1 Chapter 7 Sampling Distributions. 2 Chapter Outline Selecting A Sample Point Estimation ...

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Chapter 7

Sampling Distributions

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Chapter Outline

Selecting A Sample Point Estimation Introduction to Sampling Distributions Sampling Distribution of Sampling Distribution of

xp

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Introduction

The reason we select a sample is to collect data to answer a research question about a population.

To directly study the population is probably too costly and too time-consuming, while to study a portion of it, i.e. a sample is much more manageable.

The sample results provide only estimates of the values of the population characteristics.

With proper sampling methods, the sample results can provide ‘good’ estimates of the population characteristics.

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Sampling from A Finite Population

A simple random sample of size n from a finite population of size N is a sample selected such that each possible sample of size n has the same probability of being selected.

Replacing each sampled element before selecting subsequent elements is called sampling with replacement. Sampling without replacement is the procedure used most often.

In large sampling projects, computer-generated random numbers are often used to automate the sample selection process.

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Sampling from An Infinite Population

It’s impossible to obtain a list of all elements in an infinite population.

Populations are often generated by an ongoing process where there is no upper limit on the number of units that can be generated.

A random sample must be selected with the following conditions satisfied: Each element selected comes from the population of

interest. Each element is selected independently.

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Point Estimation

Point estimation is a form of statistical inference. In point estimation we use the data from the sample to compute a value of a sample statistic the serves as an estimate of

a population parameter.

Sample Statistics

Population Parameters

Sample mean:

Sample variance:

Sample proportion:

Population mean:

Population variance:

Population proportion:

X

2S 2

P P

Point Estimates

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Point Estimation

Example: Checking Accounts A local small bank has a total of 600 checking A local small bank has a total of 600 checking accounts. The average daily balance of all the checking accounts. The average daily balance of all the checking accounts is $310 with a standard deviation of $66. The accounts is $310 with a standard deviation of $66. The proportion of accounts with a daily balance of no less proportion of accounts with a daily balance of no less than $500 is 30%.than $500 is 30%.

To find point estimates for the population, aTo find point estimates for the population, a random random sample of 121 checking accounts at the bank are chosen, sample of 121 checking accounts at the bank are chosen, which shows an average daily balance of $306. The which shows an average daily balance of $306. The standard deviation of the sample is $61, and the sample standard deviation of the sample is $61, and the sample proportion of accounts with a daily balance of no less proportion of accounts with a daily balance of no less than $500 is 27%.than $500 is 27%.

The following table summarizes the point estimates The following table summarizes the point estimates from the sample of 121 checking accounts.from the sample of 121 checking accounts.

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Summary of Point Estimates of A Simple Random Sample of 121 Checking Accounts

PopulationPopulationParameterParameter

PointPointEstimatorEstimator

PointPointEstimateEstimate

ParameterParameterValueValue

= Population mean= Population mean account balance account balance

$310$310 $306$306

= Population std.= Population std. deviation for deviation for account balance account balance

$66$66 s s = Sample std.= Sample std. deviation fordeviation for account balance account balance

$61$61

pp = Population pro- = Population proportion of accountportion of account balance no less thanbalance no less than $500 $500

.3.3 .27.27

= Sample mean= Sample mean account balance account balance

xx

= Sample pro-= Sample proportion of accountportion of account balance no less thanbalance no less than $500 $500

pp

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Sampling Distribution of

The The sampling distribution of sampling distribution of is the is the probability distribution of probability distribution of all possible valuesall possible values of the sample mean .of the sample mean .

x

x

x

where: where: = the population mean = the population mean

EE( ) = ( ) = xx

When the expected value of the point estimatorWhen the expected value of the point estimator

equals the population parameter, we say the pointequals the population parameter, we say the point

estimator is estimator is unbiasedunbiased..

Expected Value ofExpected Value ofx

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x

x Standard Deviation of

Finite PopulationFinite Population

)(1 nN

nNx

)(1 nN

nNx

x n

x n

Infinite PopulationInfinite Population

= the standard deviation of = the standard deviation of xx xx

= the standard deviation of the population = the standard deviation of the population

nn = the sample size = the sample size

NN = the population size = the population size

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x

x Standard Deviation of

Finite PopulationFinite Population

)(1 nN

nNx

)(1 nN

nNx

x n

x n

Infinite PopulationInfinite Population

• is referred to as the is referred to as the standard standard errorerror of the of the mean.mean.

x x

• A finite population is treated as beingA finite population is treated as being infinite if infinite if nn//NN << .05. .05.

• is the is the finite populationfinite population correction factorcorrection factor..

( ) / ( )N n N 1( ) / ( )N n N 1

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Central Limit Theorem

When the population from which we are selecting When the population from which we are selecting

a random sample does not have a normal distribution,a random sample does not have a normal distribution,

the the central limit theoremcentral limit theorem is helpful in identifying the is helpful in identifying the

shape of the sampling distribution of . shape of the sampling distribution of . x

CENTRAL LIMIT THEOREMCENTRAL LIMIT THEOREM

In selecting random samples of size In selecting random samples of size nn from from a population, the a population, the sampling distribution of the sampling distribution of the sample meansample mean can be approximated by a can be approximated by a normal distribution as the sample size normal distribution as the sample size becomes large.becomes large.

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Sampling Distribution of x

When the population follows a normal distribution, the sampling distribution of is normally distributed for any sample size.

According to the Central Limit Theorem, when the sample size is large enough (at least 30 in most cases), the sampling distribution of can be approximated by a normal distribution.

The point of studying the sampling distribution of

is to better estimate the population mean .

x

x

x

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Process of Statistical InferenceProcess of Statistical Inference

The value of is used toThe value of is used tomake inferences aboutmake inferences about

the value of the value of ..

xx The sample data The sample data provide a value forprovide a value for

the sample meanthe sample mean . .xx

A simple random sampleA simple random sampleof of nn elements is selected elements is selected

from the population.from the population.

Population Population with meanwith mean

= ?= ?

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Example: Checking AccountsExample: Checking Accounts Suppose the population of checking accounts follows a normal Suppose the population of checking accounts follows a normal

distribution, a random sample of 121 checking accounts should also distribution, a random sample of 121 checking accounts should also follow a normal distribution with the mean account balance of $310 and follow a normal distribution with the mean account balance of $310 and the standard error of 6. the standard error of 6.

6121

66x

310$xEx

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Example: Checking AccountsExample: Checking Accounts What is the probability that a simple random sample of 121 checking What is the probability that a simple random sample of 121 checking

accounts will provide an point estimate of the population mean account accounts will provide an point estimate of the population mean account balance within +/- $5 of the actual population mean? (i.e. between $305 balance within +/- $5 of the actual population mean? (i.e. between $305 and $315)and $315)

x$305 $315

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Example: Checking AccountsExample: Checking Accounts Since follows a normal distribution, we can first calculate the Since follows a normal distribution, we can first calculate the z z values values

of both cutoff points as follows:of both cutoff points as follows:

x

xExz

x

83.06

310305

83.0

6

310315

x

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Example: Checking AccountsExample: Checking Accounts Next, we check the cumulative probabilities for the standard normal Next, we check the cumulative probabilities for the standard normal

distribution table for areas corresponding to the distribution table for areas corresponding to the z z values. Recall that the values. Recall that the numbers in the table represent the areas under the standard normal curve numbers in the table represent the areas under the standard normal curve to the to the LEFTLEFT of of z z values.values.

z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09

. . . . . . . . . . .

.5 .6915 .6950 .6985 .7019 .7054 .7088 .7123 .7157 .7190 .7224

.6 .7257 .7291 .7324 .7357 .7389 .7422 .7454 .7486 .7517 .7549

.7 .7580 .7611 .7642 .7673 .7704 .7734 .7764 .7794 .7823 .7852

.8 .7881 .7910 .7939 .7967 .7995 .8023 .8051 .8078 .8106 .8133

.9 .8159 .8186 .8212 .8238 .8264 .8289 .8315 .8340 .8365 .8389

. . . . . . . . . . .

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Example: Checking AccountsExample: Checking Accounts The area under the standard normal curve to the left of 0.83The area under the standard normal curve to the left of 0.83

00 0.830.83

Area = .7967Area = .7967

zz

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Example: Checking AccountsExample: Checking Accounts The area under the standard normal curve to the left of -0.83The area under the standard normal curve to the left of -0.83

Area = 1-0.7967Area = 1-0.7967 = 0.2033= 0.2033

zz00-0.83-0.83

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Example: Checking AccountsExample: Checking Accounts The area we are looking for is the area under the normal curve between The area we are looking for is the area under the normal curve between

the cutoff points. Therefore, the probability is calculated asthe cutoff points. Therefore, the probability is calculated as

PP(-.83 (-.83 << zz << .83) = .83) = PP((zz << .83) .83) -- PP((zz << -.83) -.83)

= .7967 = .7967 -- .2033 .2033= .5934= .5934

The probability that the sample mean account balance The probability that the sample mean account balance will be between $305 and $315 is:will be between $305 and $315 is:

PP(305 (305 << << 315) = .5934 315) = .5934xx

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Example: Checking AccountsExample: Checking Accounts

xx$315$315$305$305$310$310

Area = .5934Area = .5934

6x

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Relationship Between the Sample Size and the Sampling Distribution of

As the sample size increases, the standard error becomes smaller.

With a smaller standard error, the values of have less variability and tend to be closer to the population mean.

x

x

nx

x

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Relationship Between the Sample Size and the Sampling Distribution of

Example: Checking Accounts

x

310$xEx

With With n=n=121 ,121 ,

With With nn = 225, = 225,

6x4.4x

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Sampling Distribution of p

E p p( ) E p p( )

where:where:pp = the population proportion = the population proportion

The The sampling distribution of sampling distribution of is the probability is the probabilitydistribution of all possible values of the sampledistribution of all possible values of the sampleproportion .proportion .

Expected Value ofExpected Value of p

p

p

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Standard Deviation of Standard Deviation of p

n

pp

N

nNp

)1(

1

n

pp

N

nNp

)1(

1

pp p

n ( )1 p

p pn

( )1

• is referred to as the is referred to as the standard standard error oferror of the proportionthe proportion..

p p

• is the finite populationis the finite population correction factor. When correction factor. When n/N n/N is is 5%, the correction 5%, the correction factor is close to 1.factor is close to 1.

( ) / ( )N n N 1( ) / ( )N n N 1

Finite PopulationFinite Population Infinite PopulationInfinite Population

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The sampling distribution of can be The sampling distribution of can be approximated by a normal distribution whenever approximated by a normal distribution whenever the sample size is large enough to satisfy the two the sample size is large enough to satisfy the two conditions:conditions:

. . . because when these conditions are satisfied, . . . because when these conditions are satisfied, the probability distribution of the probability distribution of xx in the sample in the sample proportion, = proportion, = xx//nn, can be approximated by , can be approximated by normal distribution (and because normal distribution (and because nn is a constant). is a constant).

npnp >> 5 5 nn(1 – (1 – pp) ) >> 5 5andand

p

p

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A simple random sampleA simple random sampleof of nn elements is selected elements is selected

from the population.from the population.

Population Population with proportionwith proportion

pp = ? = ?

Making Inferences about a Population ProportionMaking Inferences about a Population Proportion

The sample data The sample data provide a value for provide a value for

thethesample sample

proportionproportion . .

pp

The value of is usedThe value of is usedto make inferencesto make inferences

about the value of about the value of pp..

pp

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Example: Checking Accounts Example: Checking Accounts

In the example, In the example, nn = 121 and = 121 and pp = .3, the sampling distribution = .3, the sampling distribution of can be approximated as a normal distribution because:of can be approximated as a normal distribution because:

npnp = 121(.30) = 36.3 = 121(.30) = 36.3 >> 5 5andand

nn(1-(1-pp) = 121(.70) = 84.7 ) = 121(.70) = 84.7 >> 5 5

p

The standard error of is: p

037.0

121

3.13.

1600

121600

p

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Example: Checking Accounts Example: Checking Accounts

3.0pE

037.0p

p

1 Chapter 7 Sampling Distributions. 2 Chapter Outline Selecting A Sample Point Estimation ...

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