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Chapter

Nine

McGraw-Hill/Irwin

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

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Chapter NineEstimation and Confidence Estimation and Confidence IntervalsIntervals

GOALSWhen you have completed this chapter, you will be able to:ONEDefine a what is meant by a point estimate.

TWO Define the term level of level of confidence.

THREEConstruct a confidence interval for the population mean when the population standard deviation is known.

FOURConstruct 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 Estimation and Confidence IntervalsIntervals

GOALSWhen you have completed this chapter, you will be able to:FIVE Construct a confidence interval for the population proportion.

SIXDetermine the sample size for attribute and variable sampling.

Goals

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Point and Interval Estimates

A confidence interval is a range of values within which the population 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 population parameter probably lies.

A point estimate is a single value (statistic) used to estimate a population value (parameter).

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Factors that Factors that determine the determine the

width of a width of a confidence confidence

intervalinterval

Point and Interval Estimates

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 1.96 standard deviations of the hypothesized population mean.

For a 95% confidence interval about 95% of the similarly constructed intervals will contain the parameter being estimated.

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Standard Error of the Sample Means

x n

x

the standard deviation of the population

Standard Error of the Sample Mean

Standard deviation of the sampling distribution of the sample means

symbol for the standard error of the sample mean

n is the size of the sample

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Standard Error of the Sample Means

n

ss x

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 errorThe standard error

If the population standard deviation is known or the sample is greater than 30 we use the z distribution.

n

szX

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Point and Interval Estimates

n

stX

The value of t for a given confidence level depends upon its degrees of freedom.

If the population standard deviation is unknown, the underlying population is approximately normal, and the sample size is less than 30 we use the t distribution.

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Characteristics of the t Characteristics of the t distributiondistribution

It is a continuous distribution.

It is bell-shaped and symmetrical.

There is a family of t There is a family of t distributions.distributions.

The t distribution is more spread out and flatter at the center than is the standard normal distribution, differences that diminish as n increases.

Point and Interval Estimates

Assumption: the population is normal

or nearly normal

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Constructing General Confidence Intervals for µ

n

sX 96.1

n

szX

Confidence interval for the mean

95% CI for the population mean

99% CI for the population mean

Xs

n2 58.

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Example 3

The value of the population mean is not known. Our best estimate of this value is the sample mean of 24.0 hours. 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 standard deviation of 4 hours. What is the population mean?

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12.100.2449

496.100.2496.1

n

sX

The confidence limits range from 22.88 to 25.12.

95 percent confidence interval for the population mean

About 95 percent of the similarly constructed intervals include the population parameter.

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Confidence Interval for a Population Proportion

n

ppzp

)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 home revealed 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 to Arizona.

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Finite-Population Correction Factor

x n

N n

N

1

fixed upper bound

Finite population

Adjust the standard errors of the sample

means and the proportion

N, total number of objectsn, sample size

Finite-Population Finite-Population Correction FactorCorrection Factor

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Finite-Population Correction Factor

1

)1(

N

nN

n

ppp

Ignore finite-population correction factor if n/N < .05.

Standard error of the sample proportionsStandard 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 The variation in the populationpopulation

3 factors that determine the size of a sample3 factors that determine the size of a sample

The degree of confidence selectedThe degree of confidence selected

The maximum allowable errorThe maximum allowable error

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2

E

szn

Selecting a Sample Size

Calculating the sample size

where

E is the allowable error

z the z- value corresponding to the selected level of confidence

s the 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 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?

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Sample Size for Proportions

n p pZ

E

( )1

2The formula for determining the

sample size in the case of a proportion is

p is the estimated proportion, based on past experience or a pilot survey

z is the z value associated with the degree of confidence selected

E is 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 that have a dog as a pet. If the club wanted the estimate to be within 3% of thepopulation 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 no

N1 +

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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An auditor wishes to survey employees in an organization to determine compliance with federal regulations. The auditor estimates that 80% of the employees would say that the organization is in compliance. The organization has 200 employees. The auditor wishes to be 95% confident in the results, with a margin of error no greater than 3%. How many employees should the auditor survey? Example 8 Optional

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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 = no

no

N1 +

=683

1 + 683200

= 155

Example 8 continued