7/2/2015 (c) 2001, ron s. kenett, ph.d.1 sampling for estimation instructor: ron s. kenett email:...

04/19/23

(c) 2001, Ron S. Kenett, Ph.D. 1

Sampling for Estimation

Instructor: Ron S. KenettEmail: [email protected]

Course Website: www.kpa.co.il/biostatCourse textbook: MODERN INDUSTRIAL STATISTICS,

Kenett and Zacks, Duxbury Press, 1998

04/19/23


Course Syllabus

•Understanding Variability•Variability in Several Dimensions•Basic Models of Probability•Sampling for Estimation of Population Quantities•Parametric Statistical Inference•Computer Intensive Techniques•Multiple Linear Regression•Statistical Process Control•Design of Experiments

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Error Sampling Nonsampling

Standard error of the mean of the proportion

Standardized individual value sample mean

Finite Population Correction (FPC)

Probability sample Simple random

sample Systematic sample Stratified sample Cluster sample

Nonprobability sample Convenience sample Quota sample Purposive sample Judgment sample

Key TermsKey Terms

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Key TermsKey Terms

Unbiased estimator

Point estimates Interval

estimates Interval limits Confidence

coefficient

Confidence level

Accuracy Degrees of

freedom (df) Maximum likely

sampling error

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Types of SamplesTypes of Samples

Simple random

Systematic

Every person has an equal chance of being selected. Best when roster of the population exists.

Randomly enter a stream of elements and sample every kth element. Best when elements are randomly ordered, no cyclic variation.

Probability, or Scientific, Samples: Each element to be sampled has a known (or calculable) chance of being selected.

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Stratified

Cluster

Randomly sample elements from every layer, or stratum, of the population. Best when elements within strata are homogeneous.

Randomly sample elements within some of the strata. Best when elements within strata are heterogeneous.

Probability, or Scientific, Samples: Each element to be sampled has a known (or calculable) chance of being selected.

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Convenience

Quota

Elements are sampled because of ease and availability.

Elements are sampled, but not randomly, from every layer, or stratum, of the population.

Nonprobability Samples: Not every element has a chance to be sampled. Selection process usually involves subjectivity.

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Purposive

Judgment

Elements are sampled because they are atypical, not representative of the population.

Elements are sampled because the researcher believes the members are representative of the population.

Nonprobability Samples: Not every element has a chance to be sampled. Selection process usually involves subjectivity.

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Distribution of the MeanDistribution of the Mean

When the population is normally distributed Shape: Regardless of sample size, the

distribution of sample means will be normally distributed.

Center: The mean of the distribution of sample means is the mean of the population. Sample size does not affect the center of the distribution.

Spread: The standard deviation of the distribution of sample means, or the standard error, is

. nx

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The Standardized MeanThe Standardized Mean

The standardized z-score is how far above or below the sample mean is compared to the population mean in units of standard error. “How far above or below” sample mean minus µ “In units of standard error” divide by

Standardized sample mean

n

xz

– error standard

mean sample

n

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When the population is not normally distributed Shape: When the sample size taken

from such a population is sufficiently large, the distribution of its sample means will be approximately normally distributed regardless of the shape of the underlying population those samples are taken from. According to the Central Limit Theorem, the larger the sample size, the more normal the distribution of sample means becomes.

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When the population is not normally distributed Center: The mean of the distribution of

sample means is the mean of the population, µ. Sample size does not affect the center of the distribution.

Spread: The standard deviation of the distribution of sample means, or the standard error, is.

nx

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Distribution of the ProportionDistribution of the Proportion

When the sample statistic is generated by a count not a measurement, the proportion of successes in a sample of n trials is p, where Shape: Whenever both n and n(1 –

) are greater than or equal to 5, the distribution of sample proportions will be approximately normally distributed.

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When the sample proportion of successes in a sample of n trials is p, Center: The center of the distribution

of sample proportions is the center of the population, .

Spread: The standard deviation of the distribution of sample proportions, or the standard error, isp ׳(1–)

n .

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The standardized z-score is how far above or below the sample proportion is compared to the population proportion in units of standard error. “How far above or below” sample p – “In units of standard error” divide by

Standardized sample proportion

n

pz)–1(

– error standard

proportion sample ׳

np)–1( ׳

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Finite Population CorrectionFinite Population Correction Finite Population Correction (FPC) Factor:

Rule of Thumb: Use FPC when n > 5%•N.

Apply to: Standard errors of mean and proportion.

FPC

N nN 1

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Unbiased Point EstimatesUnbiased Point Estimates

PopulationSampleParameterStatistic Formula

Mean, µ

Variance,

Proportion,

x x xi

n

1–

2)–( 22

nxixss

p p x successesn trials

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Confidence Intervals: Confidence Intervals: µ, Known

where = sample mean ASSUMPTION: = population standard infinite

population deviationn = sample sizez = standard normal score for area in tail = /2

nzxx

nzxx

zzz ׳׳

–:

0–:

x

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Confidence Intervals: Confidence Intervals: µ, Unknown

where = sample mean ASSUMPTION: s = sample standard Population deviation

approximately n = sample size normal

and t = t-score for area infinite in tail = /2 df = n – 1

nstxx

nstxx

ttt׳׳

–:

0–:

x

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Confidence Intervals on Confidence Intervals on

where p = sample proportion ASSUMPTION: n = sample size n•p 5,

z = standard normal score n•(1–p) 5,

for area in tail = /2 and population

infinite

nn

ppzppppzpp )–1()–1(–: ׳׳

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Confidence Intervals for Finite Confidence Intervals for Finite PopulationsPopulations

Mean:

or

Proportion:

1––

2

1––

2

NnN

nstx

NnN

nzx

1–

–)–1(

2

NnN

nppzp

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Interpretation of Confidence Interpretation of Confidence IntervalsIntervals

Repeated samples of size n taken from the same population will generate (1–)% of the time a sample statistic that falls within the stated confidence interval.

OR We can be (1–)% confident that the

population parameter falls within the stated confidence interval.

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Sample Size Determination for Sample Size Determination for Infinite PopulationsInfinite Populations

Mean: Note is known and e, the bound within which you want to estimate µ, is given. The interval half-width is e, also called

the maximum likely error:

Solving for n, we find: 2

22

e

zn

nze

׳

׳

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Sample Size Determination for Sample Size Determination for Finite PopulationsFinite Populations

Mean: Note is known and e, the bound within which you want to estimate µ, is given.

where n = required sample sizeN = population sizez = z-score for (1–)%

confidence

n 2e2z2

2N

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Sample Size Determination of Sample Size Determination of for for Infinite PopulationsInfinite Populations

Proportion: Note e, the bound within

which you want to estimate , is given. The interval half-width is e, also called

the maximum likely error:

Solving for n, we find:2

)–1(2

)–1(

eppzn

nppze

׳

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Sample Size Determination of Sample Size Determination of for for Finite PopulationsFinite Populations

Mean: Note e, the bound within which

you want to estimate , is given.

where n = required sample sizeN = population sizez = z-score for (1–)%

confidence

p = sample estimator of

n p(1– p)e2z2

p(1– p)N

7/2/2015 (c) 2001, ron s. kenett, ph.d.1 sampling for estimation instructor: ron s. kenett email:...

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