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Chapter 12

OMIS 2010 Chapter Notes - Chapter 12: Interval Estimation, Statistical Inference, Confidence Interval


Department
Operations Management and Information System
Course Code
OMIS 2010
Professor
Alan Marshall
Chapter
12

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Chapter 12: Inference about One Population
12.1 Introduction
In this chapter, we presented the statistical inference methods used when the problem objective is to
describe a single population. Sections 12.2 and 12.3 addressed the problem of drawing inferences about
a single population when the data are quantitative. In Section 12.2, we introduced the interval estimator
and the test statistic for drawing inferences about a population mean. This topic was covered earlier in
Sections 10.3 and 11.3. However, in Chapters 10 and 11 we operated under the unrealistic assumption
that the population standard deviation was known. In this chapter, the population standard deviation was
assumed to be unknown. Section 12.3 presented the methods used to make inferences about a population
variance. Finally, in Section 12.4 we addressed the problem of describing a single population when the
data are qualitative.
At the completion of this chapter, you are expected to know the following:
1. How to apply the concepts and techniques of estimation and hypothesis testing introduced in
Chapters 9 and 10, including:
a) setting up the null and alternative hypotheses
b) calculating the test statistic (by hand or using a computer)
c) setting up the rejection region
d) interpreting statistical results
2. How to recognize when the parameter of interest is a population mean.
3. How to recognize when to use the Student t distribution to estimate and test a population mean
and when to use the standard normal distribution.
4. How to recognize when the parameter of interest is a population variance.
5. How to recognize when the parameter of interest is a population proportion.
12.2 Inference About Population Mean When the Population Standard
Deviation Is Unknown
We presented the statistical techniques used when the parameter to be tested or estimated is the
population mean under the more realistic assumption that the population standard deviation is unknown.
Thus, the only difference between this section and Sections 10.3 and 11.3 is that when the population
standard deviation σ is known we use the z-statistic as the basis for the inference, whereas when σ is
unknown we use the t-statistic.
The formula for the interval estimator of the population mean when the population standard
deviation is unknown is
x
± tα/2
s
n
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The test statistic is
t=
x
−µ
s/n
The number of degrees of freedom of this Student t distribution is n – 1.
Question: How do I determine
t
α/2?
Answer: The degrees of freedom are n – 1, where n is the sample size. Turn to Table 4
in Appendix B and locate the degrees of freedom in the left column. Pick one
of
t
.100 ,
t
.050 ,
t
.025 ,
t
.010 or
t
.005 depending on the confidence level 1 – α.
The value of
t
α/2 is at the intersection of the row of the degrees of freedom
and the column of the value of α/2.
Question: Which interval estimator do I use when the degrees of freedom are greater
than 200?
Answer: If σ is unknown, the interval estimator of µ is
2
x
± tα/2
s
n
regardless of the sample size. The fact that
t
α/2 is approximately equal to
z
α/2
for d.f. > 200 does not change the interval estimator.
Example 12.1
The mean monthly sales of insurance agents in a large company is $72,000. In an attempt to im-
prove sales, a new training program has been devised. Ten agents are randomly selected to participate in
the program. At its completion, the sales of the agents in the next month are recorded as follows (in
$1,000):
63, 87, 95, 75, 83, 78, 69, 79, 103, 98
a) Do these data provide sufficient evidence at the 10% significance level to indicate that the pro-
gram is successful?
b) Estimate with 95% confidence the mean monthly sales for those agents who have taken the new
training program.
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Solution
The problem objective is to describe the population of monthly sales, which is a quantitative vari-
able. Thus, the parameter of interest is µ. We have no knowledge of the population standard deviation.
As a result we must calculate s (as well as
x
) from the data. Hence, the basis of the statistical inference
is the t-statistic.
a) To establish that the program is successful, we must show that there is enough evidence to con-
clude that µ is greater than $72 (thousand). Thus, the alternative hypothesis is
H1: µ > 72
and the null hypothesis is
H0: µ = 72
The test statistic is
t=
x
−µ
s/n
which is Student t distributed with n – 1 degrees of freedom. The rejection region is
t >
t
α,n1=
t
.10, 9 =1.383
From the data, we compute
x
= 83.0
and
s = 12.85
The value of the test statistic is
t=
x
−µ
s/n=83.0 72
12.85 / 10 =2.71
Since 2.71 > 1.383, we reject H0 and conclude that there is enough evidence to show that
µ > 72 (thousand) and hence that the program is successful.
b) Once we’ve determined that the parameter to be estimated is µ and that the population standard
deviation is unknown, we identify the interval estimator as
x ±tα/2s/n
The specified confidence level is .95. Thus,
1 α = .95
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