chapter 8 introduction to hypothesis testing ©. chapter 8 - chapter outcomes after studying the...

Chapter 8Chapter 8

Introduction to Introduction to Hypothesis TestingHypothesis Testing

©

Chapter 8 - Chapter 8 - Chapter Chapter OutcomesOutcomes

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 8 - Chapter Chapter OutcomesOutcomes

(continued)(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 Formulating the HypothesisHypothesis

The null hypothesisnull hypothesis is a statement about the population value that will be tested. The null hypothesisnull hypothesis will be rejected only if the sample data provide substantial contradictory evidence.

Formulating the Formulating the HypothesisHypothesis

The alternativealternative hypothesis hypothesis is the hypothesis that includes all population values not covered by the null hypothesis. The alternative hypothesisalternative hypothesis is deemed to be true if the null hypothesis is rejected.

Formulating the Formulating the HypothesisHypothesis

The research hypothesisresearch 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.

Types of Statistical ErrorsTypes of Statistical Errors

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

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

Types of Statistical ErrorsTypes of Statistical Errors

State of Nature

DecisionConclude Null True(Don’t reject H00)

Null Hypothesis True Null Hypothesis False

Correct Decision Type II Error

Conclude Null False(Reject H00)

Type I Error Correct decision

Establishing the Establishing the Decision RuleDecision Rule

The critical valuecritical 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 Establishing the Decision RuleDecision Rule

The significance levelsignificance level is the maximum probability of committing a Type I statistical error. The probability is denoted by the symbol .

Reject H0xx25

Do not reject H0

Sampling DistributionSampling Distribution

Maximum probability of committing a Type I error =

Establishing the Establishing the Decision RuleDecision Rule

(Figure 8-3)(Figure 8-3)

xz

25

Rejection region = 0.10

28.1z0

From the standard normal table

28.110.0 zThen

28.1z

0.5 0.4

Establishing the Critical Establishing the Critical Value as a Value as a z z -Value-Value

(Figure 8-4)(Figure 8-4)

?xz

25

Rejection region = 0.10

28.1z0

0.5 0.4

Example of Determining Example of Determining the Critical Value the Critical Value

(Figure 8-5)(Figure 8-5)

64

3

nx

x for Solving

64

328.125

nzx

48.25x

Establishing the Establishing the Decision RuleDecision Rule

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

Establishing the Establishing the Decision RuleDecision Rule

The p-valuep-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-valuep-value is also known as the observed significance level.

48.25x

z25

Rejection region = 0.10

28.1z0

0.5 0.4

Relationship Between the p-Relationship Between the p-Value and the Rejection Value and the Rejection

RegionRegion(Figure 8-6)(Figure 8-6)

69.2z26x

p-value = 0.0036

Summary of Hypothesis Summary of Hypothesis Testing ProcessTesting 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.

One-Tailed Hypothesis One-Tailed Hypothesis TestsTests

A one-tailed hypothesis one-tailed hypothesis testtest is a test in which the entire rejection region is located in one tail of the test statistic’s distribution.

Two-Tailed Hypothesis Two-Tailed Hypothesis TestsTests

A two-tailed hypothesis two-tailed hypothesis testtest is a test in which the rejection region is split between the two tails of the test statistic’s distribution.

Ux )2/(

z16

645.1z0

Two-Tailed Hypothesis Two-Tailed Hypothesis Tests Tests

(Figure 8-7)(Figure 8-7)

645.1z

Lx )2/(

05.02

05.02

Type II ErrorsType 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.

Computing 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.

Power of the TestPower of the Test

The power of the testpower of the test is the probability that the hypothesis test will reject the null hypothesis when the null hypothesis is false.

Power = 1 - Power = 1 -

Hypothesis Tests for Hypothesis Tests for ProportionsProportions

The null and alternative hypotheses are stated in terms of and the sample values become pp.

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.

Hypothesis Tests for Hypothesis Tests for ProportionsProportions

z z TEST STATISTIC FOR PROPORTIONSTEST STATISTIC FOR PROPORTIONS

where:p = Sample proportion = Hypothesized population

proportionn = Sample size

n

pz

)1(

?p01.0

H0 : 0.01

HA : > 0.01

= 0.02

p = 9/600 = 0.015

Hypothesis Tests for Hypothesis Tests for Proportions Proportions

(Example 8-13)(Example 8-13)

004.0600

)01.01(01.0

p

p for Solving)004.0)(05.2(01.002. pzp

0182.0p

= 0.02

Since p < 0.0182, do not reject H0

Hypothesis Tests for Hypothesis Tests for VariancesVariances

CHI-SQUARE TEST FOR A SINGLE CHI-SQUARE TEST FOR A SINGLE POPULATION VARIANCEPOPULATION VARIANCE

where: = Standardized chi-square

variablen = Sample sizes2 = Sample variance

2 = Hypothesized variance

2

22 )1(

sn

204.272

H0 : 2 0.25

HA : 2 > 0.25

= 0.1

Hypothesis Tests for Hypothesis Tests for Proportions Proportions

(Example 8-13)(Example 8-13)

2 for Solving

08.2525.0

)33.0(19)1(2

22

sn

Rejection region = 0.02

Since 25.08 < 27.204, do not reject HSince 25.08 < 27.204, do not reject H00

df = 19

Key TermsKey Terms

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