Lecture20

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Transcript Lecture20

Hypothesis Tests
In statistics a hypothesis is a statement that
something is true.
• Selecting the population parameter being
tested (mean, proportion, variance, ect.)
• Using p-values for hypothesis tests
• Using Critical Regions for hypothesis tests
• One tailed vs. two tailed tests
Hypothesis Tests
In a hypothesis test:
1. Identify H0 and HA
2. Select a level of significance ( )
3. Assume the null hypothesis is true
4. Take a sample and determine the
probability of that occurring. This is
called the p-value.
5. Reject or Fail to reject H0
Error in Hypothesis Tests
Type I vs Type II Error
Conclusions and Consequences for a Test of Hypothesis
True State of Nature
Conclusion
H0 True
Ha True
Accept H0
(Assume H0 True)
Correct decision
Type II error
(probability )
Reject H0
(Assume Ha True)
Type I error
(probability )
Correct decision
Population Proportion
We have seen how to conduct hypothesis
tests for a mean and we now give some
attention to proportions.
The process is completely analogous. We
use the z-score (for large samples) and we
will need to use the standard deviation
formula for a proportion. E.g.

pq
n
Example
The CEO of a large electric utility claims that at
least 80 percent of his 1,000,000 customers are
very satisfied with the service they receive. To test
this claim, the local newspaper surveyed 100
customers, using simple random sampling. Among
the sampled customers, 73 percent say they are
very satisfied. The lawyers for the newspaper says
to avoid a law suit they can accuse the CEO of
misrepresentation if they are 97% certain she is
wrong. Should they print the story?
Example - Two tailed
The CEO of a large electric utility claims that
exactly 80 percent of his 1,000,000 customers are
very satisfied with the service they receive, but has
a reputation for just making up data. To test this
claim, the local newspaper surveyed 100
customers, using simple random sampling. Among
the sampled customers, 73 percent say they are
very satisfied. The lawyers for the newspaper says
to avoid a law suit they can once again accuse the
CEO of misrepresentation if they are 97% certain
she is wrong. Should they print the story? What if
the lawyer had said 95%?
Tests of Hypothesis about a
Population Variance
• Hypotheses about the variance use the
Chi-Square distribution and statistic
n  1s
• The quantity 
has a sampling
distribution that follows the
chi-square distribution
assuming the population the
sample is drawn from is
normally distributed.
2
2
Properties of c2
• Continuous
• Right of vertical axis
• Shape varies with  = n-1 = degrees of
freedom
• Skewed (n < 30)
• Nearly normal for n>30
• See page 897/898 in text or formula sheet
on line.
Sketch of Chi-squared
distribution
c
2
25, 0.01
 44.3141
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Example
Franklin’s Aviation Parts Unlimited
manufactures aircraft altimeters with errors
normally distributed with mean of 0 ft and a
standard deviation of 43.7 ft. After
installation of a new production line, 30
altimeters were randomly selected for a
quality control test. This sample group had
errors with standard deviation of 57.4 ft.
Use a 0.05 significance level to test the
claim the new altimeters have a different
standard deviation from the old ones.
Example
Franklin’s Aviation Parts Unlimited
manufactures aircraft altimeters with errors
normally distributed with mean of 0 ft and a
standard deviation of 43.7 ft. After
installation of a new production line, 30
altimeters were randomly selected for a
quality control test. This sample group had
errors with standard deviation of 57.4 ft.
Use a 0.05 significance level to test the
claim the new altimeters have a higher
standard deviation from the old ones.
Exercises
• #8.74, 8.78 on page 403
• #8.110, 8.112 on page 416
Problems from supplementary exercises:
• # 8.125, 8.127 on page 419
• # 8.138, 8.144 on page 421
Homework
• Review Chapters 8.1-8.5, 8.7
• Read Chapter 9.1-9.5
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