value of all possible sample means

computed from all possible random samples of a given size from the population is equal to the population mean (call “Unbiased Estimator”): See “Sampling Distributions using Excel”

Sampling Distribution Properties

μμx Theorem 1Considered an

“unbiased” estimator

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The standard deviation of the possible sample means computed from all random samples of size n is equal to the population standard deviation divided by the square root of the sample size (call Standard Error): Try using “Sampling Distributions using Excel” Not Equal!

Sampling Distribution Properties

n

σσx

Theorem 2

Also called the standard error

(continued)

Finite Population Correction

Virtually all survey research, sampling is conducted without replacement from populations that are of a finite size N.

In these cases, particularly when the sample size n is not small in comparison with the population size N (i.e., more than 5% of the population is sampled)

so that n/N > 0.05, a finite population correction factor (fpc) is used to define both the standard error of the mean and the standard error of the proportion.

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Impact of fpc by Sample Size

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Finite Population Correction

Apply the Finite Population Correction (fpc) if: The sample size is greater than 5% of population size. Only with sampling without replacement

Then

See fpc by “Sampling Distributions using Excel”1NnN

n

σ

μ)x(z

Sampling Distribution Properties

As the sample size is increased, the StdDev of the sampling distribution is reduced…..

That is, the potential for extreme sampling error is reduced when larger sample size are used.

Graphical illustration: next slide

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(continued)

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The value of becomes closer to μ as n

increases):

Sampling Distribution Properties

Larger sample size

Small sample size

x

(continued)

nσ/σx

μ

x

x

Population

As n increases,

decreases

x

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If the Population is Normal

If a population is normal with mean μ and standard deviation σ, the sampling distribution

of is also normally distributed with

and

x

μμx n

σσx

Theorem 3

As n increases the data behaves more like a normal distribution

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z-value for Sampling Distribution of x

z-value for the sampling distribution of :

where: = sample mean

= population mean

= population standard deviation

n = sample size

xμσ

n

σμ)x(

z

x

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Example

Suppose that a population is known to be normally distributed with μ = 2,000 and σ = 230 and random sample of size n = 8 is selected.

Because the population is normally distributed, the sampling distribution for the mean will also be normally distributed.

What is the probability that the sample mean will exceed 2,100?

Convert to Z value Using Excel: 1-NORMSDIST(1.23) Answer on the website

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If the Population is not Normal

We can apply the Central Limit Theorem:

Even if the population is not normal, …sample means from the population will be

approximately normal as long as the sample size is large enough

…and the sampling distribution will have

andμμx n

σσx

Theorem 4

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n↑

Central Limit Theorem

As the sample size gets large enough…

the sampling distribution becomes almost normal regardless of shape of population

x

If necessary, watch the Video

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How Large is Large Enough?

For most distributions, n > 30 will give a sampling distribution that is nearly normal For fairly symmetric distributions, n > 15 is

sufficient

For normal population distributions, the sampling distribution of the mean is always normally distributed

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Example

Suppose a population (not normally distributed) has mean μ = 8 and standard deviation σ = 3 and random sample of size n = 36 (greater than 30) is selected.

What is the probability that the sample mean is between 7.8 and 8.2?

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Example

Solution (continued) -- find z-scores:(continued)

0.31080.4)zP(-0.4

363

8-8.2

nσ

μ- μ

363

8-7.8P 8.2) μ P(7.8 x

x

z7.8 8.2 -0.4 0.4

Sampling Distribution

Standard Normal Distribution 0.1554

+0.1554

Population Distribution

??

??

?????

??? Sample Standardize

8μ 8μx 0μz x x

Sampling Distribution of a Proportion

Try this by yourself! In many instances, the objective of sample is to

estimate a population proportion. An accountant may be interested in determining the proportion

of accounts payable balances that are correct. A production supervisor may wish to determine the percentage

of product that is defect free. A marketing research department might want to know the

proportion of potential customers who will purchase a prticular product.

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Population Proportions Example

If the true proportion of voters who support Proposition

A is π (population proportion) = 0.4, what is the probability

that a sample of size 200 yields a sample proportion

between 0.40 and 0.45?

i.e.: if π = 0.4 and n = 200, what is

P(0.40 ≤ p ≤ 0.45) ?

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Example

if π = .4 and n = 200, what is

P(0.40 ≤ p ≤ 0.45) ?

(continued)

.034640200

.4)0.4(10

n

π)π(1σp

1.44)zP(0

.03464

.4000.450z

.03464

.4000.400P.45)0pP(0.40

Find :

Convert to standard normal (z-values):

pσ

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Use standard normal table: P(0 ≤ z ≤ 1.44) = 0.4251

Example

z0.45 1.44

0.4251

Standardize

Sampling DistributionStandardized

Normal Distribution

if π = 0.4 and n = 200, what is

P(0.40 ≤ p ≤ 0.45) ?

(continued)

0.40 0p

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Population Proportions, π

π = the proportion of the population having

some characteristic

Sample proportion ( p ) provides an estimate of π :

If two outcomes, p is a binomial distribution

size sample

sampletheinsuccessesofnumber

n

xp

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Sampling Distribution of p

Approximated by a

normal distribution if:

where

and

(where π = population proportion)

Sampling DistributionP( p )

.3

.2

.1 0

0 . 2 .4 .6 8 1 p

πμp n

π)π(1σp

5π)n(1

5nπ

Theorem 5

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z-Value for Proportions

If sampling is without replacement

and n is greater than 5% of the

population size, then must use

the finite population correction

factor:

1N

nN

n

π)π(1σp

nπ)π(1

πp

σ

πpz

p

Standardize p to a z value with the formula:

pσ

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Using the Sample Distribution for Proportions

1. Determine the population proportion, p2. Calculate the sample proportion, p

3. Derive the mean and standard deviation of the sampling distribution

4. Define the event of interest

5. If np and n(1-p) are both > 5, then covert p to z-value

6. Use standard normal table (Appendix D) to determine the probability

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Chapter Summary

Discussed sampling error Introduced sampling distributions Described the sampling distribution of the mean

For normal populations Using the Central Limit Theorem (normality unknown)

Described the sampling distribution of a proportion

Calculated probabilities using sampling distributions

Discussed sampling from finite populations

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