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STATISTICS 200
Lecture #18
Thursday, October 20, 2016
Textbook: Sections 12.1 to 12.4
Objectives:
• Formulate null and alternative hypotheses correctly based on
the context of a testing situation.
• Distinguish between a one-sided and two-sided alternative
hypothesis.
• Calculate a test statistic using standardization.
• Find a p-value based on a test statistic along with the
direction of the alternative hypothesis.
Question from Midterm #2
Correct Answer: A
We have begun a strong focus on
Inference
Means
Proportions
One
population
proportion
Two
population
proportions
One
population
mean
Difference
between
Means
This week
Mean
difference
Now that we know how
the sample proportion
behaves, we can
compare a specific
sample estimate to this
distribution.
Key to hypothesis testing: we state a
null, get a distribution like this based on
that null, and then see how unusual our
sample estimate is in comparison.
Statistical Hypotheses
Null Hypothesis, H0:
• Nothing happening
• No change /
difference
Alternative Hypothesis, Ha:
• Something is happening
• There is a change /
difference
6
Major Study found: 3 in
10 adults acknowledged
that they have nodded off
while driving within the
past year.
Clicker Quiz # 11:
Quoted Number: (3 in 10) is
A. odds
B. relative
C. increased risk D. individ
Drowsy Driving Example:
Hypothesis Test
When considering all drivers about three in 10
indicate that they have driven while drowsy.
Research Question: Does recent evidence suggest that for
young drivers ages 16–24, this risk is actually higher than 0.3?
State hypotheses using appropriate language.
In the population:
same risk as 0.3.
H0: drivers 16–24 years old have the ______
higher risk than 0.3.
Ha: drivers 16–24 years old have a ________
This is a onesided alternative.
Drowsy Driving Example cont’d
Rewrite the hypotheses using the appropriate
Parameters:
Research Question: Recent evidence suggests that with
drivers ages 16–24, this rate is actually higher
H0: p = 0.3
Ha: p > 0.3
(Here, p represents the population proportion of drivers aged
16–24 who have nodded off while driving in the past year.)
Important! Hypotheses are always about
population parameters, never sample
statistics.
Question from Midterm #2
Correct Answer: A
Hypothesis Test: Steps
State
Hypotheses
Data Summary:
test statistic
State a
conclusion
Find the
p-value
Make a
decision
Calculate the sample statistic
A sample of 300 drivers from the 16–24 age
group found 105 who say that they have
driven while drowsy in the last year.
This is not (yet) the test
statistic. We need to
standardize it by
subtracting the mean of
p-hat and dividing by the
SD of p-hat under the
null.
Calculate the test statistic
This is not (yet) the test
statistic. We need to
standardize it by
H0: p = 0.30
subtracting the mean of
p-hat and dividing by the
SD of p-hat under the
null.
Ha: p > 0.30
General test statistic formula:
In our example, this formula leads to:
Question from Midterm #2
Correct Answer: B
Hypothesis Test: Steps
State
Hypotheses
Data Summary:
test statistic
State a
conclusion
Find the
p-value
Make a
decision
We have a test
statistic equal to
1.890.
Also, the
alternative is
Ha: p > 0.30.
We can use this
info to find the
p-value.
p-value definition
The p-value is the probability, if H0
is true, that our experiment would
give a test statistic at least as
extreme as the test statistic we
observed.
We have a test
statistic equal to
1.890.
Also, the
alternative is
Ha: p > 0.30.
“At least as extreme” means in
the direction determined by
the alternative hypothesis.
In this case, the p-value is P(Z≥1.890).
Therefore, the p-value is 0.0294.
Recall this example:
Are women more likely to have dogs?
Female
Male
Total
Has Dog
89
56.7%
66
50.8%
155
No Dog
68
43.3%
64
49.2%
132
Total
157
130
287
Your class data
Recall this example:
Are women more likely to have dogs?
Female
Male
Total
Has Dog
89
56.7%
66
50.8%
155
No Dog
68
43.3%
64
49.2%
132
Total
157
130
287
Let’s reframe this problem: Examine the difference
between two independent proportions, that is, pf–pm.
Is it zero? Let’s run a statistical hypothesis test.
Recall this example:
Are women more likely to have dogs?
Female
Male
Total
Has Dog
89
56.7%
66
50.8%
155
No Dog
68
43.3%
64
49.2%
132
Total
157
130
287
H0: pf–pm = 0
Hypotheses:
Ha: pf–pm ≠ 0
In this dataset,
This is a
two-sided
alternative.
The sampling distribution of
As long as both p-hat1 and p-hat2 are
approximately normal…
...and the two samples are independent...
Then the sampling distribution is
approximately normal with mean p1–p2 and
standard deviation
Recall the general test statistic formula:
In our example, the parameter is pf–pm.
Therefore:
• The sample estimate is
• The mean under H0 is 0
• The std dev. under H0 is
Notice: Same value of
p-hat in both fractions!
That value is the combined
sample proportion:
Recall the general test statistic formula:
In our example, the parameter is pf–pm.
Therefore:
• The sample estimate is
• The mean under H0 is 0
• The std dev. under H0 is
Conclusion: The test statistic is
p-value definition
The p-value is the probability, if H0
is true, that our experiment would
give a test statistic at least as
extreme as the test statistic we
observed.
We have a test
statistic equal to
1.00.
Also, the
alternative is
Ha: pf–pm ≠ 0.
“At least as extreme” means in
the direction determined by
the alternative hypothesis.
In this case, the p-value is P(Z≥1.00 or Z≤–1.00).
Therefore, the p-value is 0.317.
Recall result from Lecture 08 (Sept. 15):
Are women more likely to have dogs?
Female
Male
Total
Has Dog No Dog
Total
89
56.7%
66
50.8%
68
43.3%
64
49.2%
157
155
132
287
130
Chi-square statistic: 1.003
P-value: 0.317
Note: There was
a mistake in the
Sept. 15
calculation!
Question from Midterm #2
Correct Answer: C
If you understand today’s lecture…
12.25, 12.27, 12.41(b-d), 12.47, 12.55, 12.60,
12.63, 12.65, 12.66
Objectives:
• Formulate null and alternative hypotheses correctly based on
the context of a testing situation.
• Distinguish between a one-sided and two-sided alternative
hypothesis.
• Calculate a test statistic using standardization.
• Find a p-value based on a test statistic along with the
direction of the alternative hypothesis.