Transcript Chapter7 4

Section 7.4
Hypothesis Testing for Proportions
Larson/Farber 4th ed.
Section 7.4 Objectives
• Use the z-test to test a population proportion p
Larson/Farber 4th ed.
z-Test for a Population Proportion
z-Test for a Population Proportion
• A statistical test for a population proportion.
• Can be used when a binomial distribution is given
such that np  5 and nq  5.
• The test statistic is the sample proportion .
• The standardized test statistic is z.
Larson/Farber 4th ed.
Using a z-Test for a Proportion p
Verify that np ≥ 5 and nq ≥ 5
In Words
1. State the claim mathematically
and verbally. Identify the null and
alternative hypotheses.
2. Specify the level of significance.
In Symbols
State H0 and Ha.
Identify  .
3. Sketch the sampling distribution.
4. Determine any critical value(s).
Larson/Farber 4th ed.
Use Table 5 in
Appendix B.
Using a z-Test for a Proportion p
In Words
In Symbols
5. Determine any rejection
region(s).
6. Find the standardized test
statistic.
7. Make a decision to reject or
fail to reject the null
hypothesis.
8. Interpret the decision in the
context of the original claim.
Larson/Farber 4th ed.
If z is in the rejection
region, reject H0.
Otherwise, fail to
reject H0.
Example: Hypothesis Test for
Proportions
A research center estimates that no more than 40% of
US adults eat breakfast every day. IN a random sample
of 250 US adults, 41.6% say they eat breakfast every
day. At α = 0.01, is there enough evidence to reject the
researcher’s claim?
Solution:
•Verify that np ≥ 5 and nq ≥ 5.
p = 0.40 and q = 0.60
np = 250(0.40) = 100 and nq = 250(0.60) = 150
Larson/Farber 4th ed.
Solution: Hypothesis Test for
Proportions
•
•
•
•
H0: p ≤ 0.40 (claim)
Ha: p > 0.40
 = 0.01
Rejection Region:
0.01
0
2.33
0.516
Larson/Farber 4th ed.
z
• Test Statistic
• Decision: Fail to reject H0
At the 1% level of significance,
there is insufficient evidence to
reject the researcher’s claim that
no more than 40% of US adults
eat breakfast every day.
Example: Hypothesis Test for
Proportions
7.4, #12 - An environmentalist clams that more than
60% of british consumers are concerned about the use
of genetic modification in food production and want to
avoid genetically modified foods. You want to test this
claim. You find that a random sample of 100
consumers, 65% say they are concerned about the use
of genetically modified foods. At  

Solution:
•Verify that np ≥ 5 and nq ≥ 5.
np = 100(0.60) = 60 and nq = 100 (0.40) = 40
Larson/Farber 4th ed.
Solution: Hypothesis Test for
Proportions
•
•
•
•
H0: p ≤ 0.60
Ha: p > 0.60
 =
0.10
Rejection Region:
• Test Statistic
0.10
0 1.28
1.02
Larson/Farber 4th ed.
z
• Decision: Fail to Reject H0
At the 10% level of
significance, there is not enough
evidence to support the
environmentalist’s claim.
Section 7.4 Summary
• Used the z-test to test a population proportion p
• HW: 5 - 15 EO
Larson/Farber 4th ed.
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