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9.1 The Logic in Constructing Confidence intervals about a Population
Mean ( population standard deviation known)
Objectives:
 Find a point estimate and a maximum error of estimate.
 Construct and interpret confidence intervals for the population mean.
 Determine the required minimum sample size when estimating  .
We’ll begin working more with inferential statistics in this chapter. You will
learn how to make a more meaningful estimate by specifying an interval of
values on a number line, together with a statement of how confident you are
that your interval contains the population parameter.

A POINT ESTIMATE is a single value estimate for a population
parameter (we normally use the sample mean x to estimate the
population mean  .
Finding a Point Estimate
Ex 1: Market researchers use the number of sentences per advertisement
as a measure of readability for magazine advertisements. A random sample
of the number of sentences found in 30 magazine advertisements is listed.
Use this sample to find point estimate for the population mean  .
16
9
12
9
17
5
14
6
18
11
11
6
Number of Sentences
17
12
99
18
17
18
20
6
4
13
11
12
13
14
12
7
5
11
An INTERVAL ESTIMATE is a range of values used to estimate a population
parameter (interval estimates are normally used more than point estimates
because it is very unlikely that the sample mean would match exactly with
the population mean) The interval estimate uses a margin of error about the
point estimate. For example if you have a point estimate of x = 12. 25 with
a margin of error of 1.75, then the interval estimate would be 12.25  1.75 .
(10.5 to 14)
Before you find an interval estimate, you should first determine how
confident you want to be that your interval estimate contains the population
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mean.
The LEVEL OF CONFIDENCE, C, is the probability that the interval
estimate contains the population parameter.
From Chapter 8.1, you know that when n  30, the sampling distribution of
sample means is a normal distribution. The level of confidence is the area
under the standard normal curve between the critical values ( z c and z c )
Common levels of confidence are noted on the Standard Normal Distribution
Table in the front cover of your text:
80% confidence
90% confidence
95% confidence
99% confidence
(.80) is
(.90) is
(.95) is
(.99) is
z
z
z
z
= 1.28
= 1.645
= 1.96
= 2.575
Other z-scores for given confidence levels can be determined using the
techniques we covered in chapter 7.
Ex 2: Find the critical value z necessary to form a confidence interval at the given
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level of confidence.
a. c = 0.85
b. c = 0.97
The ERROR of ESTIMATE is the distance from the actual population
parameter to the point estimate, but since we do not know the population
parameter in most cases, we can determine a MAXIMUM ERROR OF
ESTIMATE, E, if know the level of confidence and the sampling distribution.
Maximum Error of Estimate: (margin or error or error tolerance)
E  z 

n
When n  30, the sample standard deviation s can be used in place of 
provided the distribution are approximate normal and there are no outliers.
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NOTE: remember that we must adjust the sample standard deviation to
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become the standard error of the mean by taking (s.d.)/(sq. rt. of number in
sample)!
Ex 3: Find the maximum error of estimate for the given values of c, s, and n.
Assume the distribution is normally distributed and there are no outliers.
a. c = 0.95, s = 3.0, n = 60
b. c = 0.975, s = 3.4, n = 100
Ex 4: Use the data in example 1 and a 95% confidence level to find the
maximum error of estimate in the mean number of sentences in a magazine
advertisement.
Confidence Intervals for the Population Mean
Using a point estimate and a margin of error, you can construct an interval
estimate of a population parameter such as  . This interval is called a
confidence interval.
A c-confidence interval for the population mean  is
x E    x E
The probability that the confidence interval contains  is c.
Guidelines for finding a Confidence Interval for a Population Mean (n  30
or  is known with a normally distributed population)
1. Find the sample statistics n and x .
2. Specify  if known. Otherwise, if n  30 , find the sample standard
deviation s (use your calculator and one-variable stats) and use it as an
estimate for  .
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3. Find the critical value z that corresponds to the given level of
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confidence. (Use methods from ch 7)
4. Find the maximum error or estimate E.

E  z
n
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5. Find the left and right hand endpoints and form the confidence
interval.
x E    x E
Left endpoint(lower bound): x  E
Right endpoint( Upper bound): x  E
Ex 5: Construct the indicated confidence interval for the population mean
.
a . c  0.95, x  31.39, n  82
b. c  0.99, x  13.5, s  1.5, n  100
Ex 6. Given the sample mean and the sample standard deviation, construct
the 90% and 95% confidence intervals for the population mean.
From a random sample of 36 days in a recent year, the closing stock
prices for Hasbro had a mean of $12.53 and a standard deviation of
$2.10.
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Ex 7. In a random sample of 50 VCR’s, the mean repair cost was $75 and the
standard deviation was $12.50. Construct a 99% confidence interval for the
population mean.
Ex 8. A random sample of 56 American beech trees has a mean diameter of
0.20 meter and a standard deviation of 0.06 meter. Construct a 95%
confidence interval for the population mean.