Estimation and Confidence Intervals

Chapter 09

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

LEARNING OBJECTIVES

LO 9-1 Define a point estimate.LO 9-2 Define confidence interval.LO 9-3 Compute a confidence interval for the population mean

when the population standard deviation is known.LO 9-4 Compute a confidence interval for a population mean

when the population standard deviation is unknown.LO 9-5 Compute a confidence interval for a population

proportion.LO 9-6 Calculate the required sample size to estimate a

population proportion or population mean.

9-2

An estimatorestimator of a population parameter is a sample statistic used to estimate the parameter. The most commonly-used estimator of the:Population Parameter Sample Statistic Mean () is the Mean (X)Variance (2) is the Variance (s2)Standard Deviation () is the Standard Deviation (s)Proportion (p) is the Proportion ( )

An estimatorestimator of a population parameter is a sample statistic used to estimate the parameter. The most commonly-used estimator of the:Population Parameter Sample Statistic Mean () is the Mean (X)Variance (2) is the Variance (s2)Standard Deviation () is the Standard Deviation (s)Proportion (p) is the Proportion ( )p

• Desirable properties of estimators include:UnbiasednessEfficiencyConsistencySufficiency

• Desirable properties of estimators include:UnbiasednessEfficiencyConsistencySufficiency

Estimators and Their Properties5-3

An estimator is said to be unbiased if its expected value is equal to the population parameter it estimates.

For example, E(X)=so the sample mean is an unbiased estimator of the population mean. Unbiasedness is an average or long-run property. The mean of any single sample will probably not equal the population mean, but the average of the means of repeated independent samples from a population will equal the population mean.

Any systematic deviation of the estimator from the population parameter of interest is called a bias.

An estimator is said to be unbiased if its expected value is equal to the population parameter it estimates.

For example, E(X)=so the sample mean is an unbiased estimator of the population mean. Unbiasedness is an average or long-run property. The mean of any single sample will probably not equal the population mean, but the average of the means of repeated independent samples from a population will equal the population mean.

Any systematic deviation of the estimator from the population parameter of interest is called a bias.

Unbiasedness5-4

An unbiasedunbiased estimator is on target on average.

A biasedbiased estimator is off target on average.

{

Bias

Unbiased and Biased Estimators5-5

不偏估計式

正偏誤估計

負偏誤估計

An estimator is efficient if it has a relatively small variance (and standard deviation).

An estimator is efficient if it has a relatively small variance (and standard deviation).

An efficientefficient estimator is, on average, closer to the parameter being estimated..

An inefficientinefficient estimator is, on average, farther from the parameter being estimated.

Efficiency5-9

估計式的相對有效性

與 Me 的相對有效性X

XX

高偏差低變異 ,高偏差高變異

低偏差高變異 , 低偏差低變異

An estimator is said to be consistent if its probability of being close to the parameter it estimates increases as the sample size increases.

An estimator is said to be consistent if its probability of being close to the parameter it estimates increases as the sample size increases.

An estimator is said to be sufficient if it contains all the information in the data about the parameter it estimates.

An estimator is said to be sufficient if it contains all the information in the data about the parameter it estimates.

n = 100n = 10

ConsistencyConsistency

Consistency and Sufficiency5-16

一致性估計式

For a normal population, both the sample mean and sample median are unbiased estimatorsunbiased estimators of the population mean, but the sample mean is both more efficientefficient (because it has a smaller variance), and sufficientsufficient. Every observation in the sample is used in the calculation of the sample mean, but only the middle value is used to find the sample median.

In general, the sample mean is the bestbest estimator of the population mean. The sample mean is the most efficient unbiased estimator of the population mean. It is also a consistent estimator.

Properties of the Sample Mean5-18

The sample variance (the sum of the squared deviations from the sample mean divided by (n-1) is an unbiased estimator of the population variance. In contrast, the average squared deviation from the sample mean is a biased (though consistent) estimator of the population variance.

The sample variance (the sum of the squared deviations from the sample mean divided by (n-1) is an unbiased estimator of the population variance. In contrast, the average squared deviation from the sample mean is a biased (though consistent) estimator of the population variance.

E s Ex x

n

Ex xn

( )( )( )

( )

22

2

22

1

Properties of the Sample Variance5-19

估計方法估計方法

點估計（ Point Estimation） 根據樣本資料所求得之單一個估計值，以推估未知的母群體參數。 單一數值無法評估其估計的準確度。

區間估計（ Interval Estimation） 根據所求得之點估計量的抽樣分配特質， 求出兩個數值以構成一區間， 並利用此一區間推估未知的母群體參數範圍。 區間估計是以區間估計值來推估母群體參數的 真實值落於該區間的機率大小。

Point Estimates

A point estimate is a single value (point) derived from a sample and used to estimate a population value.

p

s

s

X

22

LO 9-1 Define a point estimate.

9-21

區間估計的意義

台北市區 30-40坪房屋價格的調查 n=36，點估計903.92萬

Q: 903.92萬不會剛好等於母體參數 μ → 這個估計值可靠嗎 ? → 這個估計值有多精確 ? → 誤差多大 ? → 估計出一個區間，並指出該區間包含母體平均數的可

靠度 ( 機率 ) 區間估計 對未知的母體參數估計出一個上下限的區間 , 並指出該區間包含母體參數的可靠度

信賴區間

民調的結果如何解讀？ 如果民調結果 有 41%選民支持甲候選人， 39%支持乙候選人。 在 95%信心水準下，誤差為正負 3%。 表有 95%信心 支持甲的選民比例，在 (0.38,0.44)範圍內 支持乙的選民比例，在 (0.36,0.42)範圍內。

信賴區間

將「大約」具體化了 95％信賴區間 從樣本數據計算出來的一個區間， 保證在所有樣本當中， 95％會把真正的母體參數包含在區間之中。

C.I. = point estimate ± margin of error

Confidence Interval Estimates A confidence interval estimate is a range of

values constructed from sample data so that the population parameter is likely to occur within that range at a specified probability.

The specified probability is called the level of confidence.

LO 9-2 Define a confidence estimate.

9-25

Factors Affecting Confidence Interval Estimates

What determines the width of a confidence interval?

1. The sample size, n.2. The variability in the population, usually σ

estimated by s.3. The desired level of confidence.

LO 9-2

9-26

Confidence Intervals for a Mean with σ Known

sample in the nsobservatio ofnumber the

deviation standard population the

level confidence particular afor valuez

mean sample

n

σ

z

x

1. The width of the interval is determined by the level of confidence and the size of the standard error of the mean.

2. The standard error is affected by two values:- Standard deviation- Number of observations in the sample

LO 9-3 Compute a confidence interval for the population mean when the population standard deviation is known.

9-27

Interval Estimates – Interpretation

For a 95% confidence interval, about 95% of the similarly constructed intervals will contain the parameter being estimated. Also 95% of the sample means for a specified sample size will lie within 1.96 standard deviations of the hypothesized population.

LO 9-3

9-28

Example: Confidence Interval for a Mean with σ Known

The American Management Association wishes to have information on the mean income of middle managers in the retail industry. A random sample of 256 managers reveals a sample mean of $45,420. The standard deviation of this population is $2,050. The association would like answers to the following questions:

1. What is the population mean?

2. What is a reasonable range of values for the population mean?

3. What do these results mean?

LO 9-3

9-29

Example: Confidence Interval for a Mean with σ Known

The American Management Association wishes to have information on the mean income of middle managers in the retail industry. A random sample of 256 managers reveals a sample mean of $45,420. The standard deviation of this population is $2,050. The association would like answers to the following questions:

What is the population mean?

In this case, we do not know. We do know the sample mean is $45,420. Hence, our best estimate of the unknown population value is the corresponding sample statistic.

The sample mean of $45,420 is a point estimate of the unknown population mean.

LO 9-3

9-30

How to Obtain the z value for a Given Confidence Level

The 95 percent confidence refers to the middle 95 percent of the observations. Therefore, the remaining 5 percent are equally divided between the two tails.

Following is a portion of Appendix B.1.

LO 9-3

Excel Function: =NORMSINV(0.025)

9-31

Example: Confidence Interval for a Mean with σ Known

The American Management Association wishes to have information on the mean income of middle managers in the retail industry. A random sample of 256 managers reveals a sample mean of $45,420. The standard deviation of this population is $2,050. The association would like answers to the following questions:

What is a reasonable range of values for the population mean?

Suppose the association decides to use the 95 percent level of confidence:

The confidence limit are $45,169 and $45,671The ±$251 is referred to as the margin of error

LO 9-3

9-32

Example: Confidence Interval for a Mean –Interpretation

American Management Association Problem

What is the interpretation of the 95% confidence interval?

Select many samples of 256 managers, compute their 95 percent confidence intervals

About 95 percent of these confidence intervals will contain the population mean.

Conversely, about 5 percent of the intervals would not contain the population mean.

LO 9-3

9-33

Characteristics of the t Distribution1. It is, like the z distribution, a continuous distribution.2. It is, like the z distribution, bell-shaped and symmetrical.3. There is not one t distribution, but rather a family of t distributions. All t

distributions have a mean of 0, but their standard deviations differ according to the sample size, n.

4. The t distribution is more spread out and flatter at the center than the standard normal distribution. As the sample size increases, however, the t distribution approaches the standard normal distribution.

LO 9-4

9-35

Comparing the z and t Distributions When n is Small, 95% Confidence Level

LO 9-4

-

-

+

+

9-36

Confidence Interval for the Mean – Example Using the t Distribution

A tire manufacturer wishes to investigate the tread life of its tires. A sample of 10 tires driven 50,000 miles revealed a sample mean of 0.32 inches of tread remaining with a standard deviation of 0.09 inches.

Construct a 95 percent confidence interval for the population mean.

Would it be reasonable for the manufacturer to conclude that after 50,000 miles the population mean amount of tread remaining is 0.30 inches?

LO 9-4

9-37

Confidence Interval for the Mean – Example Using the t Distribution

n

stXIC

t

levelConfidence

s

x

n

..

unknown) is (sinceon distributi- theUse

%95

09.0

32.0

10

:problem in theGiven

LO 9-4

9-38

Student’s t Distribution TableLO 9-4

9-39

Student’s t Distribution Table

)384.0,256.0(

064.032.010

09.0)262.2(32.0

..

262.2d.f. 9 , c 95%for

09.0

32.0

10

:problem in theGiven

n

stXIC

levelonfidencet

s

x

n

LO 9-4

9-40

Confidence Interval Estimates for the Mean – Using Minitab

The manager of the Inlet Square Mall, near Ft. Myers, Florida, wants to estimate the mean amount spent per shopping visit by customers. A sample of 20 customers reveals the following amounts spent. Obtain the 95% confidence interval for the mean amount spent by customers of the mall.

LO 9-4

sample mean: $49.35 sample standard deviation: $9.01 9-41

Student’s t Distribution TableLO 9-4

9-42

Confidence Interval Estimates for the Mean – By Formula

)57.53$ ,13.45($

22.435.4920

01.9)093.2(35.49

..

093.2d.f. 19 , c 95%for

01.9

35.49

20

:problem in theGiven

n

stXIC

levelonfidencet

s

x

n

LO 9-4

9-43

Confidence Interval Estimates for the Mean – Using Minitab

LO 9-4

9-44

Confidence Interval Estimates for the Mean – Using Excel

LO 9-4

Data > Data Analysis > Descriptive Statistics

9-45

Confidence Interval Estimates for the Mean

LO 9-3 and 9-4

9-46

A Confidence Interval for a Proportion (π)The examples below illustrate the nominal scale of measurement.1. The career services director at Southern Technical Institute

reports that 80 percent of its graduates enter the job market in a position related to their field of study.

2. A company representative claims that 45 percent of Burger King sales are made at the drive-through window.

3. A survey of homes in the Chicago area indicated that 85 percent of the new construction had central air conditioning.

4. A recent survey of married men between the ages of 35 and 50 found that 63 percent felt that both partners should earn a living.

LO 9-5 Compute a confidence interval for a population proportion.

9-47

Using the Normal Distribution to Approximate the Binomial Distribution

Assumptions:

1. The binomial conditions, discussed in Chapter 6, have been met. These are:a. The sample data is the result of counts.b. Each trial has two outcomes. c. The probability of a success remains the same for each triald. The trials are independent.

2. The values nπ and n(1 – π) should both be greater than or equal to 5.

LO 9-5

9-48

Confidence Interval for a Population Proportion – Formula

LO 9-5

9-49

Confidence Interval for a Population Proportion – Example

The union representing the Bottle Blowers of America (BBA) is considering a proposal to merge with the Teamsters Union. According to BBA union bylaws, at least three-fourths of the union membership must approve any merger. A random sample of 2,000 current BBA members reveals 1,600 plan to vote for the merger proposal.

What is the estimate of the population proportion?

Develop a 95 percent confidence interval for the population proportion. Basing your decision on this sample information, can you conclude that the necessary proportion of BBA members favor the merger? Why?

LO 9-5

9-50

Confidence Interval for a Population Proportion – Example

n

ppzp

n

xp

)1(C.I. theCompute :3 Step

C.I. 95% for the valuez theFind :2 Step

80.02000

1,600

:proportion sample theCompute :1 Step

LO 9-5

9-51

E = 0.03 of true proportion 95% Confidence Level

Finding the z value for a 95% Confidence Interval Estimate

LO 9-5

- +

9-52

Confidence Interval for a Population Proportion – Example

)818.0 ,782.0(

018.80.2,000

)80.1(80.96.180.0

)1(C.I.

C.I. 95% theCompute :3 Step

n

ppzp

LO 9-5

9-53

Selecting an Appropriate Sample Size

There are 3 factors that determine the size of a sample, none of which has any direct relationship to the size of the population.

The level of confidence desired. The margin of error the researcher will tolerate. The variation in the population being studied.

LO 9-6 Calculate the required sample size to estimate a population proportion or population mean.

9-54

What if Population Standard Deviation is Not Known

1. Conduct a pilot study.2. Use a comparable study.3. Use a range-based approach.

LO 9-6

9-55

Sample Size for Estimating the Population Mean

2

E

zn

LO 9-6

9-56

Sample Size Determination for a Variable –Example

A student in public administration wants to determine the mean amount members of city councils in large cities earn per month as remuneration for being a council member. The error in estimating the mean is to be less than $100 with a 95 percent level of confidence. The student found a report by the Department of Labor that estimated the standard deviation to be $1,000. What is the required sample size?

Given in the problem: E, the maximum allowable error, is $100. The value of z for a 95 percent level of

confidence is 1.96. The estimate of the standard deviation is

$1,000.

385

16.384

)6.19(

100$

)000,1)($96.1(

2

2

2

E

zn

LO 9-6

9-57

Sample Size Determination for a Variable –Another Example

A consumer group would like to estimate the mean monthly electricity charge for a single family house in July within $5 using a 99 percent level of confidence. Based on similar studies, the standard deviation is estimated to be $20.00. How large a sample is required?

1075

)20)(58.2(2

n

LO 9-6

9-58

Sample Size for Estimating a Population Proportion

2

)1(

E

zn

where:n is the size of the samplez is the standard normal value corresponding to

the desired level of confidenceE is the maximum allowable error

LO 9-6

9-59

Sample Size Determination – Example

The American Kennel Club wanted to estimate the proportion of children that have a dog as a pet. If the club wanted the estimate to be within 3% of the population proportion, how many children would they need to contact? Assume a 95% level of confidence and that the club estimated that 30% of the children have a dog as a pet.

89703.

96.1)70)(.30(.

2

n

2

)1(

E

zn

LO 9-6

9-60

Another Example

A study needs to estimate the proportion of cities that have private refuse collectors. The investigator wants the margin of error to be within .10 of the population proportion, the desired level of confidence is 90 percent, and no estimate is available for the population proportion. What is the required sample size?

cities 68

65.6710.

645.1)5.1)(5(.

2

n

n

2

)1(

E

zn

LO 9-6

9-61