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