Lec 07_Sampling Method and CLT(1)

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Sampling Distribution

Outline

1 Review on Probability Distributions

2 Sampling Distribution

3 Sampling Distribution for Sample Means

Sampling DistributionOBJECTIVES

1. Define the sampling process.2. Distinguish between population, sample

and sampling distributions.3. Describe the characteristics of sampling

distribution of the mean.4. State the Central Limit Theorem.5. Apply the Central Limit Theorem.6. Convert sampling variable to standardized

variable and vice versa.


Recommended ReadingCustomised Text, Adapted from ‘StatisticalTechniques in Business & Economics by Lind,Marchal 16th Edition’ McGraw Hill

Chapter 7, Page 209 – 211, 218 – 240 excludingSimple Random Sampling

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Slide number 4

Review: Probability Distribution

A probabilitydistribution is a list of allprobabilities of ______the possible outcome in asurvey / experiment.

Number ofTests, x

Probability,P(x)

0 4/20 = 0.20

1 8/20 = 0.40

2 6/20 = 0.30

3 2/20 = 0.10

Total 1.00

All


Slide number 5

Review: Probability Distribution

Types of Probability Distribution

Distribution for values of a VARIABLE

Population Distribution

Sample Distribution


Slide number 6

Sampling Distribution A sampling distribution is a probability distribution

of all possible statistic values of a variable.

Sampling distribution for samplemeansA probability distribution of allpossible sample mean values of agiven sample size

x xxxx x

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Slide number 7

(3.1)Describing the sampling distribution forsample means

Mean of the sampling distribution

The average of all sample mean values

x

x

Refer to text book pg 220 - 222

Denoted by The mean of the sampling distribution

equals the population mean. i.e.


Slide number 8

Formula for Standard Error

Population size N is infinite, or not stated


X n

X n

N nN

1

Finite multiplier orFinite populationCorrection Factor

Population size N is finite AND 0.05


Slide number 9

Standard Error of the sample means Represents the average size of the

Sampling Errors It is the standard deviation of all the sample

mean values Measures the spread of the sampling

distribution It is smaller than the population standard

deviation


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Slide number 10

(3.2)Deciding the Shape of the SamplingDistribution for sample means

Sampling from a Normally distributed populationwith known population standard deviation The sampling distribution is a normal distribution

regardless of the sample size Sampling from Non-normally distributed population

or unknown population standard deviation The sampling distribution is approximately a

normal distribution IF the sample size is largeenough, i.e. n ≥ 30(by Central Limit Theorem)


Slide number 11

Central Limit Theorem If all samples of a particular size are selected

from any population, the sampling distributionof the sample mean is approximately a normaldistribution. This approximation improveswith larger samples.

The distribution is more “normal” for n=20then n=10, which is in turn more “normal”then n= 6

For skewed population distribution, we needn ≥ 30 so that the sampling distribution willconverge to a normal distribution

the sample size is largeenough (n 30) evenwhen the underlyingpopulation may be

non-normal

Sample meansfollow the normalprobabilitydistribution undertwo conditions:

the underlying populationfollows the normal

distribution & populationStd dev is known

OR

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Slide number 13

Sampling distribution for samplemeans: finding probabilities

Convert the sample statistic to thestandard normal random variable

XX

X XXZ

where

X n Or

X n

N nN

1

when N is finite AND

0.05


Slide number 14

Example 1

The distribution of annual earnings of all bank tellers with fiveyears' experience is skewed negatively with mean $15,000 anda standard deviation of $2,000. If we draw a random sampleof 30 tellers, what is the probability that their earnings willaverage more than $15,750 annually?

X = annual earnings of bank tellers

= $15,000 = $2,000 n = 30 Find P( > 15,750)


Slide number 15

Example 1

X = annual earnings of bank tellers

= $15,000 = $2,000 n = 30

Find P( > 15,750)

Standard Error

Do not need finite multiplier because thepopulation is not known (infinite)

= = 365.1484

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Slide number 16

Sampling Distribution:

Although population distribution is skewed, but n = 30,Therefore, by the Central Limit Theorem, the samplingdistribution is normally distributed

Example 1 - continue


Slide number 17

2.05

0.5-0.4798=0.0202

= 0.5 – 0.4798= 0.0202

Sampling Distribution:Although population distribution is skewed, n = 30, by theCentral Limit theorem, the sampling distribution isnormally distributed


0521484365

1500015750 ..x

xZ

Z

0

)15750( xP

)05.2( zP


Slide number 18

Example 2

Sarjit Singh, the owner of a chain of 25 clothingstores, has been thinking about winding up hisbusiness because it is not as profitable as before.Over the past few years, the mean net income forthe 25 stores has been $21,000 with a standarddeviation of $3,399.29. He would sell the businessif the average of the first 5 stores audited at yearend is less than $20,000. What is the probabilitythat Sarjit will sell out? (Assume that the profitsfollow a normal distribution).

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Slide number 19

Example 2 (lecture notes pg 8)

X = annual profits of the clothing stores

= $21,000 = $3,399.29 N = 25 n = 5

If Sarjit sells the store,

average of 5 stores < $20,000

Find P( average < $20,000)= P( < 20,000 )= ….


Slide number 20

Example 9.4 continue

Standard Error

Population is finite. ChecknN

525

0 2 0 05. .Should apply Finitemultiplier

75.1387125

525

5

29.3399

X

Finite multiplier


Slide number 21

Sampling Distribution:Although n < 30, the population is a normal distributionwith known population standard deviation(given). Thusthe sampling distribution is normally distributed


720751387

0002100020 ..

,,

x

xZ

P( < 20,000)= P( Z < -0.72)= 0.5 – 0.2642= 0.2358

-0.72

Z

0

0.2642

The probability that Sarjit will sellhis stores is 0.2358

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Slide number 22

FEBRUARY 2010 EXAMINATION, QUESTION 3 (25 Marks)

(a) Mr. Lee owns a fleet of taxis. Based on past data, he knows thatthe expenses incurred on each taxi is normally distributed with amean of $120 and a standard deviation of $12. (Assume that thepopulation follows the normal distribution and sixty taxis arerandomly selected.) Mr. Lee has approached you, a businessconsultant, to plan for next year’s expenses by finding thefollowing:

(i) Expected mean expenses incurred

n = 60 ; μ = $120 ; = $12

µ = = $120

(ii) Standard error of the expenses incurred= = = $1.549

Example 3


Slide number 23


(a) Mr. Lee owns a fleet of taxis. Based on past data, he knows that theexpenses incurred on each taxi is normally distributed with a mean of$120 and a standard deviation of $12. (Assume that the populationfollows the normal distribution and sixty taxis are randomly selected.)Mr. Lee has approached you, a business consultant, to plan for nextyear’s expenses by finding the following:

(iii) Probability that the mean expenses will exceed $123.80

P( ≥ 123.8)

= P( Z ≥ .. )

= P( Z ≥ 2.45) = 0.5 - 0.4929 = 0.0071

Example 3


Slide number 24


(a) Mr. Lee owns a fleet of taxis. Based on past data, he knows that theexpenses incurred on each taxi is normally distributed with a mean of$120 and a standard deviation of $12. (Assume that the populationfollows the normal distribution and sixty taxis are randomly selected.)Mr. Lee has approached you, a business consultant, to plan for nextyear’s expenses by finding the following:

(iv) Probability that the mean expenses will fall between $117 and $122

P( 117 < < 122 )

= P( < Z < )

= P( -1.94 < Z < 1.29)

= 0.4738 + 0.4015 = 0.8753

Example 3

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Slide number 25

Pg 238 Q35

93.0

60

5.28.241.25

x

xZ

P( < 25.1)= P(Z < 0.93)= 0.5 + 0.3238= 0.8238

Z

0

0.3238

The mean age at which men in USA marry for the first time follows thenormal distribution with a mean of 24.8 years. The standard deviation is2.5years. For a random sample of 60 men, what is the likelihood that the ageat which they were married for the first time is less than 25.1 years.

LHS

= 24.8 = 2.5 n = 6060

5.2X

0.5

0.93

Samplestatistics

Mean ofsamplingdistribution

where x = value of the variable

=

StandardError

Samplingdistribution

Sampling distribution is a normaldistribution if:sample size n 30 by Central Limittheorem ORpopulation distribution is a normaldistribution and is known

StandardizedFormula

x

x

nx

1

N

nN

nx

05.0NnIf

xx

x xxZ

Lec 07_Sampling Method and CLT(1)

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