Topics for Today
Hypothesis involving Proportions
Stat203
Fall 2011 – Week 9 Lecture 2
Page 1 of 14

Proportions?
Yes, proportions.
As the textbook says:
“… the logic of testing the difference between two proportions is the same as for testing the difference between means. The only change is in the symbols and formulas …”
That’s pretty much the story! Let’s review the new symbols and formulas.
Stat203
Fall 2011 – Week 9 Lecture 2
Page 2 of 14

Symbols for
Tests of Proportions
Parameters
Statistics
1-sample
Hypotheses
1-sample Test
Statistics
2-sample
Hypotheses
2-sample test statistics
Testing Means Testing
Proportions
𝑋̅
µ
, s
𝐻
0
: 𝜇 = 𝜇
0
vs 𝐻 𝑎
: 𝜇 > 𝜇
0
𝐻
0
: 𝜇 = 𝜇
0
vs 𝐻 𝑎
: 𝜇 < 𝜇
0
𝐻
0
: 𝜇 = 𝜇
0
vs 𝐻 𝑎
: 𝜇 ≠ 𝜇
0 𝑝 𝑝̂
𝐻
0
: 𝑝 = 𝑝
0
vs 𝐻 𝑎
: 𝑝 > 𝑝
0
𝐻
0
: 𝑝 = 𝑝
0
vs 𝐻 𝑎
: 𝑝 < 𝑝
0
𝐻
0
: 𝑝 = 𝑝
0
vs 𝐻 𝑎
: 𝑝 ≠ 𝑝
0 𝑡 𝑛−1
=
𝑋 − 𝜇
0
= 𝑠
𝑋
𝑋 − 𝜇
0 𝑠
⁄
√𝑛
𝐻
0
: 𝜇
1
𝐻
0
: 𝜇
1
𝐻
0
: 𝜇
1
= 𝜇
2
vs 𝐻 𝑎
: 𝜇
1
> 𝜇
2
= 𝜇
= 𝜇
2
2
vs 𝐻
vs 𝐻 𝑎 𝑎
: 𝜇
: 𝜇
1
1
< 𝜇
≠ 𝜇
2
2 𝑡 𝑛−1
=
𝑋
1 𝑠
− 𝑋
𝐷
2 and you define s
D
depending on whether you have equal variances 𝑧 = 𝑝̂ − 𝑝
0
√ 𝑝
0
(1−𝑝
0
) 𝑛
𝐻
0
: 𝑝
1
𝐻
0
: 𝑝
1
𝐻
0
: 𝑝
1
= 𝑝
2
vs 𝐻 𝑎
: 𝑝
1
= 𝑝
2
vs 𝐻 𝑎
: 𝑝
1
= 𝑝
2
vs 𝐻 𝑎
: 𝑝
1 𝑧 = 𝑝̂
1
− 𝑝̂
2
√𝑝 ∗ (1 − 𝑝 ∗
1
) ( 𝑛
1
> 𝑝
2
< 𝑝
2
≠ 𝑝
2
+
1 𝑛
2
) where 𝑝 ∗ = 𝑛
1 𝑝̂
1
+𝑛
2 𝑝̂
2
is the ‘pooled’ proportion. 𝑛
1
+𝑛
2
Stat203
Fall 2011 – Week 9 Lecture 2
Page 3 of 14

Almost.
So … it’s just a new formula?
But, notice one thing about the test statistics for proportions.
They’re z’s!
So, instead of using the t-table or the computer to look up p-values, we can go back to the z-table to identify our p-values just like we did with all the normal distribution questions from way back when.
But … certain conditions must be met in order to use these tests.
Page 4 of 14 Stat203
Fall 2011 – Week 9 Lecture 2

Conditions for Hypothesis Tests on Proportions
For 1-sample hypotheses, you can use the zstatistic if both of the following are true:
𝑛𝑝
0
> 10
𝑛(1 − 𝑝
0
) > 10
For 2-sample hypotheses, you can use the zstatistic if all of the following are true:
𝑛
1
> 10
𝑛
1 𝑝
1
and
> 5 𝑛
and
2 𝑛
2
> 10 𝑝
2
> 5
Page 5 of 14 Stat203
Fall 2011 – Week 9 Lecture 2

Example: 1-sample test of Proportions
Let’s choose one of the following 3 research hypotheses to test on this class:
The proportion of students in this class that …
1.
… are men is more than 50%.
2.
… own a car is less than 30%.
3.
… have a roommate (including family) is more than 60%.
Now, we’ll do our usual process:
Stat203
Fall 2011 – Week 9 Lecture 2
Page 6 of 14

Research Hypothesis:
Individual:
Population:
Variable:
Parameter:
Statistical Hypotheses:
Confirm Conditions for Test:
Stat203
Fall 2011 – Week 9 Lecture 2
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Test Statistic: p-value:
Reject the Null Hypothesis at

= 0.05?
Conclusion:
Stat203
Fall 2011 – Week 9 Lecture 2
Page 8 of 14

Example (one-sample test for proportions):
An experiment on the side effects of pain relievers assigned arthritis patients to take on of several pain medications. Of the 440 patients who took one particular brand, 23 suffered some ‘adverse symptom’.
Can the drug company claim that fewer than
10% of people will experience an adverse symptom?
Research Hypothesis:
Individual:
Population:
Variable:
Page 9 of 14 Stat203
Fall 2011 – Week 9 Lecture 2

Statistical Hypotheses:
Confirm Conditions for Test:
Test Statistic: p-value:
Reject Null Hypothesis at

= 0.05?
Conclusion:
Stat203
Fall 2011 – Week 9 Lecture 2
Page 10 of 14

Example (2-sample test of proportions, Q36 p262): A pollster interviews, by telephone, a statewide random sample of persons aged 18 and over about their attitudes toward stricter gun control. Using the following set of results, test the significance of the difference between the proportions of men and women who support stricter controls.
Favor
Males Females
92 120
Oppose 74 85
Research Hypothesis:
Individuals:
Populations:
Variables:
Stat203
Fall 2011 – Week 9 Lecture 2
Page 11 of 14

Parameters:
Statistical Hypotheses:
Confirm Conditions for Testing:
Test Statistic:
Stat203
Fall 2011 – Week 9 Lecture 2
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p-value:
Reject Null Hypothesis?
Conclusion:
Stat203
Fall 2011 – Week 9 Lecture 2
Page 13 of 14

Today’s Topics
Tests for Proportions
identical process as with t-tests
must confirm conditions in order to use zscore
New Reading
Chapter 9 up to pg 348
Stat203
Fall 2011 – Week 9 Lecture 2
Page 14 of 14