Statistics final exam, 40 multiple choice questions.

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Chapter ContentsChapter Contents

8.1 Sampling Variation 8 2 E ti t d S li E8.2 Estimators and Sampling Errors 8.3 Sample Mean and the Central Limit Theorem 8 4 Confidence Interval for a Mean (μ) with Known σ8.4 Confidence Interval for a Mean (μ) with Known σ 8.5 Confidence Interval for a Mean (μ) with Unknown σ 8 6 Confidence Interval for a Proportion (π)8.6 Confidence Interval for a Proportion (π) 8.7 Estimating from Finite Populations 8 8 Sample Size Determination for a Mean8.8 Sample Size Determination for a Mean 8.9 Sample Size Determination for a Proportion 8.10 Confidence Interval for a Population Variance,  2 (Optional)

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8.10 Confidence Interval for a Population Variance,  (Optional)

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Sampling Distributions and EstimationSampling Distributions and Estimation

Chapter Learning Objectives (LO’s)Chapter Learning Objectives (LO’s)

Chapter Learning Objectives (LO s)Chapter Learning Objectives (LO s)

LO8LO8 11LO8LO8--1: 1: Define sampling error, parameter, and estimator.Define sampling error, parameter, and estimator. LO8LO8--2: 2: Explain the desirable properties of estimators.Explain the desirable properties of estimators. LO8LO8--3:3: State the Central Limit Theorem for a mean.State the Central Limit Theorem for a mean. LO8LO8--4:4: Explain how sample size affects the standard error.Explain how sample size affects the standard error.LO8LO8 4:4: Explain how sample size affects the standard error.Explain how sample size affects the standard error. LO8LO8--5:5: Construct a 90, 95, or 99 percent confidence interval for Construct a 90, 95, or 99 percent confidence interval for μ.μ.

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Sampling Distributions and EstimationSampling Distributions and Estimation

Chapter Learning Objectives (LO’s)Chapter Learning Objectives (LO’s)

Chapter Learning Objectives (LO s)Chapter Learning Objectives (LO s)

LO8LO8 66LO8LO8--6:6: Know when to use Student’s Know when to use Student’s t t instead of instead of z z to estimate to estimate μ.μ. LO8LO8--7:7: Construct a 90, 95, or 99 percent confidence interval for Construct a 90, 95, or 99 percent confidence interval for π.π. LO8LO8--8:8: Construct confidence intervals for finite populations.Construct confidence intervals for finite populations. LO8LO8--9:9: Calculate sample size to estimate a mean or proportion.Calculate sample size to estimate a mean or proportion.LO8LO8 9:9: Calculate sample size to estimate a mean or proportion.Calculate sample size to estimate a mean or proportion. LO8LO8--10: 10: Construct a confidence interval for a variance (optional).Construct a confidence interval for a variance (optional).

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8 1 S li V i ti8 1 S li V i ti pter

8.1 Sampling Variation8.1 Sampling Variation

• Sample statistic – a random variable whose value depends on

which population items are included in the random sample. • Depending on the sample size, the sample statistic could either

represent the pop lation ell or differ greatl from the pop lationrepresent the population well or differ greatly from the population. • This sampling variation can easily be illustrated.

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8.1 Sampling Variation8.1 Sampling Variation 8

C id i ht d l f iC id i ht d l f i 5 f l5 f l•• Consider eight random samples of size Consider eight random samples of size nn = 5 from a large = 5 from a large population of GMAT scores for MBA applicants.population of GMAT scores for MBA applicants.

•• The sample means tend to be close to the population mean The sample means tend to be close to the population mean (( = 520.78).= 520.78).

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8.1 Sampling Variation8.1 Sampling Variation

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

than the than the individualindividual sample items. sample items.

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8.2 Estimators and Sampling Distributions8.2 Estimators and Sampling DistributionsLO8LO8--11 8

LO8LO8--1: 1: Define sampling error, parameter and estimator.Define sampling error, parameter and estimator.

E ti tE ti t t ti ti d i d f l t i f th l ft ti ti d i d f l t i f th l f

Some TerminologySome Terminology •• EstimatorEstimator –– a statistic derived from a sample to infer the value of a a statistic derived from a sample to infer the value of a

population parameter.population parameter. •• EstimateEstimate –– the value of the estimator in a particular samplethe value of the estimator in a particular sampleEstimateEstimate –– the value of the estimator in a particular sample.the value of the estimator in a particular sample. •• Population parameters are usually represented by Population parameters are usually represented by

Greek letters and the corresponding statistic Greek letters and the corresponding statistic p gp g by Roman letters.by Roman letters.

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8.2 Estimators and Sampling Distributions8.2 Estimators and Sampling DistributionsLO8LO8--11

Examples of EstimatorsExamples of Estimators

Sampling DistributionsSampling Distributions

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

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• Note: An estimator is a random variable since samples vary.

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8.2 Estimators and Sampling Distributions8.2 Estimators and Sampling DistributionsLO8LO8--11 8

• Sampling errorSampling error is the difference between an estimate and the corresponding population parameter. For example, if we use the sample

ti t f th l ti th th

BiasBias

mean as an estimate for the population mean, then the

• Bias is the difference between the expected value of the estimator and the true parameter Example for the mean

BiasBias

the true parameter. Example for the mean,

•• An estimator is An estimator is unbiasedunbiased if its expected value is the parameter being if its expected value is the parameter being estimated. The sample mean is an unbiased estimator of the population estimated. The sample mean is an unbiased estimator of the population

iimean sincemean since

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

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8.2 Estimators and Sampling Distributions8.2 Estimators and Sampling DistributionsLO8LO8--11 8

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8.2 Estimators and Sampling Distributions8.2 Estimators and Sampling DistributionsLO8LO8--22 8

LO8LO8--2: 2: Explain the desirable properties of estimators.Explain the desirable properties of estimators.

EfficiencyEfficiency Note: Also, a desirable property for an estimator is for it to be unbiased.

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

Fi 8 6•• A A more efficientmore efficient estimator has smaller variance.estimator has smaller variance.Figure 8.6

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8.2 Estimators and Sampling Distributions8.2 Estimators and Sampling DistributionsLO8LO8--22 8

LO8LO8--2: 2: Explain the desirable properties of estimators.Explain the desirable properties of estimators.

ConsistencyConsistency A consistent estimator converges toward the parameter being estimatedA consistent estimator converges toward the parameter being estimated as the sample size increases.

Fi 8 6Figure 8.6

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8.3 Sample Mean and the Central Limit Theorem8.3 Sample Mean and the Central Limit TheoremLO8LO8--33 8

LO8LO8--3: 3: State the Central Limit Theorem for a mean.State the Central Limit Theorem for a mean.

The Central Limit Theorem is a powerful result that allows us to i t th h f th li di t ib ti f th lapproximate the shape of the sampling distribution of the sample

mean even when we don’t know what the population looks like.

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8.3 Sample Mean and the Central Limit Theorem8.3 Sample Mean and the Central Limit TheoremLO8LO8--33

•• If the population is exactly If the population is exactly normal, then the sample meannormal, then the sample mean

•• As the sample size As the sample size nn increases, the increases, the distribution of sample means narrowsdistribution of sample means narrowsnormal, then the sample mean normal, then the sample mean

follows a normal distribution.follows a normal distribution. distribution of sample means narrows distribution of sample means narrows in on the population mean in on the population mean µµ..

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8.3 Sample Mean and the Central Limit Theorem8.3 Sample Mean and the Central Limit TheoremLO8LO8--33

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

normal.normal.

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8.3 Sample Mean and the Central Limit Theorem8.3 Sample Mean and the Central Limit TheoremLO8LO8--33

Illustrations of Central Limit Theorem Illustrations of Central Limit Theorem

Using the uniform and a right skewed di t ib ti

Note: distribution.

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8.3 Sample Mean and the Central Limit Theorem8.3 Sample Mean and the Central Limit TheoremLO8LO8--33

Th C t l Li it Th it t d fi i t l ithi hi h

Applying The Central Limit TheoremApplying The Central Limit Theorem

The Central Limit Theorem permits us to define an interval within which the sample means are expected to fall. As long as the sample size n is large enough, we can use the normal distribution regardless of the population shape (or any n if the population is normal to begin with).

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8.3 Sample Mean and the Central Limit Theorem8.3 Sample Mean and the Central Limit TheoremLO8LO8--44 8

LO8LO8--4: 4: Explain how sample size affects the standard error.Explain how sample size affects the standard error.

Even if the population standard deviation σ is large, the sample means Sample Size and Standard ErrorSample Size and Standard Error

p p g , p will fall within a narrow interval as long as n is large. The key is the standard error of the mean:.. The standard error decreases as n increasesincreases.

For example, when n = 4 the standard error is halved. To halve it again requires n = 16, and to halve it again requires n = 64. To halve the standard error, you must quadruple the sample size (the law of diminishing returns).

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8.3 Sample Mean and the Central Limit Theorem8.3 Sample Mean and the Central Limit Theorem

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

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

•• The population parameters are:The population parameters are:  = 1 5= 1 5  = 1 118= 1 118•• The population parameters are: The population parameters are:  = 1.5, = 1.5,  = 1.118.= 1.118.

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8.3 Sample Mean and the Central Limit Theorem8.3 Sample Mean and the Central Limit Theorem

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

• The population is uniform, yet the distribution of all possible sample means of size 2 has a peaked triangular shapesample means of size 2 has a peaked triangular shape.

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8 4 Confidence Interval for a Mean (8 4 Confidence Interval for a Mean () with) with pter 8.4 Confidence Interval for a Mean (8.4 Confidence Interval for a Mean () with ) with known (known ())LO8LO8--55

LO8LO8--5: 5: Construct a 90, 95, or 99 percent confidence interval for Construct a 90, 95, or 99 percent confidence interval for μ.μ.

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

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What is a Confidence Interval?What is a Confidence Interval? •• The confidence interval forThe confidence interval for  with knownwith known  is:is:

The confidence interval for The confidence interval for  with known with known  is:is:

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A hi h fid l l l d t id fid i t lA hi h fid l l l d t id fid i t l

Choosing a Confidence LevelChoosing a Confidence Level

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

•• Greater confidence Greater confidence implies loss of precision implies loss of precision (i t i f(i t i f(i.e. greater margin of (i.e. greater margin of error).error).

•• 95% confidence is95% confidence is•• 95% confidence is 95% confidence is most often used.most often used.

Confidence Intervals for Example 8.2

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•• A confidence interval either A confidence interval either doesdoes or or does notdoes not contain contain .. InterpretationInterpretation

 •• The confidence level quantifies the The confidence level quantifies the riskrisk.. •• Out of 100 confidence intervals, approximately 95% Out of 100 confidence intervals, approximately 95% maymay contain contain , , , pp y, pp y yy ,,

while approximately 5% while approximately 5% might notmight not contain contain  when constructing 95% when constructing 95% confidence intervals.confidence intervals.

When Can We Assume Normality?When Can We Assume Normality? • If  is known and the population is normal, then we can safely use the p p , y

formula to compute the confidence interval. • If  is known and we do not know whether the population is normal, a common

rule of thumb is that n  30 is sufficient to use the formula as long as therule of thumb is that n  30 is sufficient to use the formula as long as the distribution Is approximately symmetric with no outliers.

• Larger n may be needed to assume normality if you are sampling from a strongly• Larger n may be needed to assume normality if you are sampling from a strongly skewed population or one with outliers.

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8 5 Confidence Interval for a Mean (8 5 Confidence Interval for a Mean () with) with pter 8.5 Confidence Interval for a Mean (8.5 Confidence Interval for a Mean () with ) with Unknown (Unknown ())LO8LO8--66

LO8LO8--6: 6: Know when to use Student’s Know when to use Student’s t t instead of instead of zz to estimate to estimate ..

•• Use the Use the Student’s t distributionStudent’s t distribution instead of the normal distribution instead of the normal distribution Student’s t DistributionStudent’s t Distribution

when the population is normal but the standard deviation when the population is normal but the standard deviation  is is unknown and the sample size is small.unknown and the sample size is small.

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LO8LO8--6: 6: Know when to use Student’s Know when to use Student’s t t instead of instead of zz to estimate to estimate ..

Student’s t DistributionStudent’s t Distribution

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Student’s t DistributionStudent’s t Distribution

•• tt distributions are symmetric and shaped like the standard normaldistributions are symmetric and shaped like the standard normal

•• tt distributions are symmetric and shaped like the standard normal distributions are symmetric and shaped like the standard normal distribution.distribution.

•• The The tt distribution is dependent on the size of the sample.distribution is dependent on the size of the sample.p pp p

Comparison of Normal and St dent’sComparison of Normal and St dent’s tt

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Comparison of Normal and Student’s Comparison of Normal and Student’s tt

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Degrees of FreedomDegrees of Freedom •• Degrees of FreedomDegrees of Freedom ((d fd f ) is a parameter based on the sample) is a parameter based on the sample

•• Degrees of Freedom Degrees of Freedom ((d.fd.f.) is a parameter based on the sample .) is a parameter based on the sample size that is used to determine the value of the size that is used to determine the value of the tt statistic.statistic.

•• Degrees of freedom tell how many observations are used to Degrees of freedom tell how many observations are used to g yg y calculate calculate , less the number of intermediate estimates used in , less the number of intermediate estimates used in the calculation. The d.f for the the calculation. The d.f for the tt distribution in this case, is given distribution in this case, is given bb d fd f 11by by d.f.d.f. = = nn --1.1.

•• As As nn increases, the increases, the tt distribution approaches the shape of the distribution approaches the shape of the l di t ib til di t ib tinormal distribution. normal distribution.

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

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Comparison of z and tComparison of z and t • For very small samples t-values differ substantially from the

• For very small samples, t-values differ substantially from the normal.

• As degrees of freedom increase, the t-values approach the g , pp normal z-values.

• For example, for n = 31, the degrees of freedom, d.f. = 31 – 1 = 30.

So for a 90 percent confidence interval, we would use t = 1.697, which is only slightly larger than z = 1.645.

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8 5 Confidence Interval for a Mean (8 5 Confidence Interval for a Mean () with) with

E l GMAT S A iE l GMAT S A i

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Example GMAT Scores AgainExample GMAT Scores Again 8

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Example GMAT Scores AgainExample GMAT Scores Again C t t 90% fid i t l f th GMAT fC t t 90% fid i t l f th GMAT f

•• Construct a 90% confidence interval for the mean GMAT score of Construct a 90% confidence interval for the mean GMAT score of all MBA applicants.all MBA applicants.

x = 510 s = 73.77

•• Since Since  is unknown, use the Student’s is unknown, use the Student’s tt for the confidence interval for the confidence interval with with d.f.d.f. = 20 = 20 –– 1 = 19.1 = 19.

•• First find First find tt/2/2 = = tt..0505 = 1.729 = 1.729 from Appendix D.from Appendix D.

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•• For a 90% confidence For a 90% confidence interval, use Appendixinterval, use Appendix

interval, use Appendix interval, use Appendix D to find tD to find t0.050.05 = 1.729 = 1.729 with with d.f.d.f. = 19.= 19.

Note: One can use Excel, Minitab, etc. to obtain these values as well as to construct confidence Intervals.

We are 90 percent confident that the true mean GMAT score might be within the

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g interval [481.48, 538.52]

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Confidence Interval WidthConfidence Interval Width • Confidence interval width reflects

• Confidence interval width reflects - the sample size, - the confidence level and - the standard deviation.

• To obtain a narrower interval and more precision i th l i- increase the sample size or

- lower the confidence level (e.g., from 90% to 80% confidence).

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Using Appendix DUsing Appendix D

•• Beyond Beyond d.f. d.f. = 50, Appendix D shows = 50, Appendix D shows d.f. d.f. in steps of 5 or 10.in steps of 5 or 10. •• If the table does not give the exact degrees of freedom, use the If the table does not give the exact degrees of freedom, use the g g ,g g ,

tt--value for the next lower degrees of freedom.value for the next lower degrees of freedom. •• This is a conservative procedure since it causes the interval to be This is a conservative procedure since it causes the interval to be

slightly wider.slightly wider. • A conservative statistician may use the t distribution for

confidence intervals when σ is unknown becauseconfidence intervals when σ is unknown because using z would underestimate the margin of error.

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8.6 Confidence Interval for a Proportion (8.6 Confidence Interval for a Proportion ())LO8LO8--77 8

LO8LO8--7: 7: Construct a 90, 95, or 99 percent confidence interval for Construct a 90, 95, or 99 percent confidence interval for π.π.

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

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8.6 Confidence Interval for a Proportion (8.6 Confidence Interval for a Proportion ())LO8LO8--77

Applying the CLTApplying the CLT

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

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8.6 Confidence Interval for a Proportion (8.6 Confidence Interval for a Proportion ())LO8LO8--77

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

•• Rule of Thumb: Rule of Thumb: The sample proportion The sample proportion pp = = xx//nn may be assumed to may be assumed to be normal if both be normal if both nn 10 and 10 and nn(1(1-- ) ) 10. 10.

Sample size to assume normality:y

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8.6 Confidence Interval for a Proportion (8.6 Confidence Interval for a Proportion ())LO8LO8--77

Confidence Interval for Confidence Interval for 

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

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8.6 Confidence Interval for a Proportion (8.6 Confidence Interval for a Proportion ())LO8LO8--77

Example AuditingExample Auditing

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8.7 Estimating from Finite Population8.7 Estimating from Finite PopulationLO8LO8--88 8

LO8LO8--8: 8: Construct Confidence Intervals for Finite PopulationsConstruct Confidence Intervals for Finite Populations.

N = population size; n = sample size

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8.8 Sample Size determination for a Mean8.8 Sample Size determination for a MeanLO8LO8--99 8

LO8LO8--9: 9: Calculate sample size to estimate a mean or proportionCalculate sample size to estimate a mean or proportion.

Sample Size to Estimate Sample Size to Estimate  •• To estimate a population mean with a precision of To estimate a population mean with a precision of ++ E E (allowable (allowable

error), you would need a sample of size. Now, error), you would need a sample of size. Now,

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8.8 Sample Size determination for a Mean8.8 Sample Size determination for a MeanLO8LO8--99

How to Estimate How to Estimate ??

•• Method 1: Method 1: Take a Preliminary SampleTake a Preliminary Sample Take a small preliminary sample and use the sample Take a small preliminary sample and use the sample ss in place of in place of  in the sample size formulain the sample size formula in the sample size formula.in the sample size formula.

•• Method 2: Method 2: Assume Uniform PopulationAssume Uniform Population Estimate rough upper and lower limitsEstimate rough upper and lower limits aa andand bb and setand setEstimate rough upper and lower limits Estimate rough upper and lower limits aa and and bb and set and set  = [(= [(bb--aa)/12])/12]½½. .

•• Method 3: Method 3: Assume Normal PopulationAssume Normal Populatione od 3e od 3 ssu e o a opu a ossu e o a opu a o Estimate rough upper and lower limits Estimate rough upper and lower limits aa and and bb and set and set  = (= (bb--aa)/4. )/4. This assumes normality with most of the data with This assumes normality with most of the data with  ±± 22 so the so the

i 4i 4range is 4range is 4..

•• Method 4: Method 4: Poisson ArrivalsPoisson Arrivals

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In the special case when In the special case when  is a Poisson arrival rate, then is a Poisson arrival rate, then  = = 

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8.9 Sample Size determination for a Proportion8.9 Sample Size determination for a ProportionLO8LO8--99

•• To estimate a population proportion with a precision of To estimate a population proportion with a precision of ±± E E (allowable error), you would need a sample of size (allowable error), you would need a sample of size

•• Since Since  is a number between 0 and 1, the allowable error is a number between 0 and 1, the allowable error EE is is

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also between 0 and 1. also between 0 and 1.

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8.9 Sample Size determination for a Proportion8.9 Sample Size determination for a ProportionLO8LO8--99

How to Estimate How to Estimate ??

•• Method 1Method 1:: Assume that Assume that  = .50 = .50 This conservative method ensures the desired precision HoweverThis conservative method ensures the desired precision HoweverThis conservative method ensures the desired precision. However, This conservative method ensures the desired precision. However, the sample may end up being larger than necessary.the sample may end up being larger than necessary.

•• Method 2Method 2: : Take a Preliminary SampleTake a Preliminary Sample T k ll li i l d th lT k ll li i l d th l i l fi l fTake a small preliminary sample and use the sample Take a small preliminary sample and use the sample pp in place of in place of  in the sample size formula.in the sample size formula.

•• Method 3Method 3:: Use a Prior Sample or Historical DataUse a Prior Sample or Historical DataMethod 3Method 3: : Use a Prior Sample or Historical DataUse a Prior Sample or Historical Data How often are such samples available? Unfortunately, How often are such samples available? Unfortunately,  might be might be different enough to make it a questionable assumption. different enough to make it a questionable assumption.

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8.10 Confidence Interval for a Population Variance (8.10 Confidence Interval for a Population Variance (22))LO8LO8--1010

LO8LO8--10: 10: Construct a confidence interval for a variance (optional).Construct a confidence interval for a variance (optional).

If th l ti i l th th l iIf th l ti i l th th l i 22

ChiChi--Square DistributionSquare Distribution

•• If the population is normal, then the sample variance If the population is normal, then the sample variance ss22 follows the follows the chichi--square distributionsquare distribution ((22) with degrees of ) with degrees of freedom freedom d.f.d.f. = = nn –– 1.1.eedoeedo dd

•• Lower (Lower (22LL) and upper () and upper (22UU) tail percentiles for the chi) tail percentiles for the chi-- square distribution can be found using Appendix Esquare distribution can be found using Appendix E..

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8.10 Confidence Interval for a Population Variance (8.10 Confidence Interval for a Population Variance (22))LO8LO8--1010

LO8LO8--10: 10: Construct a confidence interval for a variance (optional).Construct a confidence interval for a variance (optional).

U i th l iU i th l i 22 th fid i t l ith fid i t l i

Confidence IntervalConfidence Interval

•• Using the sample variance Using the sample variance ss22, the confidence interval is, the confidence interval is

•• To obtain a confidence interval for the standard deviation To obtain a confidence interval for the standard deviation , just take the square root of the interval bounds., just take the square root of the interval bounds.

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8.10 Confidence Interval for a Population Variance (8.10 Confidence Interval for a Population Variance (22))LO8LO8--1010

You can use Appendix E to find critical chi-square values.

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8.10 Confidence Interval for a Population Variance (8.10 Confidence Interval for a Population Variance (22))LO8LO8--1010

Caution: Assumption of NormalityCaution: Assumption of Normality

•• The methods described for confidence interval estimation of the The methods described for confidence interval estimation of the variance and standard deviation depend on the population having a variance and standard deviation depend on the population having a normal distributionnormal distributionnormal distribution.normal distribution.

•• If the population does not have a normal distribution, then the If the population does not have a normal distribution, then the confidence interval should not be considered accurateconfidence interval should not be considered accurateconfidence interval should not be considered accurate.confidence interval should not be considered accurate.

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