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Chapter 12

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

by OC9945

School

York UniversityDepartment

Operations Management and Information SystemCourse Code

OMIS 2010Professor

Alan MarshallChapter

12This

**preview**shows pages 1-3. to view the full**15 pages of the document.**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 Ïƒ is known we use the z-statistic as the basis for the inference, whereas when Ïƒ 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

Â± tÎ±/2

s

n

137

Only pages 1-3 are available for preview. Some parts have been intentionally blurred.

The test statistic is

t=

x

âˆ’Âµ

s/n

The number of degrees of freedom of this Student t distribution is n â€“ 1.

Question: How do I determine

t

Î±/2?

Answer: The degrees of freedom are n â€“ 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 â€“ Î±.

The value of

t

Î±/2 is at the intersection of the row of the degrees of freedom

and the column of the value of Î±/2.

Question: Which interval estimator do I use when the degrees of freedom are greater

than 200?

Answer: If Ïƒ is unknown, the interval estimator of Âµ is

2

x

Â± tÎ±/2

s

n

regardless of the sample size. The fact that

t

Î±/2 is approximately equal to

z

Î±/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.

138

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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 Âµ. 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 Âµ is greater than $72 (thousand). Thus, the alternative hypothesis is

H1: Âµ > 72

and the null hypothesis is

H0: Âµ = 72

The test statistic is

t=

x

âˆ’Âµ

s/n

which is Student t distributed with n â€“ 1 degrees of freedom. The rejection region is

t >

t

Î±,nâˆ’1=

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

âˆ’Âµ

s/n=83.0 âˆ’72

12.85 / 10 =2.71

Since 2.71 > 1.383, we reject H0 and conclude that there is enough evidence to show that

Âµ > 72 (thousand) and hence that the program is successful.

b) Once weâ€™ve determined that the parameter to be estimated is Âµ and that the population standard

deviation is unknown, we identify the interval estimator as

x Â±tÎ±/2s/n

The specified confidence level is .95. Thus,

1 â€“ Î± = .95

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