prepared by lloyd r. jaisingh

McGraw-Hill/Irwin Copyright © 2013 by The McGraw-Hill Companies, Inc. All rights reserved.

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Applied Statistics in Business & Economics, 4th edition

David P. Doane and Lori E. Seward

Prepared by Lloyd R. Jaisingh

8-2

Sampling Distributions and Estimation

Chapter Contents

8.1 Sampling Variation

8.2 Estimators and Sampling Errors

8.3 Sample Mean and the Central Limit Theorem

8.4 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)

8.7 Estimating from Finite Populations

8.8 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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Chapter Learning Objectives (LO’s)

LO8-1: Define sampling error, parameter, and estimator.

LO8-2: Explain the desirable properties of estimators.

LO8-3: State the Central Limit Theorem for a mean.

LO8-4: Explain how sample size affects the standard error.

LO8-5: Construct a 90, 95, or 99 percent confidence interval for μ.

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Chapter Learning Objectives (LO’s)

LO8-6: Know when to use Student’s t instead of z to estimate μ.

LO8-7: Construct a 90, 95, or 99 percent confidence interval for π.

LO8-8: Construct confidence intervals for finite populations.

LO8-9: Calculate sample size to estimate a mean or proportion.

LO8-10: Construct a confidence interval for a variance (optional).

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• 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 population well or differ greatly from the population.

• This sampling variation can easily be illustrated.

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

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

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• The dot plots show that the sample means have much less variation than the individual sample items.

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• Estimator – a statistic derived from a sample to infer the value of a population parameter.

• Estimate – the value of the estimator in a particular sample.• Population parameters are usually represented by

Greek letters and the corresponding statistic by Roman letters.

Some Terminology

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LO8-1: Define sampling error, parameter and estimator.

8.2 Estimators and Sampling DistributionsLO8-1

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Examples of Estimators

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Sampling 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.

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


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

Bias

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

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

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


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

• A more efficient estimator has smaller variance.

Efficiency

Figure 8.6

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


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ConsistencyA consistent estimator converges toward the parameter being estimated

as the sample size increases.

Figure 8.6

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LO8-3: State the Central Limit Theorem for a mean.

The Central Limit Theorem is a powerful result that allows us toapproximate the shape of the sampling distribution of the sample mean even when we don’t know what the population looks like.

LO8-3

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

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

8.3 Sample Mean and the Central Limit TheoremLO8-3

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• If the sample is large enough, the sample means will have approximately a normal distribution even if your population is not normal.

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Illustrations of Central Limit Theorem

Note:

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Using the uniformand a right skewed distribution.

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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).

Applying The Central Limit Theorem

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Even if the population standard deviation σ is large, the sample means 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 increases.

Sample Size and Standard Error

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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 thestandard error, you must quadruple the sample size (the law of diminishing returns).

LO8-4: Explain how sample size affects the standard error.

LO8-4

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

• The population parameters are: m = 1.5, s = 1.118.

Illustration: All Possible Samples from a Uniform Population

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

Illustration: All Possible Samples from a Uniform Population

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What is a Confidence Interval?

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8.4 Confidence Interval for a Mean () with known ()LO8-5

LO8-5: Construct a 90, 95, or 99 percent confidence interval for μ.

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

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

Choosing a Confidence Level

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

• 95% confidence is most often used.

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Confidence Intervals for Example 8.2

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• A confidence interval either does or does not contain m.• The confidence level quantifies the risk.• Out of 100 confidence intervals, approximately 95% may contain

m, while approximately 5% might not contain when constructing 95% confidence intervals.

Interpretation

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

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 the distributionIs approximately symmetric with no outliers.

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

Student’s t Distribution

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8.5 Confidence Interval for a Mean () with Unknown ()LO8-6

LO8-6: Know when to use Student’s t instead of z to estimate .

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LO8-6: Know when to use Student’s t instead of z to estimate .

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

• The t distribution is dependent on the size of the sample.

Figure 8.11

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Comparison of Normal and Student’s t

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

size that is used to determine the value of the t statistic.• Degrees of freedom tell how many observations are used to

calculate s, less the number of intermediate estimates used in the calculation. The d.f for the t distribution in this case, is given by d.f. = n -1.

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• As n increases, the t distribution approaches the shape of the normal distribution.

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


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

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

normal z-values.• For example, for n = 31, the degrees of freedom, d.f. = 31 – 1 =

30.

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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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Example GMAT Scores Again

Figure 8.13

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Example GMAT Scores Again

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

x = 510 s = 73.77

• Since s is unknown, use the Student’s t for the confidence interval with d.f. = 20 – 1 = 19.

• First find t/2 = t.05 = 1.729 from Appendix D.

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• For a 90% confidence interval, use Appendix D to find t0.05 = 1.729 with d.f. = 19.

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

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We are 90 percent confident that the true mean GMAT score might be within the interval [481.48, 538.52]


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Confidence Interval Width• Confidence interval width reflects

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

• To obtain a narrower interval and more precision- increase the sample size or - lower the confidence level (e.g., from 90% to 80% confidence).

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Using Appendix D

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

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

slightly wider.• A conservative statistician may use the t distribution for confidence intervals when σ is unknown because using z would underestimate the margin of error.

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• 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 ()

LO8-7: Construct a 90, 95, or 99 percent confidence interval for π.

LO8-7

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

Applying the CLT

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8.6 Confidence Interval for a Proportion ()LO8-7

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• Rule of Thumb: The sample proportion p = x/n may be assumed to be normal if both np 10 and n(1- p) 10.

When is it Safe to Assume Normality of p?

Sample size to assume normality:

Table 8.9

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Confidence Interval for p

• Since p is unknown, the confidence interval for p = x/n (assuming a large sample) is

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Example Auditing

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8.7 Estimating from Finite Population

LO8-8: Construct Confidence Intervals for Finite Populations.

LO8-8

N = population size; n = sample size

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• To estimate a population mean with a precision of + E (allowable error), you would need a sample of size. Now,

Sample Size to Estimate m

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8.8 Sample Size determination for a Mean

LO8-9: Calculate sample size to estimate a mean or proportion.

LO8-9

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

• Method 2: Assume Uniform PopulationEstimate rough upper and lower limits a and b and set s = [(b-a)/12]½.

How to Estimate s?

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• Method 3: Assume Normal PopulationEstimate rough upper and lower limits a and b and set s = (b-a)/4. This assumes normality with most of the data with m ± 2s so the range is 4s.

• Method 4: Poisson ArrivalsIn the special case when m is a Poisson arrival rate, then s = m.

8.8 Sample Size determination for a MeanLO8-9

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

• Since p is a number between 0 and 1, the allowable error E is also between 0 and 1.

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8.9 Sample Size determination for a ProportionLO8-9

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

• Method 2: Take a Preliminary SampleTake a small preliminary sample and use the sample p in place of p in the sample size formula.

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

How to Estimate p?

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8.9 Sample Size determination for a ProportionLO8-9

• If the population is normal, then the sample variance s2 follows the chi-square distribution (c2) with degrees of freedom d.f. = n – 1.

• Lower (c2L) and upper (c2

U) tail percentiles for the chi-square distribution can be found using Appendix E.

Chi-Square Distribution


8.10 Confidence Interval for a Population Variance (2)LO8-10

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• Using the sample variance s2, the confidence interval is

Confidence Interval

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



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


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

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

Caution: Assumption of Normality


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