Download 8 - Cerritos College

Chapter 8
Hypothesis Testing
for Population
Proportions
Copyright © 2014 Pearson Education, Inc. All rights reserved
Learning Objectives




8- 2
Know how to test hypotheses concerning a
population proportion and hypotheses concerning
the comparison of two population proportions.
Understand the meaning of p-value and how it is
used.
Understand the meaning of significance level and
how it is used.
Know the conditions required for calculating a
p-value and significance level.
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8.1
The Main Ingredients
of Hypothesis Testing
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Hypothesis Testing
How do you determine if someone is
cheating when they toss a coin?
 If the toss is fair
P(Heads) = p = 0.5
 Otherwise
P(Heads) = p ≠ 0.5
 If only 15 Heads come up out of 40 tosses, is
the person cheating?

8- 4
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Null and Alternative Hypotheses

H0: p = 0.5



Ha: p ≠ 0.5
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8- 5
“H naught” or “the Null Hypothesis”
The status quo, no cheating, no surprises, no change, no
effect
“The Alternative Hypothesis”
This is what we hope or guess is true.
Note that p is the population proportion.
Hypothesis testing is always for the population
parameter, never the statistic.
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Choosing the Null and Alternative
Hypothesis

First come up with an idea or hypothesis.
 The
dealer is palming aces.
 Babies are born less often on Tuesdays.

Write down the null hypothesis.
 H0:
p = 1/4
 H0: p = 1/7

Write down the alternative hypothesis.
 Ha:
p > 1/4
 Ha: p < 1/7
8- 6
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Level of Significance

The Level of Significance, a, is the
probability of rejecting H0 when H0 is true.
 Concluding
that the coin tossing is not fair when
it is fair.
 Saying that the defendant is guilty but he did not
commit the crime.
 Concluding that the person has psychic abilities
when she was just guessing.
8- 7
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Typical Levels of Significance

Since the level of significance is the
probability of making an error, it should be
small.
a
= 0.05 or 5% is the most typical.
 a = 0.01 or 1% is used when making this error
has very bad repercussions.
 a = 0.10 or 10% is used when an error is less an
issue than making no conclusion when Ha is true.
8- 8
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The Test Statistic
In conducting a hypothesis test one takes a
sample and finds p̂ .
 The Central Limit Theorem for proportions:


N  p0 ,


8- 9
p0 (1  p0 ) 

n

The Test Statistic is the z-score:
pˆ  p0
Observed  Null
z

SE
p0 (1  p0 )
n
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The Meaning of the Test Statistic
The test statistic tells us how unlikely that
sample proportion could have happened by
random chance had the null hypothesis been
true.
 If the null hypothesis is true, then the
test statistic should be close to 0. Therefore,
the farther the test statistic is from 0, the
more the null hypothesis is discredited.

8 - 10
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The p-Value

8 - 11
The p-value is a probability. Assuming the
null hypothesis is true, the p-value is the
probability that if the experiment were
repeated, you would get a test statistic as
extreme as or more extreme than the one you
actually got. A small p-value suggests that a
surprising outcome has occurred and
discredits the null hypothesis.
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p-Value Example
8 - 12

A hypothesis test was conducted to see if 10% of the
time that the weather report states a 10% chance of rain
that there is rain. Of 300 randomly selected days with a
10% chance of rain, 6% had rain. The p-value was
found to be 0.02. Interpret this
p-value.

If another 300 days with 10% chance of rain were
randomly selected, then there is only a 2% chance that
the number of days of rain would be less than 18 (6%)
or more than 42 (14%).
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Hypothesis Testing Summary
1.
Hypothesize
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2.
Prepare
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8 - 13
Come up with an idea or hypothesis.
Write down H0 and Ha.
Choose the level of significance a.
Select the test statistic
Check Sampling Distributions’ Conditions
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Hypothesis Testing Summary
(Continued)
3.
Compute to Compare


4.
Interpret
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8 - 14
Compute the test statistic.
Compute the p-Value.
If p-Value < a, reject H0.
If p-Value > a, fail to reject H0.
State the conclusion in the context of the study.
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8.2
Characterizing
p-values
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Three Types of Hypothesis Tests

Left Tailed Hypothesis
 H0:
p = p0
 Ha: p < p0

Right Tailed Hypothesis
 H0:
p = p0
 Ha: p > p0

Two Tailed Hypothesis
 H0:
p = p0
 Ha: p ≠ p0
8 - 16
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Left Tailed Hypothesis
H0: p = p0
 Ha: p < p0

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8 - 17
The p-value represents the probability that
if p = p0 and another random sample is
taken with the same sample size then the
new sample proportion will be less than
observed sample proportion.
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Right Tailed Hypothesis
H0: p = p0
 Ha: p > p0

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8 - 18
The p-value represents the probability that
if p = p0 and another random sample is
taken with the same sample size then the
new sample proportion will be greater
than observed sample proportion.
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Two Tailed Hypothesis
H0: p = p0
 Ha: p ≠ p0


8 - 19
The p-value represents the probability that
if p = p0 and another random sample is
taken with the same sample size then the
new sample proportion will be farther
from p0 than observed sample proportion.
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p-Value Example
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8 - 20
45% of Americans have type “O” blood. A study
was done to see if the percent differs for college
students. 47% of the 1000 randomly selected
students had type O blood. The p-Value was 0.2.
Two tailed test: “differs”
If 45% of college students do have type “O” blood
and if another 1000 randomly selected students are
tested, then there is a 20% chance that either fewer
than 43% or greater than 47% of this new group of
students will have type “O” blood.
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Requirements for a Valid p-Value
1.
2.
3.
4.
5.
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Random sample: The sample is collected randomly.
Large enough sample size: The sample size, n, is
large enough that the sample has at least 10 expected
successes and 10 expected failures
np0 ≥10 and n(1 - p0) ≥ 10
Without replacement: If the sample is collected
without replacement, then the population size is at
least 10 times bigger than the sample size.
Independence: Each observation or measurement
must have no influence on any others.
Null hypothesis : The null hypothesis is true.
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StatCrunch and the p-Value
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8 - 22
Have more than 70% of all dogs had their rabies
shots? 592 of the 800 dogs examined had their
rabies shots.
H0: p = 0.7
Ha: p > 0.7
Stat→Proportions→One sample→with summary
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StatCrunch and the p-Value

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8 - 23
Have more than 70% of all dogs had their rabies
shots? 592 of the 800 dogs examined had their
rabies shots.
H0: p = 0.7
Ha: p > 0.7
If 70% of all dogs have had their rabies shots and if
another 800 randomly selected dogs are examined,
then there is a 0.68% chance that more than 592 of
them will have had their rabies shots.
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Summary of the p-Value
Tells us how surprising the sample data is if
the null hypothesis is true.
 A very small p-Value, less than 0.05 for
example, indicates that the results that were
obtained would be very surprising.
 A larger p-Value, greater than 0.05 for
example, indicates that the results that were
obtained would not be very surprising.

8 - 24
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8.3
Hypothesis Testing in
Four Steps
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Four Steps to Hypothesis Testing
1.
2.
3.
4.
8 - 26
Hypothesize: State your hypothesis about the
population parameter.
Prepare: Choose a significance level and test
statistic, check requirements, and state
assumptions.
Compute to Compare: Compute the test statistic
and find the p-value to measure the surprise.
Interpret: Reject or fail to reject the null
hypothesis. State the conclusions in the context of
the study.
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Hypothesize

Each year 5% of the world tiger population is
killed by poachers. Now that there is a
campaign to educate people on the problem,
has this percent gone down? 300 tigers were
observed and 18 were killed by poachers.
 H0:
p = 0.05
 Ha: p < 0.05
8 - 27
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Hypothesize

90% of all restaurants fail after one year. Is
that number different for Chinese restaurants.
Of the 108 new Chinese restaurants 87 had
failed after one year.
 H0:
p = 0.90
 Ha: p ≠ 0.90
8 - 28
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Prepare

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8 - 29
Each year 5% of the world tiger population is killed
by poachers. Now that there is a campaign to
educate people on the problem, has this percent
gone down? 300 tigers were observed and 18 were
killed by poachers.
Use the standard a = 0.05.
np0 = (300)(0.05) = 15 ≥ 10
n(1-p0) = (300)(0.95) = 285 ≥ 10
Can use the z-statistic.
Assume that the 300 tigers were randomly selected
and that there are more than 3000 tigers in the
world.
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Prepare


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8 - 30
90% of all restaurants fail after one year. Is that
number different for Chinese restaurants. Of the
108 new Chinese restaurants 87 had failed after one
year.
Use the standard a = 0.05.
np0 = (108)(0.9) = 97.2 ≥ 10
n(1-p0) = (108)(0.1) = 10.8 ≥ 10
Can use the z-statistic.
Assume that the 108 Chinese restaurants were
randomly selected and that there are more than 1080
Chinese restaurants in existence.
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Compute to Compare
8 - 31

Each year 5% of the world tiger population is killed by
poachers. Now that there is a campaign to educate people on
the problem, has this percent gone down? 300 tigers were
observed and 18 were killed by poachers.


z ≈ 0.795
p-value ≈ 0.79

pˆ  0.06

If 5% of the tigers are killed by poaching and if
another sample is observed, there is a 79% chance
of getting a sample proportion less than 6%. 79% is
very high.
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Compute to Compare
8 - 32

90% of all restaurants fail after one year. Is that number
different for Chinese restaurants. Of the 108 new Chinese
restaurants 87 had failed after one year.


z ≈ -3.27
p-value ≈ 0.001

pˆ  0.81

If 90% of all Chinese restaurants fail after one year
and if another sample is observed, there is a 0.1%
chance of getting a sample proportion less than
81%. 0.1% is a very small chance.
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Rules to Interpret
1.
2.
8 - 33
If p-value < a, reject H0 and accept Ha and
state the conclusion.
If p-value > a, fail to reject H0 and just state
that we do not have statistically significant
evidence to make a conclusion.
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Interpret


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8 - 34
Each year 5% of the world tiger population is killed
by poachers. Now that there is a campaign to
educate people on the problem, has this percent
gone down? 300 tigers were observed and 18 were
killed by poachers.
z ≈ 0.795, p-value ≈ 0.79, pˆ  0.06
Since p-value ≈ 0.79 > 0.05 = a, fail to reject H0.
There is statistically insignificant evidence to
support the claim that since the campaign began,
less than 5% of the world tiger population is killed
by poachers.
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Interpret



8 - 35
90% of all restaurants fail after one year. Is that
number different for Chinese restaurants. Of the
108 new Chinese restaurants 87 had failed after one
year.
z ≈ -3.27, p-value ≈ 0.001, pˆ  0.81
Since p-value ≈ 0.001 < 0.05 = a, reject H0 and
accept Ha. There is statistically significant evidence
to support the claim that the percent of Chinese
restaurants that fail after one year is different from
90%.
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8.4
Comparing Proportions
from Two Populations
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Comparing Two Populations
Comparing two groups, men vs. women,
blacks vs. whites, with treatment vs. without
treatment, etc.
 Instead of the sample size, n, there will be
two sample sizes, n1 and n2.
 p1 and p2 instead of a single population
proportion, p.
 p̂1 and p̂2 instead of the sample proportion, p̂

8 - 37
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Hypothesis Tests

The Null Hypothesis will be:
 H0:

p1 = p 2
Three possibilities for the Alternative
Hypothesis:
 Ha:
p1 < p 2
 Ha: p1 > p2
 Ha: p1 ≠ p2
8 - 38
(Left Tailed Test)
(Right Tailed Test)
(Two Tailed Test)
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The Test Statistic
estimator  null value
z
SE
 The estimator is just the difference between
the sample proportions: pˆ1  pˆ 2

The null value is typically 0.
 SE 

8 - 39
1 1
Total # of Successes
pˆ 1  pˆ     , pˆ 
Total # of Trials
 n1 n2 
StatCrunch is easier to use than the formula.
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Checking Conditions



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8 - 40
Large Sample:
Note the use of the pooled proportion.
n1 pˆ  10, n1 1  pˆ   10, n2 pˆ  10, n2 1  pˆ   10
Random Sample or at least close to random.
The two samples are independent of each other.
Independent Within Samples: The observations
within each sample must be independent of one
another.
The Null Hypothesis is true.
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Using StatCrunch

8 - 41
Stat→Proportions→Two sample→with summary
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Caffeine Therapy For Premature Infants
Apnea of prematurity occurs
when premature babies have
shallow breathing or stop
breathing for more than
20 seconds.
 Treatment group received
caffeine therapy while the
other group received a placebo.

8 - 42
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Does caffeine therapy lower the rate of bad events?
Of the 937 infants given the therapy, 377 suffered
from death or disability. The placebo group had 932
infants, and of these, 431 suffered from death or
disability.
1.
Hypothesize

2.
Prepare

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
8 - 43
H0: p1 = p2, Ha: p1 < p2
Use a = 0.05.
z-statistic
937 x 0.43, 937 x 0.57, 932 x 0.43, 932 x 0.57
are all greater than or equal to 10. Assume that the
infants were randomly and independently selected.
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Does caffeine therapy lower the rate of bad events?
Of the 937 infants given the therapy, 377 suffered
from death or disability. The placebo group had 932
infants, and of these, 431 suffered from death or
disability.
3.
Compute to Compare

4.
StatCrunch → z = -2.62, p-value = 0.004
Interpret
 p-value

8 - 44
377  431
pˆ 
 0.43
937  932
= 0.004 < 0.05 = a
Reject H0, Accept Ha. There is statistically significant
evidence to conclude that a lower proportions of babies
will die or suffer with this therapy than without this
therapy.
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Is there a difference between the proportions of men
and women who play guitar? 82 of the 400 men and
54 of the 300 women surveyed played guitar.

Hypothesize


Prepare



8 - 45
H0: p1 = p2, Ha: p1 ≠ p2
Use a = 0.05.
Large Sample?
400 x 0.19, 400 x 0.81, 300 x 0.19, 300 x 0.81
are all greater than or equal to 10.
Assume that the men and women were randomly and
independently selected.
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Is there a difference between the proportions of men
and women who play guitar? 82 of the 400 men and
54 of the 300 women surveyed played guitar.

Compute to Compare


StatCrunch → z = 0.83, p-value = 0.41
Interpret
 p-value

8 - 46
82  54
pˆ 
 0.19
400  300
= 0.41 > 0.05 = a
Fail to Reject H0. There is statistically insufficient
evidence to make a conclusion about there being a
difference between the proportions of men and women
who own a guitar.
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8.5
Understanding
Hypothesis Testing
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What to do if Conditions Fail

The sample size is too small
 Redo
the study with a larger sample size.
 Use an advanced test, e.g. Fisher’s Exact Test.

The samples are not independent
 Take
an advance statistics class or
 Consult a statistician.

Samples not random
 State
8 - 48
the conclusion for the sample only.
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Power
The Power of a hypothesis test is the
probability of rejecting the null hypothesis
when the null hypothesis is false.
 Example: If the alternative hypothesis is
that the person cheated, then the power is the
probability that if the person is cheating then
we will conclude correctly that cheating
occurred.
 Always strive for a large power.

8 - 49
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Hypotheses, Power, and Level of
Significance



8 - 50
Before the data is collected you can choose to make
it more likely to reject H0 or more likely to fail to
reject H0, but not both.
Both choices lead to an increased chance of one of
the “Bad” outcomes.
Only increasing the sample size decreases the
chance of both “Bad” outcomes.
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Power vs. Level of Significance
Increasing the level of significance decreases
the power.
 If you want a better chance of rejecting H0
when H0 is false, you can decrease the level
of significance.
 The only way to have a large power and a
large level of significance is to have a large
sample size.

8 - 51
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Statistical Significance vs. Practical
Significance
The term Statistical Significance is used
when the null hypothesis is rejected. There is
a small probability that if H0 is correct then
results as extreme as were obtained would
happen randomly.
 The term Practical Significance means that
the results obtained are clearly far from the
hypothesized value.

8 - 52
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Statistical Significance and Practical
Significance

A poll to see if the incumbent will get more than
50% of the votes results resulted in a sample
proportion 58%. a = 0.05 and the p-value = 0.06.


A study to see if there is a difference in pass rates
for male vs. female statistics students resulted in a
sample proportion of men: 72.1% and women:
72.3%. a = 0.05 and the p-value = 0.04.

8 - 53
This is practically significant but statistically
insignificant.
This is practically insignificant but statistically
significant.
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Hypotheses Should Never Be Changed
After the Data is Analyzed
The null and alternative hypotheses must be
written down before the data is analyzed.
 Never adjust H0 and Ha to fit your results.
 If you change your mind based on the data,
you must collect a new data set in order to
support the adjusted hypotheses.

8 - 54
Copyright © 2014 Pearson Education, Inc. All rights reserved
Using Proper Language
If the p-value is less than the level of
significance, a, then state that we “reject H0”
and “accept Ha”. Do not state that the
alternative hypothesis is proven or true.
 If the p-value is greater than a then state that
we “fail to reject H0” and that no conclusion
can be made. Do not state that H0 is
accepted or true. It is possible that the power
is too small and that H0 is still false.

8 - 55
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Conclude Only the Inequality


8 - 56
A two tailed test was used to see whether there is a
difference between the proportion of genetically
modified seeds that germinate and natural seeds that
germinate. The sample proportions were 0.68 and
0.43 and H0 was rejected.
State that there is evidence for a difference in the
population proportions. Do not state that for all
seeds, the ones that are genetically modified are
25% more likely to germinate.
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Chapter 8
Case Study
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TV Violence Leads to Actual Violence?

1.
A study was done to see if children who view
violent TV shows are more likely to become
violent towards their spouses when they grow
up.
Hypothesize
 H0:
p1 = p 2
 Ha: p1 > p2
2.
Prepare
a
8 - 58
= 0.05
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TV Violence Leads to Actual Violence?
8 - 59
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Checking Conditions
We will assume that the participants were
selected randomly and are independent from
each other.
 The population sizes are certainly large
compared to the sample sizes.


8 - 60
66
pˆ 
 0.25 :
329
66  0.25, 66  .75, 263  0.25, 263  .75  10
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3. Compute to Compare
8 - 61
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4. Interpret
Since p-value = 0.0033 < 0.05 = a, we reject
the null hypothesis and accept the alternative
hypotheses.
 There is statistically significant evidence to
conclude that children who grow up watching
violent TV are more likely to become violent
towards their spouses when they grow up.
 We cannot conclude based solely on the data
that watching violent TV as children causes
adults to become violent towards their
spouses.

8 - 62
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Chapter 8
Guided Exercise 1
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Gun Control


8 - 64
Historically, the percentage of U.S. residents who
support stricter gun control laws has been 52%. A
recent Gallup Poll of 1011 people showed 495 in
favor of stricter gun control laws . Assume the poll
was given to a random sample of people.
QUESTION Test the claim that the proportion of
those favoring stricter gun control has changed from
0.52. Perform a hypothesis test, using a significance
level of 0.05.
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Hypothesize

Test the claim that the proportion of those
favoring stricter gun control has changed
from 0.52.
 H0:
The population proportion that supports gun
control is 0.52, p = 0.52
 Ha: p ≠ 0.52
8 - 65
Copyright © 2014 Pearson Education, Inc. All rights reserved
Prepare
Choose the one-proportion z-test.
 Random sample: yes
 Sample size:

= 1011(0.52) ≈ 526 ≥ 10
 n(1 - p0) = 1011(0.48) ≈ 526 ≥ 10
 np0

8 - 66
Population size is more than 10 times 1011.
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Compute and Compare
8 - 67
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Interpret

Reject H0 (if the p-value is 0.05 or less) or do not
reject H0 and choose one of the following
conclusions:
1.
2.

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The percentage is not significantly different from 52%.
(A significant difference is one for which the p-value is
less than or equal to 0.05.)
The percentage is significantly different from 52%.
Fail to reject H0. There is insufficient evidence to
make a conclusion about the percentages being
different.
Copyright © 2014 Pearson Education, Inc. All rights reserved
Chapter 8
Guided Exercise 2
Copyright © 2014 Pearson Education, Inc. All rights reserved

8 - 70
Perform a hypothesis test to test whether
those who take Vioxx have a greater rate of
heart attack than those who take a placebo.
Use a level of significance of 0.05. Can we
conclude that Vioxx causes an increased risk
of heart attack?
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The Study

8 - 71
In the fall of 2004, drug manufacturer Merck
Pharmaceutical withdrew Vioxx, a drug that had
been used for arthritis pain, from the market after a
study revealed that its use was associated with an
increase in the risk of heart attack. The experiment
was placebo-controlled, randomized, and doubleblind. Out of 1287 people taking Vioxx there were
45 heart attacks, and out of 1299 people taking the
placebo there were 25 heart attacks (Source: Los
Angeles Times , October 23, 2004).
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Sample Proportions



8 - 72
Find the proportion of people in the sample
taking Vioxx who had a heart attack and the
proportion of people in the sample taking a
placebo who had a heart attack. Compare
these proportions.
45
pˆ1 
 0.035
1287
25
pˆ 2 
 0.019
1299
There is a clear practical significance with
3.5% much higher than 1.9%.
Copyright © 2014 Pearson Education, Inc. All rights reserved
Hypotheses

Let pV be the population proportion of those
taking Vioxx who had a heart attack, and let
pp be the population proportion of those
taking the placebo who had a heart attack.
 H0:
p V = pp
 Ha: pV > pp
8 - 73
Copyright © 2014 Pearson Education, Inc. All rights reserved
Prepare

Choose the two-proportion z -test. Although we
don’t have a random sample, we have random
assignment to groups. The pooled proportion of
heart attacks is
45  25
70
pˆ 
1287  1299
n1 pˆ  1287  0.027  34.7  10
n1 1  pˆ   1287  0.973  10
n2 pˆ  1299  0.027  10
n2 1  pˆ   1299  0.973  10
8 - 74
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
2586
 0.027
Compute and Compare
z ≈ 2.46
 p-value = 0.0069

8 - 75
Copyright © 2014 Pearson Education, Inc. All rights reserved
Interpret

Reject or do not reject H0 and choose i or ii:




8 - 76
Those taking Vioxx did not have a significantly different
rate of heart attack than those taking the placebo.
Those who took Vioxx had a significantly higher heart
attack rate than those who took the placebo.
Since p-value = 0.0069, we reject H0 and accept Ha.
There is statistically significant evidence to support
the claim that Vioxx users have a higher heart attack
rate compared to non-Vioxx users.
Note that this does not mean that Vioxx causes heart
attacks.
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