Agresti/Franklin Statistics, 1 of 122 Chapter 8 Statistical inference: Significance Tests About...
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Transcript of Agresti/Franklin Statistics, 1 of 122 Chapter 8 Statistical inference: Significance Tests About...
Agresti/Franklin Statistics, 1 of 122
Chapter 8Statistical inference: Significance Tests
About Hypotheses
Learn ….
To use an inferential method called
a Significance Test
To analyze evidence that data provide
To make decisions based on data
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Two Major Methods for Making Statistical Inferences about a Population
Confidence Interval
Significance Test
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Questions that Significance Tests Attempt to Answer
Does a proposed diet truly result in weight loss, on the average?
Is there evidence of discrimination against women in promotion decisions?
Does one advertising method result in better sales, on the average, than another advertising method?
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Section 8.1
What Are the Steps For Performing a Significance
Test?
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Hypothesis A hypothesis is a statement about a
population, usually of the form that a certain parameter takes a particular numerical value or falls in a certain range of values
The main goal in many research studies is to check whether the data support certain hypotheses
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Significance Test
A significance test is a method of using data to summarize the evidence about a hypothesis
A significance test about a hypothesis has five steps
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Step 1: Assumptions
A (significance) test assumes that the data production used randomization
Other assumptions may include:• Assumptions about the sample size
• Assumptions about the shape of the population distribution
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Step 2: Hypotheses
Each significance test has two hypotheses:
• The null hypothesis is a statement that the parameter takes a particular value
• The alternative hypothesis states that the parameter falls in some alternative range of values
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Null and Alternative Hypotheses
The value in the null hypothesis usually represents no effect
• The symbol Ho denotes null hypothesis
The value in the alternative hypothesis usually represents an effect of some type
• The symbol Ha denotes alternative hypothesis
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Null and Alternative Hypotheses A null hypothesis has a single
parameter value, such as Ho: p = 1/3
An alternative hypothesis has a range of values that are alternatives to the one in Ho such as
• Ha: p ≠ 1/3 or
• Ha: p > 1/3 or
• Ha: p < 1/3
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Step 3: Test Statistic
The parameter to which the hypotheses refer has a point estimate: the sample statistic
A test statistic describes how far that estimate (the sample statistic) falls from the parameter value given in the null hypothesis
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Step 4: P-value To interpret a test statistic value, we use a
probability summary of the evidence against the null hypothesis, Ho
• First, we presume that Ho is true
• Next, we consider the sampling distribution from which the test statistic comes
• We summarize how far out in the tail of this sampling distribution the test statistic falls
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Step 4: P-value
We summarize how far out in the tail the test statistic falls by the tail probability of that value and values even more extreme
• This probability is called a P-value
• The smaller the P-value, the stronger the evidence is against Ho
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Step 4: P-value
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Step 4: P-value
The P-value is the probability that the test statistic equals the observed value or a value even more extreme
It is calculated by presuming that the null hypothesis H is true
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Step 5: Conclusion
The conclusion of a significance test reports the P-value and interprets what it says about the question that motivated the test
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Summary: The Five Steps of a Significance Test
1. Assumptions
2. Hypotheses
3. Test Statistic
4. P-value
5. Conclusion
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Is the Statement a Null Hypothesis or an Alternative Hypothesis?
In Canada, the proportion of adults who favor legalize gambling is 0.50.
a. Null Hypothesis
b. Alternative Hypothesis
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Is the Statement a Null Hypothesis or an Alternative Hypothesis?
The proportion of all Canadian college students who are regular smokers is less than 0.24, the value it was ten years ago.
a. Null Hypothesis
b. Alternative Hypothesis
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Section 8.4
Decisions and Types of Errors in Significance Tests
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Type I and Type II Errors
When H0 is true, a Type I Error occurs when H0 is rejected
When H0 is false, a Type II Error occurs when H0 is not rejected
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Significance Test Results
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An Analogy: Decision Errors in a Legal Trial
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P(Type I Error) = Significance Level α
Suppose H0 is true. The probability of rejecting H0, thereby committing a Type I error, equals the significance level, α, for the test.
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P(Type I Error)
We can control the probability of a Type I error by our choice of the significance level
The more serious the consequences of a Type I error, the smaller α should be
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Type I and Type II Errors
As P(Type I Error) goes Down, P(Type II Error) goes Up
• The two probabilities are inversely related
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A significance test about a proportion is conducted using a significance level of 0.05.
The test statistic is 2.58. The P-value is 0.01. If Ho is true, for what probability of a Type I error was the test designed?
a. .01
b. .05
c. 2.58
d. .02
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A significance test about a proportion is conducted using a significance level of 0.05.
The test statistic is 2.58. The P-value is 0.01. If this test resulted in a decision error, what type of error was it?
a. Type I
b. Type II