Transcript Chapter 8 - McGraw Hill Higher Education

A PowerPoint Presentation Package to Accompany
Applied Statistics in Business &
Economics, 4th edition
David P. Doane and Lori E. Seward
Prepared by Lloyd R. Jaisingh
McGraw-Hill/Irwin
Copyright © 2013 by The McGraw-Hill Companies, Inc. All rights reserved.
Chapter 8
Sampling Distributions and Estimation
Chapter Contents
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.9
8.10
Sampling Variation
Estimators and Sampling Errors
Sample Mean and the Central Limit Theorem
Confidence Interval for a Mean (μ) with Known σ
Confidence Interval for a Mean (μ) with Unknown σ
Confidence Interval for a Proportion (π)
Estimating from Finite Populations
Sample Size Determination for a Mean
Sample Size Determination for a Proportion
Confidence Interval for a Population Variance,  2 (Optional)
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Chapter 8
Sampling Distributions and Estimation
Chapter Learning Objectives
LO8-1:
LO8-2:
LO8-3:
LO8-4:
LO8-5:
LO8-6:
LO8-7:
LO8-8:
LO8-9:
LO8-10:
Define sampling error, parameter, and estimator.
Explain the desirable properties of estimators.
State the Central Limit Theorem for a mean.
Explain how sample size affects the standard error.
Construct a 90, 95, or 99 percent confidence interval for μ.
Know when to use Student’s t instead of z to estimate μ.
Construct a 90, 95, or 99 percent confidence interval for π.
Construct confidence intervals for finite populations.
Calculate sample size to estimate a mean or proportion.
Construct a confidence interval for a variance (optional).
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Chapter 8
8.1 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 population well or differ greatly from the population.
This sampling variation can easily be illustrated.
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Chapter 8
8.1 Sampling Variation
•
•
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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Chapter 8
8.1 Sampling Variation
•
The dot plots show that the sample means have much less variation
than the individual sample items.
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Chapter 8
LO8-1 8.2 Estimators and Sampling Errors
LO8-1: Define sampling error, parameter and estimator.
Some Terminology
•
•
•
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 represented by Greek letters and the corresponding
statistic by Roman letters.
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Chapter 8
LO8-1
8.2 Estimators and Sampling Errors
Examples of Estimators
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.
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Chapter 8
LO8-1
8.2 Estimators and Sampling Errors
• Sampling error is the difference between an estimate and the
corresponding population parameter. Example for the sample mean.

•
On average, an unbiased estimator neither overstates nor understates the true
parameter.
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Chapter 8
LO8-2
8.2 Estimators and Sampling Errors
LO8-2: Explain the desirable properties of estimators.
Note: Unbiasness is also a desirable property.
Efficiency
•
•
Efficiency refers to the variance of the estimator’s sampling distribution.
A more efficient estimator has smaller variance.
Consistency
Figure 8.6
A consistent estimator converges toward the parameter being estimated as
the sample size increases.
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Chapter 8
8.3 Sample Mean and the Central Limit Theorem
LO8-3
LO8-3: State the Central Limit Theorem for a mean.
The Central Limit Theorem is a powerful result that allows us to approximate
the shape of the sampling distribution of the sample mean even when we don’t
know what the population looks like.
•
If the population is exactly normal, then the sample mean follows a
normal distribution for any sample size.
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Chapter 8
LO8-3
8.3 Sample Mean and the Central Limit Theorem
Illustrations of Central Limit Theorem
Using the uniform
and a right skewed
distribution.
Note:
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Chapter 8
LO8-4
8.3 Sample Mean and the Central Limit Theorem
LO8-4: Explain how sample size affects the standard error.
Sample Size and Standard Error
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.
LO8-5: Construct a 90, 95, or 99 percent confidence interval for μ.
What is a Confidence Interval?
•
•
•
A confidence interval for the mean is a range mlower < m < mupper
The confidence level is the probability that the confidence interval contains the
true population mean.
The confidence level (usually expressed as a %) is the area under the curve of
the sampling distribution.
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Chapter 8
LO8-5
8.4 Confidence Interval for a Mean (m) with
known 
What is a Confidence Interval?
•
The confidence interval for m with known  is:
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Chapter 8
LO8-6
8.4 Confidence Interval for a Mean (m) with
known 
LO8-6: Know when to use Student’s t instead of z to estimate m.
Student’s t Distribution
•
Use the Student’s t distribution instead of the normal distribution when the
population is normal but the standard deviation  is unknown and the sample size
is small.
•
The confidence interval for m
(unknown ) can be rewritten as
Note: The degrees of freedom for the t distribution is n – 1.
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Chapter 8
LO8-7
8.6 Confidence Interval for a Proportion 
LO8-7: Construct a 90, 95, or 99 percent confidence interval for π.
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Chapter 8
LO8-8
8.7 Estimating from Finite Populations
LO8-8: Construct Confidence Intervals for Finite Populations.
N = population size; n = sample size
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Chapter 8
LO8-9
8.8 Sample Size Determination for a Mean
LO8-9: Calculate sample size to estimate a mean or proportion.
Sample Size to Estimate m
•
To estimate a population mean with a precision of ± E (allowable error),
you would need a sample of size. Now,
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LO8-9
8.9 Sample Size Determination for a Proportion
•
To estimate a population proportion with a precision of ± E
(allowable error), you would need a sample of size
•
Since  is a number between 0 and 1, the allowable error E is
also between 0 and 1.
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LO8-10 8.10 Confidence Interval for a Population Variance 2
LO8-10: Construct a confidence interval for a variance (optional).
Chi-Square Distribution
•
•
•
•
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 (c2U) tail percentiles for the chi-square distribution can be
found using Appendix E.
Using the sample variance s2, the confidence interval is
To obtain a confidence interval for the standard deviation , just take
the square root of the interval bounds.
8-20