Transcript Lecture 21 - Statistics

Today
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Today: Chapter 10
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Sections from Chapter 10: 10.1-10.4
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Recommended Questions: 10.1, 10.2, 10-8, 10-10, 10.17, 10.19
Small Sample Hypothesis Test for the Population Mean
• Have a random sample of size n ; x1, x2, …, xn
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H 0 :   0
• Test Statistic:
t
X 
S/ n
Small Sample Hypothesis Test for the Population Mean
(cont.)
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P-value depends on the alternative hypothesis:
– :    : p - value  P(T  t )
H
1
0
– :    : p - value  P(T  t )
H
1
0
H– 1 :   0 : p - value  2P(T  | t |)
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Where T represents the t-distribution with (n-1 ) degrees of freedom
Example:
• An ice-cream company claims its product contains 500 calories per
pint on average
• To test this claim, 24 of the company’s one-pint containers were
randomly selected and the calories per pint measured
• The sample mean and standard deviation were found to be 507 and 21
calories
• At the 0.01 level of significance, test the company’s claim
• What assumptions do we make when using a t-test?
• Can use t procedures even when population distribution is not normal.
Why?
Relationships Between Tests and CI’s
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Confidence interval gives a plausible range of values for a population
parameter based on the sample data
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Hypothesis Test assesses whether data gives evidence that a hypothesized
value of the population parameter is plausible or implausible
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Seem to be doing something similar
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For testing:
H 0 :   0 vs. H1 :   0
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If the test rejects the null hypothesis, then
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If the null hypothesis is not rejected,
Example (3.96)
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Based on a random sample of size 18 from a normal population, an
investigator computes a 95% confidence interval for the mean and gets [27.1,
39.3]
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What is the conclusion of the t-test at the 5% level for:
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H 0 :   29 vs. H1 :   29
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H 0 :   26.8 vs. H1 :   26.8
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Suppose we reject the second null hypothesis at the 5% level
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Another experimenter wishes to perform the test at the 10% level…would they
reject the null hypothesis
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Another experimenter wishes to perform the test at the 1% level…would they
reject the null hypothesis
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What does changing the significance level do to the range of values for which
we would reject the null hypothesis
Large Sample Inferences for Proportions
Example:
• Consider 2 court cases:
– Company hires 40 women in last 100 hires
– Company hires 400 women in last 1000 hires
• Is there evidence of discrimination?
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Can view hiring process as a Bernoulli distribution:
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Want to test:
Situation:
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Want to estimate the population proportion (probability of a “success”), p
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Select a random sample of size n
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Record number of successes, X
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Estimate of the sample proportion is:
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If n is large, what is distribution of
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Can use this distribution to test hypotheses about proportions
p̂
Large Sample Hypothesis Test for the Population Proportion
• Have a random sample of size n
• H 0 : p  p0
•
pˆ 
X
n
• Test Statistic:
Z
pˆ  p0
p0 q0 / n
• P-value depends on the alternative hypothesis:
–
H1 : p  p0 : p - value  P(Z  z)
–
H1 : p  p0 : p - value  P(Z  z)
–
H1 : p  p0 : p - value  2P(Z  | z |)
• Where Z represents the standard normal distribution
• What assumptions must we make when doing large sample hypotheses
tests about proportions?
• Example revisited:
Example
• For both court cases, find a 95% confidence interval for the probability
that the company hires a woman