Statistical techniques in business ecohomics chap009

Chapter NineEstimation and Confidence Intervals Intervals GOALS When you have completed this chapter, you will be able to: Construct a confidence interval for the population mean when

Chapter Nine

Estimation and Confidence Intervals

Intervals GOALS

When you have completed this chapter, you

will be able to:

Construct a confidence interval for the population mean when

the population standard deviation is known.

FOUR

Construct a confidence interval for the population mean when

the population standard deviation is unknown Goals

Chapter Nine continued

Estimation and Confidence Intervals

GOALS

When you have completed this chapter, you

will be able to:

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

Point and Interval Estimates

The sample size, n

The variability in the population,

usually estimated by s

The desired level of confidence

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.

For a 95% confidence

interval about 95% of

the similarly constructed

intervals will contain the

parameter being

estimated.

Standard Error of the Sample

 the standard deviation of the population

Standard Error of the Sample Mean

means symbol for the standard error of the sample mean

n is the size of the sample

Standard Error of the Sample

The standard error

If the population standard deviation is known or the sample is greater than 30

we use the z distribution.

n

s z

Point and Interval Estimates

n

s t

normal, and the

sample size is less

than 30 we use the

t distribution.

differences that diminish as n

Constructing General Confidence

Confidence interval for the mean

95% CI for the population mean

99% CI for the population mean

n

2 58

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?

.24

49

496

.100

.2496

.1

Confidence Interval for a Population Proportion

n

p

p z

p ( 1  )



The confidence interval for a

population proportion

Example 4

0497

35

500

) 65 )(.

35

(.

33

2 35

A sample of 500

executives who own

their own home

executives that plan to

sell and move to

Arizona.

Finite-Population Correction Factor

Adjust the standard

errors of the sample

means and the proportion

N, total number of objects

n, sample size

Finite-Population Correction Factor

Finite-Population Correction

Factor

1

) 1

EXAMPLE 4 revisited

0648

1 00

24

) 1 500

49

500 )(

49

4 (

96 1

are only 500 students on campus

Use finite population correction factor

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

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

Example 6

107 5

) 20 )(

58 2

would like to estimate

the mean monthly

electricity charge for a

single family house in

July within $5 using a

Sample Size for Proportions

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

Example 7

897 03

.

96

1 ) 70 )(.

30 (.

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

What happens when the population has less members than the

sample size calculated requires?

Step One: Calculate the sample

size as before.

n =

no

n o N

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.

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