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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.

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Chapter 8

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Chapter 8 l.jpg

Chapter 8

Introduction to Hypothesis Testing


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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.


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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.


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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.


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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.


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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.


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Types of Statistical Errors

  • Type I Error - This type of statistical error occurs when the null hypothesis is true and is rejected.

  • Type II Error - This type of statistical error occurs when the null hypothesis is false and is not rejected.


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Types of Statistical Errors


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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.


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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 .


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Establishing the Decision Rule(Figure 8-3)

Sampling Distribution

Maximum probability of committing a Type I error = 

Do not reject H0

Reject H0


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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


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Example of Determining the Critical Value (Figure 8-5)

Rejection region  = 0.10

0.5

0.4

0


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Establishing the Decision Rule

The test statistic is a function of the sampled observations that provides a basis for testing a statistical hypothesis.


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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.


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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


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Summary of Hypothesis Testing Process

The hypothesis testing process can be summarized in 6 steps:

  • Determine the null hypothesis and the alternative hypothesis.

  • Determine the desired significance level ().

  • Define the test method and sample size and determine a critical value.

  • Select the sample, calculate sample mean, and calculate the z-value or p-value.

  • Establish a decision rule comparing the sample statistic with the critical value.

  • Reach a conclusion regarding the null hypothesis.


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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.


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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.


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Two-Tailed Hypothesis Tests (Figure 8-7)

0


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Type II Errors

  • A Type II error occurs when a false hypothesis is accepted.

  • The probability of a Type II error is given by the symbol .

  •  and  are inversely related.


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Computing 

  • Draw a picture of the hypothesized sampling distribution showing acceptance/rejection regions and with the mean equal to the value specified by H0.

  • Determine the critical value(s).

  • Below the hypothesized distribution, draw the sampling distribution whose mean is that for which you want to determine .

  • Extend the critical values from the hypothesized distribution down to the sampling distribution under HA and shade the rejection region.

  • The unshaded area in the sampling distribution is the graphical representation of beta - find this area.


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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 - 


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Hypothesis Tests for Proportions

  • The null and alternative hypotheses are stated in terms of  and the sample values become p.

  • The null hypothesis should include an equality.

  • The significance level determines the size of the rejection region.

  • The test can be one- or two-tailed depending on the situation being addressed.


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Hypothesis Tests for Proportions

z TEST STATISTIC FOR PROPORTIONS

where:

p = Sample proportion

 = Hypothesized population proportion

n = Sample size


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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


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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


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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


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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

Key Terms


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