Chapter 8. Introduction to Hypothesis Testing. Chapter 8 - Chapter Outcomes. After studying the material in this chapter, you should be able to: Formulate null and alternative hypotheses for applications involving a single population mean, proportion, or variance.
Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.
Chapter 8
Introduction to Hypothesis Testing
Chapter 8 - Chapter Outcomes
After studying the material in this chapter, you should be able to:
Formulate null and alternative hypotheses for applications involving a single population mean, proportion, or variance.
Correctly formulate a decision rule for testing a null hypothesis.
Know how to use the test statistic, critical value, and p-value approach to test the null hypothesis.
Chapter 8 - Chapter Outcomes(continued)
After studying the material in this chapter, you should be able to:
Know what Type I and Type II errors are.
Compute the probability of a Type II error.
Formulating the Hypothesis
The null hypothesis is a statement about the population value that will be tested. The null hypothesis will be rejected only if the sample data provide substantial contradictory evidence.
Formulating the Hypothesis
The alternativehypothesis is the hypothesis that includes all population values not covered by the null hypothesis. The alternative hypothesis is deemed to be true if the null hypothesis is rejected.
Formulating the Hypothesis
The research hypothesis is the hypothesis the decision maker attempts to demonstrate to be true. Since this is the hypothesis deemed to be the most important to the decision maker, it will not be declared true unless the sample data strongly indicates that it is true.
Establishing the Decision Rule
The critical value is the value of a statistic corresponding to a given significance level. This cutoff value determines the boundary between the samples resulting in a test statistic that leads to rejecting the null hypothesis and those that lead to a decision not to reject the null hypothesis.
Establishing the Decision Rule
The significance level is the maximum probability of committing a Type I statistical error. The probability is denoted by the symbol .
Sampling Distribution
Maximum probability of committing a Type I error =
Do not reject H0
Reject H0
Establishing the Critical Value as a z -Value(Figure 8-4)
From the standard normal table
Then
Rejection region = 0.10
0.5
0.4
0
Example of Determining the Critical Value (Figure 8-5)
Rejection region = 0.10
0.5
0.4
0
Establishing the Decision Rule
The test statistic is a function of the sampled observations that provides a basis for testing a statistical hypothesis.
Establishing the Decision Rule
The p-value refers to the probability (assuming the null hypothesis is true) of obtaining a test statistic at least as extreme as the test statistic we calculated from the sample. The p-value is also known as the observed significance level.
Relationship Between the p-Value and the Rejection Region(Figure 8-6)
Rejection region = 0.10
p-value = 0.0036
0.5
0.4
0
The hypothesis testing process can be summarized in 6 steps:
One-Tailed Hypothesis Tests
A one-tailed hypothesis test is a test in which the entire rejection region is located in one tail of the test statistic’s distribution.
Two-Tailed Hypothesis Tests
A two-tailed hypothesis test is a test in which the rejection region is split between the two tails of the test statistic’s distribution.
Two-Tailed Hypothesis Tests (Figure 8-7)
0
Power of the Test
The power of the test is the probability that the hypothesis test will reject the null hypothesis when the null hypothesis is false.
Power = 1 -
Hypothesis Tests for Proportions
z TEST STATISTIC FOR PROPORTIONS
where:
p = Sample proportion
= Hypothesized population proportion
n = Sample size
Hypothesis Tests for Proportions (Example 8-13)
H0 : 0.01
HA : > 0.01
= 0.02
p = 9/600 = 0.015
= 0.02
Since p < 0.0182, do not reject H0
Hypothesis Tests for Variances
CHI-SQUARE TEST FOR A SINGLE POPULATION VARIANCE
where:
= Standardized chi-square variable
n = Sample size
s2 = Sample variance
2 = Hypothesized variance
Hypothesis Tests for Proportions (Example 8-13)
H0 : 2 0.25
HA : 2 > 0.25
= 0.1
df = 19
Rejection region = 0.02
Since 25.08 < 27.204, do not reject H0
Alternative Hypothesis
Critical Value(s)
Hypothesis
Null Hypothesis
One-Tailed Hypothesis Test
p-Value
Power
Research Hypothesis
Significance Level
States of Nature
Statistical Inference
Test Statistic
Two-Tailed Hypothesis Test
Type I Error
Type II Error