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2. Chapter Eight Hypothesis Testing 8-2 of 23 McGraw-Hill/Irwin
Copyright © 2003 by The McGraw-Hill Companies, Inc. All rights reserved.
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3. Hypothesis Testing
8.1 Null and Alternative Hypotheses and Errors in Testing
8.2 Large Sample Tests about a Mean: Testing a One-Sided Alternative Hypothesis
8.3 Large Sample Tests about a Mean: Testing a Two-Sided Alternative Hypothesis
8.4 Small Sample Tests about a Population Mean
8.5 Hypothesis Tests about a Population Proportion
8.6* Type II Error Probabilities and Sample Size Determination
8.7* The Chi-Square Distribution
8.8* Statistical Inference for a Population Variance
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4. 8.1 Null and Alternative Hypotheses
The null hypothesis, denoted H0, is a statement of the basic proposition being tested. The statement generally represents the status quo and is not rejected unless there is convincing sample evidence that it is false.
The alternative or research hypothesis, denoted Ha, is an alternative (to the null hypothesis) statement that will be accepted only if there is convincing sample evidence that it is true.
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5. Types of Hypothesis One-Sided, Greater Than
H0:   50 Ha:  > 50 (Trash Bag)
One-Sided, Less Than
H0:   19.5 Ha:  < 19.5 (Accounts Receivable)
Two-Sided, Not Equal To
H0:  = 4.5 Ha:   4.5 (Camshaft)
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6. Type I and Type II Errors
Type I Error Rejecting H0 when it is true
Type II Error Failing to reject H0 when it is false
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7. 8. 2 Large Sample Tests about a Mean:
8.2 Large Sample Tests about a Mean: Testing a One-Sided Alternative Hypothesis
If the sampled population is normal or if n is large, we can reject H0:  = 0 at the  level of significance (probability of Type I error equal to ) if and only if the appropriate rejection point condition holds.
Test Statistic
Alternative
Reject H0 if: : z
> z : m
<
> a H a z
< - z a
If  unknown and n is large, estimate  by s.
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8. Example: One-Sided, Greater Than
Testing H0:   50 versus Trash Bag
Ha:  > 50 for  = 0.05 and  = 0.01
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9. Example: The p-Value for “Greater Than”
Testing H0:   50 vs Ha:  > 50 using rejection points and p-value.
Trash Bag
The p-value or the observed level of significance is the probability of observing a value of the test statistic greater than or equal to z when H0 is true. It measures the weight of the evidence against the null hypothesis and is also the smallest value of  for which we can reject H0.
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10. Example: One-Sided, Less Than
Testing H0:   19.5 versus Accts Rec
Ha:  < 19.5 for  = 0.01
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11. Large Sample Tests about Mean: p-Values
If the sampled population is normal or if n is large, we can reject H0:  = 0 at the  level of significance (probability of Type I error equal to ) if and only if the appropriate rejection point condition holds or, equivalently, if the corresponding p-value is less than .
Alternative
Reject H0 if:
p-Value
Test Statistic
If  unknown and n is large, estimate  by s.
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12. Five Steps of Hypothesis Testing
Determine null and alternative hypotheses
Specify level of significance (probability of Type I error) 
Select the test statistic that will be used. Collect the sample data and compute the value of the test statistic.
Use the value of the test statistic to make a decision using a rejection point or a p-value.
Interpret statistical result in (real-world) managerial terms
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13. Example: Two-Sided, Not Equal to
Testing H0:  = 4.5 versus Camshaft
Ha:   4.5 for  = 0.05
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14. 8.5 Small Sample Tests about a Population Mean
If the sampled population is normal, we can reject H0:  = 0 at the  level of significance (probability of Type I error equal to ) if and only if the appropriate rejection point condition holds or, equivalently, if the corresponding p-value is less than .
Alternative
Reject H0 if:
p-Value
Test Statistic
t, t/2 and p-values are based on n – 1 degrees of
freedom.
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15. Example: Small Sample Test about a Mean
Testing H0:   18.8 vs Ha:  < 18.8 using rejection points and p-value.
Credit Card Interest Rates
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16. 8.5 Hypothesis Tests about a Population Proportion
If the sample size n is large, we can reject H0: p = p0 at the  level of significance (probability of Type I error equal to ) if and only if the appropriate rejection point condition holds or, equivalently, if the corresponding p-value is less than .
Alternative
Reject H0 if:
p-Value
Test Statistic
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17. Example: Hypothesis Tests about a Proportion
Testing H0: p  0.70 versus Ha: p > 0.70
using rejection points and p-value.
Using Phantol, proportion of patients with reduced severity and duration of viral infections.
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18. *8.6 Type II Error Probabilities
Testing H0:   vs Ha:  < 3
(Amount of Coffee in 3-Pound Can)
, Probability of Type II Error,
Given  = 2.995,  = 0.05.
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19. How Type II Error Varies Against Alternatives
Testing H0:   vs Ha:  < 3
(Amount of Coffee in 3-Pound Can)
, Probability of Type II Error
( = 0.05)
Given  = 2.995,
Given  = 2.990,
Given  = 2.985,
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20. *8.7 The Chi-Square Distribution
The chi-square distribution depends on the number of degrees of freedom.
A chi-square point is the point under a chi-square distribution that gives right-hand tail area .
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21. *8.8 Statistical Inference for Population Variance
If s2 is the variance of a random sample of n measurements from a normal population with variance 2, then the sampling distribution of the statistic (n - 1) s2 / 2 is a chi-square distribution with (n – 1) degrees of freedom and
100(1-)% confidence
interval for 2
All chi-square points are based
on n – 1 degrees of freedom
Test of H0: 2 = 20
Test Statistic
Reject H0 in favor of
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22. Summary: Selecting an Appropriate Test Statistic for a Test about a Population Mean
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23. Hypothesis Testing Summary:
8.1 Null and Alternative Hypotheses and Errors in Testing
8.2 Large Sample Tests about a Mean: Testing a One-Sided Alternative Hypothesis
8.3 Large Sample Tests about a Mean: Testing a Two-Sided Alternative Hypothesis
8.4 Small Sample Tests about a Population Mean
8.5 Hypothesis Tests about a Population Proportion
8.6* Type II Error Probabilities and Sample Size Determination
8.7* The Chi-Square Distribution
8.8* Statistical Inference for a Population Variance
Summary:
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