chapter 09 estimation and confidence intervals

8/11/2019 Chapter 09 Estimation and Confidence Intervals

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Chapter

Nine

McGraw-Hill/Irwin 2005 The McGraw-Hill Companies, Inc., All Rights Reserved.


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Chapter Nine

Estimation and Confidence Intervals

GOALS

When you have completed this chapter, you will be able to:

ONE

Define a what is meant by a point estimate.

TWO

Define the term level of level of confidence.

THREE

Construct a confidence interval for the population mean whenthe population standard deviation is known.

FOUR

Construct a confidence interval for the population mean when

the population standard deviation is unknown.Goals


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Chapter Nine continued

Estimation and Confidence Intervals

GOALS

When you have completed this chapter, you will be able to:

FIVE

Construct a confidence interval for the population proportion.

SIX

Determine the sample size for attribute and variable sampling.

Goals


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Point and Interval Estimates

A confidence intervalis

a range o f va lues

w i t h i n w h i c h t h epopulation parameter is

expected to occur.

The two confidence

intervals that are used

extensively are the

95% and the 99%.

An Interval Estimate

states the range within

which a populationparameter probably

l i e s .

A point estimate is

a s i n g l e v a l u e

(statistic) used toe s t i m a t e a

population value

( p a r a m e t e r ) .


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Factors that

determine the

width of a

confidence

interval


The sample size, n

The variability in the population,

usually estimated by s

The desired level of confidence


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Interval Estimates

For the 99% confidence

interval, 99% of the sample

means for a specified sample

size will lie within 2.58 standard

deviations of the hypothesized

population mean.

95% of the sample means

for a specified sample

size will lie within 1.96standard deviations of

the hypothesized

population mean.

For a 95% confidence

interval about 95% of

the similarly constructedintervals will contain the

parameter being

estimated.


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Standard Error of the Sample

Means

xn

x

the standard deviation of the population

Standard Error of the Sample Mean

Standarddeviation of

the sampling

distribution of

the samplemeans

symbol for the standard errorof the sample mean

nis the size of the sample


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Standard Error of the Sample

Means

n

ssx

If s is not known and n

>30, the standard

deviation of the sample,designated s, is used to

approximate the

population standard

deviation.

The standard error

If the population standard

deviation is known or the

sample is greater than 30

we use the z distribution.n

s

zX


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n

stX

The value of t for a given confidence level depends

upon its degrees of freedom.

If the population

standard deviationis unknown, the

underlying

population is

approximatelynormal, and the

sample size is less

than 30 we use the

t distribution.


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Characteristics of the t

distribution

It is a continuous

distribution.

It is bell-shaped and

symmetrical.

There is a family of t

distributions.

The t distribution is morespread out and flatter at the

center than is the standard

normal distribution,

differences that diminish as nincreases.


Assumption: the

population is normal

or nearly normal


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Constructing General Confidence

Intervals for

n

sX 96.1

n

s

zX

Confidence interval for the mean

95% CI for the population mean

99% CI for the population mean

Xs

n 2 5 8.


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Example 3

The value of thepopulation mean is not known. Ourbest estimate of this value is the sample mean of 24.0hours. This value is called a point estimate.

The Dean of the Business

School wants to estimate the

mean number of hours worked

per week by students. A sample

of 49 students showed a mean

of 24 hours with a standarddeviation of 4 hours. What is

the population mean?


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12.100.24

49

496.100.2496.1

n

sX

The confidence

limits range from22.88 to 25.12.

95 percent confidence interval

for the population mean

About 95 percent of the similarlyconstructed intervals include the

population parameter.


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Confidence Interval for a

Population Proportion

nppzp )1(

The confidence interval for a

population proportion


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Example 4

0497.35.500

)65)(.35(.33.235.

A sample of 500

executives who own

their own homerevealed 175 planned to

sell their homes and

retire to Arizona.

Develop a 98%confidence interval for

the proportion of

executives that plan to

sell and move toArizona.


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Finite-Population

Correction Factor

x

n

N n

N

1

fixed upper bound

Finite population

Adjust the standarderrors of the sample

means and the proportion

N, total number of objects

n, sample size

Finite-Population

Correction Factor


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Finite-Population Correction

Factor

1

)1(

N

nN

n

ppp

Ignore finite-population

correction factor ifn/N< .05.

Standard error of the sample proportions


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EXAMPLE 4 revisited

0648.100.24)1500

49500)(

49

4(96.124

n/N= 49/500 = .098 > .05

95% confidence interval for the mean number of

hours worked per week by the students if there

are only 500 students on campus

Use finite population correction factor


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Selecting a Sample Size

The variation in the population

3 factors that determine the size of a sample

The degree of confidence selected

The maximum allowable error


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2

E

szn

Selecting a Sample Size

Calculating the sample size

where

Eis the allowable error

zthe z-value corresponding to the selected level

of confidence

sthe sample deviation of the pilot survey


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Example 6

1075

)20)(58.2( 2

n

A consumer group

would like to estimate

the mean monthlyelectricity charge for a

single family house in

July within $5 using a

99 percent level of

confidence. Based on

similar studies the

standard deviation isestimated to be $20.00.

How large a sample is

required?


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Sample Size for Proportions

n p pZ

E

( )1

2The formula for

determining the

sample size in the caseof a proportion is

pis the estimated proportion, based on past

experience or a pilot survey

zis the zvalue associated with the degree of

confidence selectedEis the maximum allowable error the

researcher will tolerate

where


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Example 7

89703.

96.1)70)(.30(.

2

n

The American Kennel Club

wanted to estimate the

proportion of children thathave 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.


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What happens when

the population has less

members than the

sample size

calculated requires?

Step One: Calculate the sample

size as before.

n =

no

noN

1 +

where no is the sample size

calculated in step one. Optional method, not covered in text:Sample Size for Small Populations

Step Two: Calculate

the new sample size.


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n p pZ

E

( )1

2

Step One

Calculate the sample size as before.

= (.80)(.20) 1.96

.03

2

= 683

Step Two

Calculate the new sample size.

n= nonoN

1 += 683

1 + 683

200

= 155

Example 8 continued

chapter 09 estimation and confidence intervals

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