McGraw-Hill/Irwin Copyright © 2009 by The McGraw-Hill Companies, Inc. Sampling Distributions and Estimation (Part 1) Chapter88 Sampling Variation Estimators.

McGraw-Hill/Irwin Copyright © 2009 by The McGraw-Hill Companies, Inc.

Sampling Distributions Sampling Distributions and Estimation and Estimation (Part 1)(Part 1)

Chapter8888

Sampling Variation

Estimators and Sampling Distributions

Sample Mean and the Central Limit Theorem

Confidence Interval for a Mean () with Known

Confidence Interval for a Mean () with Unknown

Confidence Interval for a Proportion ()

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Sampling VariationSampling Variation

• Sample statisticSample statistic – a – a random variablerandom variable whose value depends on which whose value depends on which population items happen to be included in population items happen to be included in the the random samplerandom sample..

• Depending on the Depending on the sample sizesample size, the sample , the sample statistic could either represent the statistic could either represent the population well or differ greatly from the population well or differ greatly from the population.population.

• This This sampling variationsampling variation can easily be can easily be illustrated.illustrated.

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• Consider eight random samples of size Consider eight random samples of size nn = 5 = 5 from a large population of GMAT scores for from a large population of GMAT scores for MBA applicants.MBA applicants.

• The sample means ( The sample means ( xxii ) tend to be close to the ) tend to be close to the

population mean (population mean ( = 520.78). = 520.78).

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• The dot plots show that the sample The dot plots show that the sample meansmeans have much less variation than the have much less variation than the individualindividual sample items. sample items.


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Estimators and Sampling Estimators and Sampling DistributionsDistributions

• EstimatorEstimator – a statistic derived from a sample to – a statistic derived from a sample to infer the value of a population infer the value of a population parameterparameter..

• EstimateEstimate – the value of the estimator in a – the value of the estimator in a particular sample.particular sample.

• Population parameters are represented by Population parameters are represented by Greek letters and the corresponding statistic Greek letters and the corresponding statistic by Roman letters.by Roman letters.

Some TerminologySome Terminology

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Examples of EstimatorsExamples of Estimators

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• The The sampling distributionsampling distribution of an estimator is of an estimator is the probability distribution of all possible the probability distribution of all possible values the statistic may assume when a values the statistic may assume when a random sample of size random sample of size nn is taken. is taken.

• An estimator is a An estimator is a random variablerandom variable since since samples vary.samples vary.

Sampling DistributionsSampling Distributions

• Sampling errorSampling error = = – – ^̂

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• BiasBias is the difference between the expected is the difference between the expected value of the estimator and the true value of the estimator and the true parameter.parameter.

BiasBias

• BiasBias = E( = E( ) – ) – ^ ^

• An estimator is An estimator is unbiasedunbiased if E( if E( ) = ) = ^̂

• On averageOn average, an unbiased estimator neither , an unbiased estimator neither overstates nor understates the true overstates nor understates the true parameter.parameter.

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• Sampling error is Sampling error is randomrandom whereas bias is whereas bias is systematicsystematic..

BiasBias

• An unbiased estimator avoids systematic An unbiased estimator avoids systematic error.error.

Figure 8.4

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• EfficiencyEfficiency refers to the variance of the refers to the variance of the estimator’s sampling distribution.estimator’s sampling distribution.

• A A more efficientmore efficient estimator has smaller variance. estimator has smaller variance.

EfficiencyEfficiency

Figure 8.5

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• A A consistent consistent estimator converges toward estimator converges toward the parameter being estimated as the the parameter being estimated as the sample sizesample size

increases. increases.

ConsistencyConsistency

Figure 8.6

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Sample Mean and the Sample Mean and the Central Limit TheoremCentral Limit Theorem

• If a random sample of size If a random sample of size nn is drawn from is drawn from a population with mean a population with mean and standard and standard deviation deviation , the distribution of the sample , the distribution of the sample mean mean xx approaches a normal distribution approaches a normal distribution with mean with mean and standard deviation and standard deviation xx = = / /

nn as the sample size increase. as the sample size increase. • If the population is normal, the distribution If the population is normal, the distribution

of the sample mean is normal regardless of of the sample mean is normal regardless of sample size.sample size.

Central Limit Theorem (CLT) for a MeanCentral Limit Theorem (CLT) for a Mean

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• If the population is exactly normal, then the If the population is exactly normal, then the sample mean follows a normal distribution.sample mean follows a normal distribution.

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• As the sample size As the sample size nn increases, the increases, the distribution of sample means narrows in on distribution of sample means narrows in on the population mean the population mean µµ..

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• If the sample is large enough, the sample means If the sample is large enough, the sample means will have approximately a normal distribution will have approximately a normal distribution even if your population is even if your population is notnot normal. normal.

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Illustrations of Central Limit Theorem Illustrations of Central Limit Theorem

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Illustrations of Central Limit TheoremIllustrations of Central Limit Theorem

• Symmetric populationSymmetric population

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Illustrations of Central Limit TheoremIllustrations of Central Limit Theorem

• Skewed populationSkewed population

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Example - Bottle Filling: Variation in XExample - Bottle Filling: Variation in X

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Make the intervalsmall by increasing n.

+ z n


• The standard error declines as The standard error declines as nn increases, increases, but at a decreasing rate.but at a decreasing rate.

Sample Size and Standard ErrorSample Size and Standard Error

The distribution of sample means collapses at the true population mean as n increases.

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• Consider a discrete uniform population Consider a discrete uniform population consisting of the integers {0, 1, 2, 3}.consisting of the integers {0, 1, 2, 3}.

• The population parameters are: The population parameters are: = 1.5, = 1.5, = 1.118 = 1.118


Illustration: All Possible Samples from a Illustration: All Possible Samples from a Uniform PopulationUniform Population

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• All possible samples of size All possible samples of size nn = 2, with = 2, with replacement, are given below along with replacement, are given below along with their means. their means.


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• The population is uniform, yet the The population is uniform, yet the distribution of all possible sample means distribution of all possible sample means has a peaked triangular shape.has a peaked triangular shape.


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• The CLT’s predictions for the mean and The CLT’s predictions for the mean and standard error arestandard error are


x = = 1.5 and

x = / n = 1.118/ 2 = 0.7905

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• xx the mean of means is the mean of means is

x = 1(0.0) + 2(.05) + 3(1.0) + 4(1.5) + 3(2.0) + 2(2.5) + 1(3.0) = 1.5 16

• The standard deviation of the means isThe standard deviation of the means is

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Confidence Interval for a Confidence Interval for a Mean (Mean () with Known ) with Known

• A sample mean A sample mean xx is a is a point estimatepoint estimate of the of the population mean population mean ..

What is a Confidence Interval?What is a Confidence Interval?

• A A confidence intervalconfidence interval for the mean is a range for the mean is a range lowerlower < < < < upperupper

• The The confidence levelconfidence level is the probability that the is the probability that the confidence interval contains the true population confidence interval contains the true population mean.mean.

• The confidence level (usually expressed as a %) The confidence level (usually expressed as a %) is the area under the curve of the sampling is the area under the curve of the sampling distribution.distribution.

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What is a Confidence Interval?What is a Confidence Interval?• The confidence interval for The confidence interval for with known with known is: is:

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• A higher confidence level leads to a wider A higher confidence level leads to a wider confidence interval.confidence interval.

Choosing a Confidence LevelChoosing a Confidence Level

• Greater Greater confidence confidence implies loss of implies loss of precision.precision.

• 95% confidence 95% confidence is most often is most often used.used.

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• A confidence interval either A confidence interval either doesdoes or or does does notnot contain contain ..

• The confidence level quantifies the The confidence level quantifies the riskrisk..• Out of 100 confidence intervals, Out of 100 confidence intervals,

approximately 95% approximately 95% wouldwould contain contain , while , while approximately 5% approximately 5% would notwould not contain contain ..

InterpretationInterpretation

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• Yes, but not very often.Yes, but not very often.• In quality control applications with ongoing In quality control applications with ongoing

manufacturing processes, assume manufacturing processes, assume stays stays the same over time.the same over time.

• In this case, confidence intervals are used In this case, confidence intervals are used to construct control charts to track the to construct control charts to track the mean of a process over time.mean of a process over time.

Is Is Ever Known? Ever Known?

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Confidence Interval for a Confidence Interval for a Mean (Mean () with Unknown ) with Unknown

• Use the Use the Student’s t distributionStudent’s t distribution instead of instead of the normal distribution when the population the normal distribution when the population is normal but the standard deviation is normal but the standard deviation is is unknown and the sample size is small.unknown and the sample size is small.

Student’s t DistributionStudent’s t Distribution

xx ++ tt ssnn • The confidence interval for The confidence interval for (unknown (unknown ) )

isisxx - - tt ss

nn

xx + + tt ssnn

< < < <

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Student’s t DistributionStudent’s t Distribution

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Student’s t DistributionStudent’s t Distribution• tt distributions are symmetric and shaped distributions are symmetric and shaped

like the standard normal distribution.like the standard normal distribution.• The The tt distribution is dependent on the size distribution is dependent on the size

of the sample.of the sample.

Figure 8.11

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Degrees of FreedomDegrees of Freedom• Degrees of Freedom Degrees of Freedom (d.f.) is a parameter (d.f.) is a parameter

based on the sample size that is used to based on the sample size that is used to determine the value of the determine the value of the tt statistic. statistic.

• Degrees of freedom tell how many Degrees of freedom tell how many observations are used to calculate observations are used to calculate , less , less the number of intermediate estimates used the number of intermediate estimates used in the calculation.in the calculation.

= = nn - 1 - 1

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Degrees of FreedomDegrees of Freedom

• As As nn increases, the increases, the tt distribution distribution approaches the shape of the normal approaches the shape of the normal distribution. distribution.

• For a given confidence level, For a given confidence level, tt is always is always larger than larger than zz, so a confidence interval , so a confidence interval based on based on tt is always wider than if is always wider than if zz were were used.used.

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Comparison of z and tComparison of z and t• For very small samples, For very small samples, tt-values differ -values differ

substantially from the normal.substantially from the normal.• As degrees of freedom increase, the As degrees of freedom increase, the tt--

values approach the normal values approach the normal zz-values.-values.• For example, for For example, for nn = 31, the degrees of = 31, the degrees of

freedom are:freedom are:• What would the What would the tt-value be for a 90% -value be for a 90%

confidence interval? confidence interval?

= 31 – 1 = 30= 31 – 1 = 30

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Comparison of z and tComparison of z and t

For For = 30, the corresponding = 30, the corresponding zz-value is 1.645.-value is 1.645.

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Example GMAT Scores AgainExample GMAT Scores Again• Here are the GMAT scores from 20 Here are the GMAT scores from 20

applicants to an MBA program:applicants to an MBA program:

Figure 8.13

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Example GMAT Scores AgainExample GMAT Scores Again• Construct a 90% confidence interval for the Construct a 90% confidence interval for the

mean GMAT score of all MBA applicants.mean GMAT score of all MBA applicants.

xx = 510 = 510 ss = 73.77 = 73.77

• Since Since is unknown, use the Student’s is unknown, use the Student’s tt for for the confidence interval with the confidence interval with = 20 – 1 = 19 = 20 – 1 = 19 d.f.d.f.

• First find First find tt0.900.90 from Appendix D. from Appendix D.

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• For a 90% For a 90% confidence confidence interval, use interval, use Appendix D to find Appendix D to find tt0.050.05 = 1.729 = 1.729

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Example GMAT Scores AgainExample GMAT Scores Again• The 90% confidence interval is:The 90% confidence interval is:

x - t sn

x + t sn

< <

513 – 1.729 73.77 20

< < 513 + 1.729 73.77 20

513 – 28.52 < < 513 + 28.52

• We are 90% certain that the true mean GMAT We are 90% certain that the true mean GMAT score is within the interval 481.48 < score is within the interval 481.48 < < 538.52. < 538.52.

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Confidence Interval WidthConfidence Interval Width• Confidence interval width reflects Confidence interval width reflects

- the sample size, - the sample size, - the confidence level and - the confidence level and - the standard deviation.- the standard deviation.

• To obtain a narrower interval and more To obtain a narrower interval and more precisionprecision- increase the sample size or - increase the sample size or - lower the confidence level (e.g., from 90% - lower the confidence level (e.g., from 90%

to 80% confidence)to 80% confidence)

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A “Good” SampleA “Good” Sample

• Here are five different samples of 25 births Here are five different samples of 25 births from a population of from a population of N N = 4,409 births and = 4,409 births and their 95% CIs.their 95% CIs.

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A “Good” SampleA “Good” Sample• An examination of the samples shows that An examination of the samples shows that

sample 5 has an outlier. sample 5 has an outlier.

• The outlier is a warning that the resulting The outlier is a warning that the resulting confidence interval possibly could not be confidence interval possibly could not be trusted.trusted.

• In this case, a larger sample size is needed. In this case, a larger sample size is needed.

Figure 8.15

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Using Appendix DUsing Appendix D• Beyond Beyond = 50, Appendix D shows = 50, Appendix D shows in in

steps of 5 or 10.steps of 5 or 10.• If the table does not give the exact degrees If the table does not give the exact degrees

of freedom, use the of freedom, use the tt-value for the next -value for the next lower lower ..

• This is a conservative procedure since it This is a conservative procedure since it causes the interval to be slightly wider.causes the interval to be slightly wider.

• For d.f. above 150, use the For d.f. above 150, use the zz-value.-value.

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Using ExcelUsing Excel• Use Excel’s function =TINV(probability, d.f.) Use Excel’s function =TINV(probability, d.f.)

to obtain a two-tailed value of to obtain a two-tailed value of tt. Here, . Here, “probability” is 1 minus the confidence “probability” is 1 minus the confidence level.level.

Figure 8.17

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Using MegaStatUsing MegaStat• MegaStat give you a choice of MegaStat give you a choice of z z or or tt and and

does all calculations for you.does all calculations for you.

Figure 8.18

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Using MINITABUsing MINITAB

• MINITAB MINITAB also gives also gives confidence confidence intervals intervals for the for the median and median and standard standard deviation.deviation.

Figure 8.19

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Confidence Interval for a Confidence Interval for a Proportion (Proportion ())

• A proportion is a mean of data whose only A proportion is a mean of data whose only value is 0 or 1.value is 0 or 1.

• The Central Limit Theorem (CLT) states that The Central Limit Theorem (CLT) states that the distribution of a sample proportion the distribution of a sample proportion pp = = xx//nn approaches a normal distribution with approaches a normal distribution with mean mean and standard deviation and standard deviation

• pp = = xx//nn is a is a consistent consistent estimator of estimator of ..

pp = = (1-(1-))nn

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• Management of the Pan-Asian Hotel System Management of the Pan-Asian Hotel System tracks the percent of hotel reservations tracks the percent of hotel reservations made over the Internet.made over the Internet.

• The binary data are:The binary data are:1 Reservation is made over the Internet1 Reservation is made over the Internet0 Reservation is not made over the Internet0 Reservation is not made over the Internet

• After data was collected, it was determined After data was collected, it was determined that the proportion of Internet reservations that the proportion of Internet reservations is is = .20. = .20.

Illustration: Internet Hotel ReservationsIllustration: Internet Hotel Reservations

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• Here are five random samples of Here are five random samples of nn = 20. = 20. Each Each pp is a point estimate of is a point estimate of ..

Illustration: Internet Hotel ReservationsIllustration: Internet Hotel Reservations

• Notice the sampling variation in the value Notice the sampling variation in the value of of pp..

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• The distribution of a sample proportion The distribution of a sample proportion pp = = xx//n n is symmetric if is symmetric if = .50 and regardless of = .50 and regardless of , , approaches symmetry as approaches symmetry as nn increases. increases.

Applying the CLTApplying the CLT

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• As As nn increases, the statistic increases, the statistic pp = = xx//nn more more closely resembles a continuous random closely resembles a continuous random variable.variable.

• As As nn increases, the distribution becomes increases, the distribution becomes more symmetric and bell shaped.more symmetric and bell shaped.

• As As nn increases, the range of the sample increases, the range of the sample proportion proportion pp = = xx//nn narrows. narrows.

• The sampling variation can be reduced by The sampling variation can be reduced by increasing the sample size increasing the sample size nn..

Applying the CLTApplying the CLT

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• Rule of Thumb: The sample proportion Rule of Thumb: The sample proportion pp = = xx//nn may be assumed to be normal if both may be assumed to be normal if both

• nn >> 10 and 10 and nn(1-(1-) ) >> 10. 10.

When is it Safe to Assume Normality?When is it Safe to Assume Normality?

Sample size to assume

normality:

Table 8.9

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• The standard error The standard error of the proportion of the proportion pp

depends on depends on , as , as well as well as nn. .

• It is largest when It is largest when is near .50 and is near .50 and smaller when smaller when is is near 0 or 1.near 0 or 1.

Standard Error of the ProportionStandard Error of the Proportion

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• The formula for the standard error is symmetric.The formula for the standard error is symmetric.


Figure 8.22

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• Enlarging Enlarging nn reduces the standard error reduces the standard error pp

but at a diminishing rate.but at a diminishing rate.


Figure 8.23

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• The confidence interval for The confidence interval for is is Confidence Interval for Confidence Interval for

(1-)n

+ z

• Since Since is unknown, the confidence is unknown, the confidence interval for interval for pp = = xx//nn (assuming a large (assuming a large sample) issample) is

pp(1-(1-pp))nn

pp ++ zz

Where z is based on the desired confidence.

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• zz can be chosen for any confidence level. can be chosen for any confidence level. For example,For example,

Confidence Interval for Confidence Interval for

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• A sample of 75 retail in-store purchases A sample of 75 retail in-store purchases showed that 24 were paid in cash. What is showed that 24 were paid in cash. What is pp??

Example AuditingExample Auditing

p = x/n = 24/75 = .32

• Is Is pp normally distributed? normally distributed?

np = (75)(.32) = 24

n(1-p) = (75)(.88) = 51

Both are Both are >> 10, so we may conclude 10, so we may conclude normality.normality.

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• The 95% confidence interval for the proportion of The 95% confidence interval for the proportion of retail in-store purchases that are paid in cash is:retail in-store purchases that are paid in cash is:

Example AuditingExample Auditing

p(1-p)n

p + z = .32(1-.32)

.32 + 1.96

= .32 + .106

.214 < < .426• We are 95% confident that this interval We are 95% confident that this interval

contains the true population proportion.contains the true population proportion.

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• The width of the confidence interval for The width of the confidence interval for depends ondepends on- the sample size- the sample size- the confidence level- the confidence level- the sample proportion - the sample proportion pp

• To obtain a narrower interval (i.e., more To obtain a narrower interval (i.e., more precision) eitherprecision) either- increase the sample size- increase the sample size- reduce the confidence level- reduce the confidence level

Narrowing the IntervalNarrowing the Interval

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• To find a confidence interval for a To find a confidence interval for a proportion in Excel, use (for example)proportion in Excel, use (for example)

=0.15-NORMSINV(.95)*SQRT(0.15*(1-0.15)/200)=0.15-NORMSINV(.95)*SQRT(0.15*(1-0.15)/200)

=0.15+NORMSINV(.95)*SQRT(0.15*(1-0.15)/=0.15+NORMSINV(.95)*SQRT(0.15*(1-0.15)/200)200)

Using Excel and MegaStatUsing Excel and MegaStat

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• In MegaStat, enter In MegaStat, enter pp and and nn to obtain the to obtain the confidence interval for a proportion.confidence interval for a proportion.


• MegaStat always assumes normality.MegaStat always assumes normality.

Figure 8.23

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• If the sample is small, the distribution of If the sample is small, the distribution of pp may not be well approximated by the may not be well approximated by the normal. normal.

• Confidence limits around Confidence limits around pp can be constructed by using the binomial distribution. can be constructed by using the binomial distribution.


Figure 8.24

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• In polls and surveys, the confidence interval width when In polls and surveys, the confidence interval width when = .5 is called the = .5 is called the margin of errormargin of error..

• Below are some margins of error for 95% confidence interval assuming Below are some margins of error for 95% confidence interval assuming = .50. = .50.

Polls and Margin of ErrorPolls and Margin of Error

• Each reduction in the margin of error Each reduction in the margin of error requires a disproportionately larger sample requires a disproportionately larger sample size.size.

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• If in If in nn independent trials, no events occur, the upper independent trials, no events occur, the upper 95% confidence bound is approximately 95% confidence bound is approximately 3/n3/n..

Rule of ThreeRule of Three

• A Very Quick Rule (VQR) for a 95% A Very Quick Rule (VQR) for a 95% confidence interval when confidence interval when pp is near .50 is is near .50 is

Very Quick RuleVery Quick Rule

pp ++ 1/ 1/ nn

McGraw-Hill/Irwin Copyright © 2009 by The McGraw-Hill Companies, Inc.

Applied Statistics in Applied Statistics in Business & EconomicsBusiness & Economics

End of Chapter 8AEnd of Chapter 8A

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McGraw-Hill/Irwin Copyright © 2009 by The McGraw-Hill Companies, Inc. Sampling Distributions and Estimation (Part 1) Chapter88 Sampling Variation Estimators.

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