Estimation and Confidence Intervals

Chapter 9

McGraw-Hill/Irwin Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved.
 GOALS

1. Define a point estimate.

2. Define level of confidence.
3. Construct a confidence interval for the population
mean when the population standard deviation is
known.
4. Construct a confidence interval for a population
mean when the population standard deviation is
unknown.
5. Construct a confidence interval for a population
proportion.
6. Determine the sample size for attribute and
variable sampling.

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 Sampling and Estimates
Why Use Sampling?
1. To contact the entire population is too time
consuming.

2. The cost of studying all the items in the

population is often too expensive.

3. The sample results are usually adequate.

4. Certain tests are destructive.

5. Checking all the items is physically impossible.

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 Point Estimate

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 Confidence Interval Estimate

 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.

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 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 mean.

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 INTERPRETATION OF:
95% Confidence Interval for Population Mean for
a sample size of 300
 INTERPRETATION:
– If you construct 100 confidence intervals for a sample size of 300, 95 of
those confidence intervals will contain the population mean.
– You could take many samples of size 300 (based upon what we studied
previously) and if we construct confidence intervals for all those samples,
95% of those confidence intervals will contain the population mean.

 SAMPLING ERROR:
– But, a particular confidence interval either contains the population
mean or it does not.

– In probability terms, we feel safe that if 95 out of 100 confidence

intervals will contain the population mean, hopefully ours does
also.

– There is a 5% chance of sampling error and this error is a risk we

9-7 assume when we select a level of confidence.
 How to Obtain 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.

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9-10
 Confidence Level and Z-value

Confidence Level (%) Z-value

80% 1.29
85% 1.44
90% 1.65
95% 1.96
99% 2.58

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 Point Estimates and Confidence Intervals for a
Mean – σ Known

x  sample mean

z  z - value for a particular confidence level

σ  the population standard deviation

n  the number of observatio ns in the sample

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 Margin of Error

1. The width of the interval is determined by the level of

confidence and the standard error of the mean.

2. The standard error is affected by two values:

- Standard deviation of Population (σ)
- Number of observations in the sample (n)

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 Width of Confidence Interval

What are the factors that determine 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. (which

affects the Z-value)

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 Example
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?

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.

2. What is a reasonable range of values for the population mean? (Use 95%
confidence level)

The confidence limit are $45,169 and $45,671

The ±$251 is referred to as the margin of error

3. What do these results mean?

If we select many samples of 256 managers, and for each sample we compute the mean
and then construct a 95 percent confidence interval, we could expect about 95 percent of
these confidence intervals to contain the population mean.

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 Population Standard Deviation
σ Unknown – The t-Distribution
In most sampling situations the population standard deviation (σ) is not known.
Below are some examples where it is unlikely the population standard
deviations would be known.

1. The Dean of the Business College wants to estimate the mean number
of hours full-time students work at paying jobs each week. He selects a
sample of 30 students, contacts each student and asks them how many
hours they worked last week.

2. The Dean of Students wants to estimate the distance the typical

commuter student travels to class. She selects a sample of 40
commuter students, contacts each, and determines the one-way distance
from each student’s home to the center of campus.

3. The manager of a store wants to estimate the mean amount spent per
shopping visit by customers. A sample of 20 customers is taken.
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 t-Distribution
CHARACTERISTICS OF THE t-Distribution

1. 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

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 Z-distribution vs. t-distribution

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 Confidence Interval Estimates for the Mean

Use Z-distribution Use t-distribution

If the population standard deviation is If the population standard deviation is
known or the sample is greater than unknown and the sample is less
30. than 30.

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 When to use Z or t-distribution

Use Z-Distribution Use t-Distribution

Use Z-Distribution Use t-distribution

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 Confidence Interval for the Mean – Example
using the t-distribution
EXAMPLE
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 inch of tread remaining with a
standard deviation of 0.09 inch.
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?

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9-22
9-23
 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.

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 Proportion

 Proportion:
– Fraction, ratio or percent indicating the part of the
sample having a particular trait of interest

 There are only two possible outcomes.

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 Confidence Interval for Population Proportion
Using the Normal Distribution to Approximate the Binomial Distribution

To develop a confidence interval for a proportion, we need to meet the following assumptions.
1. The binomial conditions, discussed in Chapter 6, have been met. Briefly, these conditions are:
a. The sample data is the result of counts.
b. There are only two possible outcomes.
c. The probability of a success remains the same from one trial to the next.
d. The trials are independent. This means the outcome on one trial does not affect the outcome on
another.

2. The values n π and n(1-π) should both be greater than or equal to 5. This condition allows us to
invoke the central limit theorem and employ the standard normal distribution, that is, z, to
complete a confidence interval.

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 Confidence Interval for a Population
Proportion- Example
First, compute the sample proportion :
x 1,600
EXAMPLE
p   0.80
n 2000
The union representing the Bottle
Blowers of America (BBA) is
considering a proposal to merge Compute the 95% C.I.
with the Teamsters Union.
According to BBA union bylaws, at p( 1  p )
least three-fourths of the union C.I.  p  z / 2
membership must approve any n
merger. A random sample of 2,000 .80( 1  .80 )
current BBA members reveals  0.80  1.96  .80  .018
1,600 plan to vote for the merger 2,000
proposal. What is the estimate of
the population proportion?  ( 0.782 , 0.818 )
`
Develop a 95 percent confidence
interval for the population Conclude : The merger proposal will likely pass
proportion. Basing your decision because the interval estimate includes values greater
on this sample information, can
you conclude that the necessary than 75 percent of the union membership .
proportion of BBA members favor
the merger? Why?
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 Selecting an Appropriate Sample Size

There are 3 factors that

determine the size of a
sample, none of which has
 z  
2
any direct relationship to the n 
size of the population.  E 

 The level of confidence

desired.
 The margin of error the
researcher will tolerate.
 The variation in the
population being Studied.

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 Example: Finding Sample Size for Estimating A
Population Mean

A student in public administration wants to determine the mean amount

that 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: 2
 E, the maximum allowable error, is $100  z  
n 
 The value of z for a 95 percent level of confidence is 1.96,  E 
 The estimate of the standard deviation is $1,000. 2
 ( 1.96 )($ 1,000 ) 
 
 $100 
 ( 19.6 )2
 384.16
 385

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 Sample Size for Estimating a
Population Proportion
2
EXAMPLE 1
Z The American Kennel Club wanted to
n  p (1  p) 
E 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
where: would they need to contact? Assume a
n is the size of the sample 95% level of confidence and that the club
estimated that 30% of the children have a
z is the standard normal value dog as a pet.
corresponding to the desired level 2
of confidence  1.96 
n  (. 30)(. 70)   897
 .03 
E is the maximum allowable error

NOTE:
use p = 0.5 if no initial information on
the probability of success is available

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 Sample Size for Estimating a Population
Proportion

EXAMPLE 2
A study needs to estimate the
proportion of cities that have private
refuse collectors. The investigator 2
 1.65 
n  (.5)(1  .5)   68.0625
wants the margin of error to be  . 10 
within .10 of the population n  69 cities
proportion, the desired level of
confidence is 90 percent, and no
estimate is available for the
population proportion. What is the
required sample size?

use p = 0.5 if no initial information on

the probability of success is
available
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 Finite-Population Correction Factor

 A population that has a fixed upper bound is said to be finite.

 For a finite population, where the total number of objects is N and the size of the sample is n, the following
adjustment is made to the standard errors of the sample means and the proportion:

Standard Error of the Mean Standard Error of the Proportion

 N n p (1  p ) N n
x  p 
n N 1 n N 1

 However, if n/N < .05, the finite-population correction factor may be ignored.
Why? See what happens to the value of the correction factor in the table below
when the fraction n/N becomes smaller

 The FPC approaches 1 when n/N becomes smaller!

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 Finite-Population Correction Factor (FPCF)

 Applied, when:
– N is known
– (n/N)>0.05 i.e. the sample is more than 5% of the
population

– FPCF affects the standard error of the mean and

the standard error of the proportion as shown below:

Standard Error of the Mean Standard Error of the Proportion

 N n p (1  p ) N n
x  p 
n N 1 n N 1

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 CI for Mean with FPC - Example
s N n
EXAMPLE X t
There are 250 families in Scandia, n N 1
Pennsylvania. A random sample of 40 of
these families revealed the mean annual
church contribution was $450 and the
standard deviation of this was $75.
 Construct a 95% confidence interval for $75 250  40
the population mean?  $450  t.10 / 2, 401
40 250  1
 Could the population mean be $445 or
$425? $75 250  40
 $450  2.023
40 250  1
Given in Problem:
N – 250
n – 40  $450  $22.03
s - $75
 ($427.96, $472.03)
Since n/N = 40/250 = 0.16, the finite
population correction factor It is likely th at the population mean is more than $427.96
must be used.
but less than $472.03
The population standard To put it another way, could the population mean be $445?
deviation is not known, Yes, but it is not likely tha t it is $425 because the value
hence we use the t-
distribution $445 is within th e confidence interval and $425 is not within
the confidence interval.

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