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Tests of Hypotheses Using Statistics
Adam Massey*and Steven J. Miller †
Mathematics Department
Brown University
Providence, RI 02912
Abstract
We present the various methods of hypothesis testing that one typically encounters in a mathematical statistics course. The focus will be on conditions for using each test, the hypothesis tested by each test, and the appropriate (and inappropriate) ways of using each test. We conclude by summarizing the different tests (what conditions must be met to use them, what the test statistic is, and what the critical region is).
Contents
1 Types of Hypotheses and Test Statistics 2
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Types of Hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Types of Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
$2 z$-Tests and $t$-Tests $\mathbf{5}$
2.1 Testing Means I: Large Sample Size or Known Variance . . . . . . . . . . . . . . . . 5
2.2 Testing Means II: Small Sample Size and Unknown Variance . . . . . . . . . . . . 9
3 Testing the Variance 12
4 Testing Proportions 13
4.1 Testing Proportions I: One Proportion . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.2 Testing Proportions II: K Proportions . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.3 Testing $r \times c$ Contingency Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.4 Incomplete $r \times c$ Contingency Tables Tables . . . . . . . . . . . . . . . . . . . . . . . 18
5 Normal Regression Analysis 19
6 Non-parametric Tests 21
6.1 Tests of Signs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
6.2 Tests of Ranked Signs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
6.3 Tests Based on Runs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
*E-mail: amassey3102@ucla.edu
${ }^{\dagger}$ E-mail: sjmiller@math.brow.edu
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