Introduction to Hypothesis Testing

1
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 alternative hypothesis 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
Rejection region ? 0.10
Then
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 statistics 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 statistics 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 ? gt 0.01 ? 0.02 p 9/600
0.015
? 0.02
Since p lt 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 gt 0.25 ? 0.1
Rejection region ? 0.02
df 19
Since 25.08 lt 27.204, do not reject H0
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Key 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