Page 281 - DMTH404_STATISTICS

Richa Nandra, Lovely Professional University Unit 21: Confidence Intervals

Unit 21: Confidence Intervals Notes

CONTENTS

Objectives
Introduction
21.1 Some Common Tests of Hypothesis for Normal Populations
21.2 Confidence Intervals

21.3 Summary
21.4 Keywords
21.5 Self Assessment
21.6 Review Questions

21.7 Further Readings

Objectives

After studying this unit, you will be able to:

 Discuss statistic for various testing of hypotheses problems as well as to derive power
functions
 Explain confidence intervals for parameters of various distributions

 Describe large sample tests.
Introduction

You have been introduced to the problem of testing of hypothesis and also to some basic
concepts of the theory of testing of hypothesis. There you have studied two important procedures
fortesting statistical hypotheses,viz. using Neyman-Pearson Lemma and the likelihood ratio
test. In this unit, you will be exposed to the problem of testing statistical hypotheses involving
the parameters of some important distributions through some selected examples. In this unit,
you will also be exposed to the problem of constructing confidence intervals for parameters of
some important distributions through some selected examples. You will also learn the use of
chi-square test for goodness of fit.

21.1 Some Common Tests of Hypothesis for Normal Populations

We have already described with examples two procedures for testing statistical hypotheses.
In this section we will employ Neyman-Pearson Lamma and likelihood ratio test for testing of
hypothesis related to a normal population.

Example 1: Let X , . . . , X and Y , . . . , Y be independent random samples from N ( ,  )
2
1 m 1 n l
2
and N ( ,  ), respectively. It is desired to obtain a test statistic for testing H :  =  against
2 0 1 2
2
 :    when  ( > 0 ) is unknown.
1 1 2
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