Transcript Chapter Hypothesis Tests Regarding a Parameter Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc.

Chapter
10
Hypothesis
Tests Regarding
a Parameter
Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc.
Section 10.2
Hypothesis
Tests for a
Population
Proportion
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Objectives
1. Explain the logic of hypothesis testing
2. Test the hypotheses about a population
proportion
3. Test hypotheses about a population
proportion using the binomial probability
distribution.
10-3
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Objective
• Hypothesis Testing Using the P-Value
Approach
If the probability of getting a sample proportion
as extreme or more extreme than the one
obtained is small under the assumption the
statement in the null hypothesis is true, reject the
null hypothesis
10-4
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A researcher obtains a random sample of 1000 people
and finds that 534 are in favor of the banning cell
phone use while driving, so p = 534/1000. Does this
suggest that more than 50% (p > 0.5)of people favor
the policy? Or is it possible that the true proportion of
registered voters who favor the policy is some
proportion less than 0.5 and we just happened to
survey a majority in favor of the policy? In other
words, would it be unusual to obtain a sample
proportion of 0.534 or higher(p>0.534) from a
population whose proportion is 0.5? What is
convincing, or statistically significant, evidence?
^
10-5
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The Logic of the P-Value Approach
A second criterion we may use for testing
hypotheses is to determine how likely it is to obtain
a sample proportion of 0.534 or higher from a
population whose proportion is 0.5. If a sample
proportion of 0.534 or higher is unlikely (or
unusual), we have evidence against the statement in
the null hypothesis. Otherwise, we do not have
sufficient evidence against the statement in the null
hypothesis.
10-6
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We can compute the probability of obtaining a
sample proportion of 0.534 or higher from a
population whose proportion is 0.5 using the
normal model.
10-7
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z
Recall
0.534  0.5


 2.15
0.5 1 0.5
(test statistics)
1000
So, we compute (right tailed test)


P pö  0.534  P(z  2.15)  0.0158
The value 0.0158 is called the P-value, which means about
2 samples in 100 will give a sample proportion as high
or higher than the one we obtained if the population
proportion really is 0.5. Because these results are
unusual, we take this as evidence against the statement
in the null hypothesis.
10-8
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Hypothesis Testing
Using the P-value Approach
If the probability of getting a sample proportion
as extreme or more extreme than the one
obtained is small under the assumption the
statement in the null hypothesis is true, reject the
null hypothesis.
10-9
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Objective 2
• Test hypotheses about a population proportion.
10-10
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Recall:
•
The best point estimate of p, the proportion
of the population with a certain
characteristic, is given by
x
pˆ 
n
where x is the number of individuals in the
sample with the specified characteristic and n
is the sample size.

10-11
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Recall:
•
The sampling distribution of pˆ is approximately
normal, with mean  pˆ  p and standard deviation
 pˆ 
p(1 p)
n

provided 
that the following requirements are
satisfied:
1. The
sample is a simple random sample.
2. np(1-p) ≥ 10.
3. The sampled values are independent of each
other.
10-12
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Testing Hypotheses Regarding a
Population Proportion, p
To test hypotheses regarding the population
proportion, we can use the steps that follow,
provided that:
•The sample is obtained by simple random
sampling.
• np0(1 – p0) ≥ 10.
•The sampled values are independent of each
other.
10-13
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Step 1: Determine the null and alternative
hypotheses. The hypotheses can be
structured in one of three ways:
10-14
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Step 2: Select a level of significance, α, based
on the seriousness of making a
Type I error.
10-15
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P-Value Approach
By Hand Step 3: Compute the test statistic.
z0 
pˆ  p0
p0 (1 p0 )
n

10-16
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P-Value Approach
Use Table V to estimate the P-value.
Two-Tailed
10-17
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P-Value Approach
Left-Tailed
10-18
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P-Value Approach
Right-Tailed
10-19
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P-Value Approach
Technology Step 3: Use a statistical spreadsheet
or calculator with statistical capabilities to
obtain the P-value.
The directions for obtaining the P-value
using the TI-83/84 Plus graphing calculator,
MINITAB, Excel, and StatCrunch are in the
Technology Step-by-Step in the text.
10-20
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P-Value Approach
Step 4: If the P-value < α, reject the null
hypothesis.
10-21
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Step 5: State the conclusion.
10-22
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Parallel Example 1: Testing a Hypothesis about a
Population Proportion: Large Sample Size
In 1997, 46% of Americans said they did not trust the
media “when it comes to reporting the news fully,
accurately and fairly”. In a 2007 poll of 1010 adults
nationwide, 525 stated they did not trust the media. At
the α = 0.05 level of significance, is there evidence to
support the claim that the percentage of Americans that
do not trust the media to report fully and accurately has
increased since 1997?
Source: Gallup Poll
10-23
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Solution
We want to know if p > 0.46. First, we must verify the
requirements to perform the hypothesis test:
1. This is a simple random sample.
2. np0(1 – p0) = 1010(0.46)(1 – 0.46) = 250.8 > 10
3. Since the sample size is less than 5% of the
population size, the assumption of independence is
met.
10-24
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Solution
Step 1: H0: p = 0.46
versus
H1: p > 0.46
Step 2: The level of significance is α = 0.05.
525
Step 3: The sample proportion is pˆ 
 0.52.
1010
The test statistic is then
z0 
10-25
0.52  0.46

 3.83
0.46(1 0.46)
1010
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Solution: P-Value Approach
Step 4: Since this is a right-tailed test, the Pvalue is the area under the standard
normal distribution to the right of the
test statistic z0=3.83. That is, P-value =
P(Z > 3.83) ≈ 0.
Step 5: Since the P-value is less than the level of
significance, we reject the null
hypothesis.
10-26
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Solution
Step 6: There is sufficient evidence at the
α = 0.05 level of significance to
conclude that the percentage of Americans that
do not trust the media to report fully and
accurately has increased since 1997.
10-27
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Problem
Based on information form the National Cyber Security
Alliance, 93% of computer owners believe that they have
antivirus programs installed on their computers. In a
random sample of 400 scanned computers, it is found that
380 of them actually have antivirus programs. Use the
sample data from the scanned computers to test the claim
that 93% of computers have antivirus programs.
10-28
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Solution- P-value approach
Meets the requirements….
Step 1: H0: p = 0.93
H1: p  0.93
versus
Step 2: The level of significance is α = 0.05.
Step 3: The Test Statistics
^
z0 
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p p
p 1  p 
n

^
p
380
 0.95
400
0.95  0.93
0.93  0.07 
400
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 1.57
Solution: P-Value Approach
Step 4: Since this is a two-tailed test, the Pvalue is twice the area to the right of the test statistic
z0=1.57. That is, P-value = 2*P(Z > 1.57) =2*(10.9418)=2*0.0582=0.1164.
Step 5: Since the P-value is greater than the level of
significance, we fail to reject the null
hypothesis.
10-30
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Step 6: There is not sufficient evidence at the
α = 0.05 level of significance to reject
the claim that 93% of the computers have
antivirus programs
10-31
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Objective 3
• Test hypotheses about a population proportion
using the binomial probability distribution.
10-32
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For the sampling distribution of pˆ to be
approximately normal, we require np(1– p) be at
least 10. What if this requirement is not met?
We stated that an event was unusual if the
 the event was less than
probability of observing
0.05. This criterion is based on the P-value
approach to testing hypotheses; the probability
that we computed was the P-value. We use this
same approach to test hypotheses regarding a
population proportion for small samples.
10-33
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Parallel Example 4: Hypothesis Test for a Population
Proportion: Small Sample Size
In 2006, 10.5% of all live births in the United
States were to mothers under 20 years of age.
A sociologist claims that births to mothers
under 20 years of age is decreasing. She
conducts a simple random sample of 34 births
and finds that 3 of them were to mothers under
20 years of age. Test the sociologist’s claim at
the α = 0.01 level of significance.
10-34
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Parallel Example 4: Hypothesis Test for a Population
Proportion: Small Sample Size
Approach:
Step 1: Determine the null and alternative hypotheses
Step 2: Check whether np0(1–p0) is greater than or
equal to 10, where p0 is the proportion stated in
the null hypothesis. If it is, then the sampling
distribution of pö is approximately normal and
we can use the steps for a large sample size.
Otherwise we use the following Steps 3 and 4.
10-35
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Parallel Example 4: Hypothesis Test for a Population
Proportion: Small Sample Size
Approach:
Step 3: Compute the P-value. For right-tailed tests, the
P-value is the probability of obtaining x or
more successes. For left-tailed tests, the Pvalue is the probability of obtaining x or fewer
successes. The P-value is always computed
with the proportion given in the null hypothesis.
Step 4: If the P-value is less than the level of
significance, α, we reject the null hypothesis.
10-36
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Solution
Step 1: H0: p = 0.105 versus H1: p < 0.105
Step 2: From the null hypothesis, we have
p0 = 0.105. There were 34 mothers
sampled, so np0(1– p0)=3.57 < 10. Thus,
the sampling distribution of pˆ is not
approximately normal.

10-37
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Solution
Step 3: Let X represent the number of live births in
the United States to mothers under 20 years
of age. We have x = 3 successes in n = 34
trials so pˆ = 3/34= 0.088. We want to
determine whether this result is unusual if
the population mean is truly 0.105. Thus,

P-value = P(X ≤ 3 assuming p=0.105 )
= P(X = 0) + P(X =1)
+ P(X =2) + P(X = 3)
= 0.51
10-38
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Solution
Step 4: The P-value = 0.51 is greater than the
level of significance so we do not reject
H0. There is insufficient evidence to
conclude that the percentage of live
births in the United States to
mothers
under the age of 20 has decreased
below
the 2006 level of 10.5%.
10-39
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