DISTRIBUSI SAMPLING

Pengertian Distribusi SamplingPengertian Distribusi Sampling Distribusi Sampling Rata-RataDistribusi Sampling Rata-Rata Distribusi Sampling proporsiDistribusi Sampling proporsi

Why Study Sampling Distributions

Sample statistics are used to estimate population parameters

e.g.: ;estimates the population mean:

Problems: Different samples provide different estimates

Large samples give better estimates, but large sample costs more

How good is the estimate? Approach to solution: Theoretical basis is

sampling distribution

50X

• An estimatorestimator of a population parameter is a sample statistic used to estimate or predict the population parameter.

• An estimateestimate of a parameter is a particular numerical value of a sample statistic obtained through sampling.

• A point estimatepoint estimate is a single value used as an estimate of a population parameter.

• An estimatorestimator of a population parameter is a sample statistic used to estimate or predict the population parameter.

• An estimateestimate of a parameter is a particular numerical value of a sample statistic obtained through sampling.

• A point estimatepoint estimate is a single value used as an estimate of a population parameter.

A population parameterpopulation parameter is a numerical measure of a summary characteristic of a population.

Sample Statistics as Sample Statistics as Estimators of Population ParametersEstimators of Population Parameters

A sample statisticsample statistic is a numerical measure of a summary characteristic

of a sample.

• The The sample meansample mean, , is the most common , , is the most common estimator of the estimator of the population meanpopulation mean, ,

• The The sample variancesample variance, , ss22, is the most common , is the most common estimator of the estimator of the population variancepopulation variance, , 22..

• The The sample standard deviationsample standard deviation, , ss, is the most , is the most common estimator of the common estimator of the population standard population standard deviationdeviation, , . .

• The The sample proportionsample proportion, , is the most common , , is the most common estimator of the estimator of the population proportionpopulation proportion, , pp..

• The The sample meansample mean, , is the most common , , is the most common estimator of the estimator of the population meanpopulation mean, ,

• The The sample variancesample variance, , ss22, is the most common , is the most common estimator of the estimator of the population variancepopulation variance, , 22..

• The The sample standard deviationsample standard deviation, , ss, is the most , is the most common estimator of the common estimator of the population standard population standard deviationdeviation, , . .

• The The sample proportionsample proportion, , is the most common , , is the most common estimator of the estimator of the population proportionpopulation proportion, , pp..

EstimatorsEstimators

X

p̂

• The population proportionpopulation proportion is equal to the number of elements in the population belonging to the category of interest, divided by the total number of elements in the population:

• The sample proportionsample proportion is the number of elements in the sample belonging to the category of interest, divided by the sample size:

Population and Sample Population and Sample ProportionsProportions

pxn

pXN

• The sampling distributionsampling distribution of a statistic is the probability distribution of all possible values the statistic may assume, when computed from random samples of the same size, drawn from a specified population.

• The sampling distribution of sampling distribution of XX is the probability distribution of all possible values the random variable may assume when a sample of size n is taken from a specified population.

X

Sampling DistributionsSampling Distributions

The expected value of the sample meanexpected value of the sample mean is equal to the population mean:

E XX X

( )

The variance of the sample meanvariance of the sample mean is equal to the population variance divided by the sample size:

V XnX

X( ) 2

2

The standard deviation of the sample mean, known as the standard error of standard deviation of the sample mean, known as the standard error of the meanthe mean, is equal to the population standard deviation divided by the square root of the sample size:

SD XnX

X( )

Relationships between Population Relationships between Population Parameters and the Sampling Parameters and the Sampling

Distribution of the Sample MeanDistribution of the Sample Mean

When sampling from a population with mean and finite standard deviation , the sampling distribution of the sample mean will tend to a normal distribution with mean and standard deviation as the sample size becomes large(n >30).

For “large enough” n:

When sampling from a population with mean and finite standard deviation , the sampling distribution of the sample mean will tend to a normal distribution with mean and standard deviation as the sample size becomes large(n >30).

For “large enough” n:

n

)/,(~ 2 nNX

P( X)

X

0.25

0.20

0.15

0.10

0.05

0.00

n = 5

P( X)

0.2

0.1

0.0 X

n = 20

f ( X)

X

-

0.4

0.3

0.2

0.1

0.0

Large n

The Central Limit TheoremThe Central Limit Theorem

• Comparing the population distribution and the sampling distribution of the mean: The sampling distribution is

more bell-shaped and symmetric.

Both have the same center. The sampling distribution of

the mean is more compact, with a smaller variance.

• Comparing the population distribution and the sampling distribution of the mean: The sampling distribution is

more bell-shaped and symmetric.

Both have the same center. The sampling distribution of

the mean is more compact, with a smaller variance.

87654321

0.2

0.1

0.0

X

P(X

)

Uniform Distribution (1,8)

X8.07.57.06.56.05.55.04.54.03.53.02.52.01.51.0

0.10

0.05

0.00

P( X

)

Sampling Distribution of the Mean

Properties of the Sampling Distribution of the Sample Mean

Developing Sampling Developing Sampling DistributionsDistributions

Assume there is a population …

Population size N=4

Random variable, X,

is age of individuals

Values of X: 18, 20,

22, 24 measured in

years A

B C

D

1

2

1

18 20 22 2421

4

2.236

N

ii

N

ii

X

N

X

N

.3

.2

.1

0 A B C D (18) (20) (22) (24)

Uniform Distribution

P(X)

X

Developing Sampling DistributionsDeveloping Sampling Distributions(continued)

Summary Measures for the Population Distribution

1st 2nd Observation Obs 18 20 22 24

18 18,18 18,20 18,22 18,24

20 20,18 20,20 20,22 20,24

22 22,18 22,20 22,22 22,24

24 24,18 24,20 24,22 24,24

All Possible Samples of Size n=2

16 Samples Taken with Replacement

16 Sample Means

1st 2nd Observation Obs 18 20 22 24

18 18 19 20 21

20 19 20 21 22

22 20 21 22 23

24 21 22 23 24

Developing Sampling Distributions(continued)

1st 2nd Observation Obs 18 20 22 24

18 18 19 20 21

20 19 20 21 22

22 20 21 22 23

24 21 22 23 24

Sampling Distribution of All Sample Means

18 19 20 21 22 23 240

.1

.2

.3

P(X)

X

Sample Means Distribution

16 Sample Means

_

Developing Sampling Developing Sampling DistributionsDistributions

(continued)

1

2

1

2 2 2

18 19 19 2421

16

18 21 19 21 24 211.58

16

N

ii

X

N

i Xi

X

X

N

X

N

Summary Measures of Sampling Distribution

Developing Sampling Distributions

(continued)

Comparing the Population with its Comparing the Population with its

Sampling DistributionSampling Distribution

18 19 20 21 22 23 240

.1

.2

.3 P(X)

X

Sample Means Distributionn = 2

A B C D (18) (20) (22) (24)

0

.1

.2

.3

PopulationN = 4

P(X)

X_

21 2.236 21 1.58X X

Properties of Summary MeasuresProperties of Summary Measures

For sampling with replacement:

X

Xn

When the Population is Normal

Central TendencyCentral Tendency

Variation

Sampling with Sampling with ReplacementReplacement

Population Distribution

Sampling Distributions

X

Xn

X50X

4

5X

n

16

2.5X

n

50

10

Mercury makes a 2.4 liter V-6 engine, the Laser XRi, used in Mercury makes a 2.4 liter V-6 engine, the Laser XRi, used in speedboats. The company’s engineers believe the engine delivers speedboats. The company’s engineers believe the engine delivers an an average power of 220 average power of 220 horsepower and that horsepower and that the standard the standard deviation of power delivered is 15 HPdeviation of power delivered is 15 HP. A potential buyer intends to . A potential buyer intends to sample 100 engines sample 100 engines (each engine is to be run a single time). What (each engine is to be run a single time). What is the probability that the sample mean will is the probability that the sample mean will be less than 217 HPbe less than 217 HP??

Mercury makes a 2.4 liter V-6 engine, the Laser XRi, used in Mercury makes a 2.4 liter V-6 engine, the Laser XRi, used in speedboats. The company’s engineers believe the engine delivers speedboats. The company’s engineers believe the engine delivers an an average power of 220 average power of 220 horsepower and that horsepower and that the standard the standard deviation of power delivered is 15 HPdeviation of power delivered is 15 HP. A potential buyer intends to . A potential buyer intends to sample 100 engines sample 100 engines (each engine is to be run a single time). What (each engine is to be run a single time). What is the probability that the sample mean will is the probability that the sample mean will be less than 217 HPbe less than 217 HP??

P X PX

n n

P Z P Z

P Z

( )

( ) .

217217

217 22015

100

217 2201510

2 0 0228

The Central Limit TheoremThe Central Limit Theorem

Population ProportionsPopulation Proportions

Categorical variableCategorical variable e.g.: Gender, voted for bush, college degreee.g.: Gender, voted for bush, college degree

Proportion of population that has a Proportion of population that has a

characteristiccharacteristic

Sample proportion provides an estimateSample proportion provides an estimate

number of successes

sample sizeS

Xp

n

p

p

Sampling Distribution of Sample Sampling Distribution of Sample

ProportionProportion Approximated by

normal distribution

Mean:

Standard error:

p = population proportion

Sampling DistributionSampling Distribution

P(ps).3.2.1 0

0 . 2 .4 .6 8 1ps

5np

1 5n p

Sp p

1Sp

p p

n

In recent years, convertible sports coupes have become very popular in Japan. In recent years, convertible sports coupes have become very popular in Japan. Toyota is currently shipping Celicas to Los Angeles, where a customizer does Toyota is currently shipping Celicas to Los Angeles, where a customizer does a roof lift and ships them back to Japan. a roof lift and ships them back to Japan. Suppose that 25% of all Japanese in a Suppose that 25% of all Japanese in a given income and lifestyle category are interested in buying Celica given income and lifestyle category are interested in buying Celica convertibles. A convertibles. A random sample of 100 Japanese random sample of 100 Japanese consumers in the category of consumers in the category of interest is to be selected. What is the probability that interest is to be selected. What is the probability that at least 20% at least 20% of those in of those in the sample will express an interest in a Celica convertiblethe sample will express an interest in a Celica convertible?

In recent years, convertible sports coupes have become very popular in Japan. In recent years, convertible sports coupes have become very popular in Japan. Toyota is currently shipping Celicas to Los Angeles, where a customizer does Toyota is currently shipping Celicas to Los Angeles, where a customizer does a roof lift and ships them back to Japan. a roof lift and ships them back to Japan. Suppose that 25% of all Japanese in a Suppose that 25% of all Japanese in a given income and lifestyle category are interested in buying Celica given income and lifestyle category are interested in buying Celica convertibles. A convertibles. A random sample of 100 Japanese random sample of 100 Japanese consumers in the category of consumers in the category of interest is to be selected. What is the probability that interest is to be selected. What is the probability that at least 20% at least 20% of those in of those in the sample will express an interest in a Celica convertiblethe sample will express an interest in a Celica convertible?

n

p

np E p

p p

nV p

p p

nSD p

100

0 25

100 0 25 25

1 25 75

1000 001875

10 001875 0 04330127

.

( )( . ) ( )

( ) (. )(. ). ( )

( ). . ( )

P p Pp p

p p

n

p

p p

n

P z P z

P z

( . )

( )

.

( )

. .

(. )(. )

.

.

( . ) .

0 201

20

1

20 25

25 75

100

05

0433

1 15 0 8749

Sample ProportionSample Proportion

If the population standard deviation, , is unknownunknown, replace with the sample standard deviation, s. If the population is normal, the resulting statistic: has a t distribution t distribution with with (n - 1) degrees of freedom(n - 1) degrees of freedom.

If the population standard deviation, , is unknownunknown, replace with the sample standard deviation, s. If the population is normal, the resulting statistic: has a t distribution t distribution with with (n - 1) degrees of freedom(n - 1) degrees of freedom.

• The t is a family of bell-shaped and symmetric distributions, one for each number of degree of freedom.

• The expected value of t is 0.

• The variance of t is greater than 1, but approaches 1 as the number of degrees of freedom increases. The t is flatter and has fatter tails than does the standard normal.

• The t distribution approaches a standard normal as the number of degrees of freedom increases.

• The t is a family of bell-shaped and symmetric distributions, one for each number of degree of freedom.

• The expected value of t is 0.

• The variance of t is greater than 1, but approaches 1 as the number of degrees of freedom increases. The t is flatter and has fatter tails than does the standard normal.

• The t distribution approaches a standard normal as the number of degrees of freedom increases.

nsXt

/

Standard normal

t, df=20t, df=10

Student’s t Distribution

Sampling from Finite SampleSampling from Finite Sample

Modify standard error if sample size (n) is large relative to population size (N ) Use finite population correction factor (FPC)

Standard error with FPC

1X

N n

Nn

1

1SP

p p N n

n N

.05 or / .05n N n N