Section 10.1b Comparing Two Proportions-Significance …€¦ · · 2018-03-15Section 10.1b...

Section 10.1b Comparing Two Proportions-Significance Tests

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Significance Tests for 1 2p p

The National Sleep Foundation asked a random sample of 1010 US adults questions about their sleep habits.

The sample was selected in the fall of 2010 from random telephone numbers, stratified by region and sex, and

an equal number of men and women were interviewed.

One of the questions asked was about snoring. Of the 995 respondents, 37% of adults reported that they snored

at least a couple nights a week during the past year. Would you expect that percentage to be the same for all

age groups? When split into two age categories, 26% of the 184 people under 30 snored, compared with 39%

of the 811 in the over 30 group. Is this difference of 13% real, or simply due to natural variation in the

samples?

To answer this question we need to conduct a hypothesis test for the difference of two proportions.

1. a. State the inference test to be conducted including the null and alternative hypothesis both in symbols and

words.

In order to conduct this test we need to understand the sampling distribution for the difference of two

proportions 1 2ˆ ˆp p . But we did this already in section 10.1a when we constructed confidence intervals. In

that investigation we found that the standard error for the difference of the two proportions 1 2ˆ ˆp p was

1 1 2 2

1 2

ˆ ˆ ˆ ˆ1 1p p p pSE

n n

. This formula will be different for a hypothesis test. Recall that in a

hypothesis test we assume that the null hypothesis 1 2 0p p is true. This implies that the two population

proportions are equal. Therefore, assuming random samples from each population we can also assume that

1 2ˆ ˆp p . As a result, there should be a single value for p̂ that we could use in the formula for SE. The

question is, how do we find this single p̂ for our snoring problem? We will determine this in part b below.

b. How many snorers are in the under 30 group?

http://www.sleepfoundation.org/


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How many where in the over 30 group?

What is the combined proportion of snorers in both groups?

We call this combined proportion from part b the pooled sample proportion and it’s the value we will use for

calculating the standard error SE. In general, the pooled proportion which we designate ˆCp . This combined

proportion is found using the following formula.

1 2

1 2

ˆC

Success Successp

n n

Where 1Success and 2Success are the number of success in groups 1 and 2 respectively. So ˆCp is the overall

proportion of success.

The z statistic then (assuming the conditions have been met) for a two-proportion z hypotheses test is as follows:

1 2

1 2

ˆ ˆ 0

ˆ ˆ ˆ ˆ1 1C C C C

p pz

p p p p

n n

Where ˆCp is the pooled sample proportion.

What does the 0 in the test statistic formula above represent?

Fortunately the conditions for a significance test for comparing two proportions are the same as the conditions

for constructing the confidence interval.


of 14

c. Verify that the conditions have been met and perform the two proportion z test for the snoring problem.

What would you conclude?


of 14

TECHNOLOGY

As you probably expected you can conduct a two proportion z test of significance on your graphing calculator.

From the TEST screen below left select option 6: 2-PropZTest and enter the appropriate data in the screen

below right.

Verify your results in the preceding problem. Use your graphing calculator from this point on to do the

necessary calculations. As always write the appropriate recipes.

Let’s summarize this test:

Two-Sample z Test for the Difference between Two Proportions

Suppose the Random, Normal, and Independent conditions are met. To test the hypothesis H0: p1 − p2 = 0, first

find the pooled proportion ˆCp of successes in both samples combined. Then compute the z statistic

1 2

1 2

ˆ ˆ 0


p pz

p p p p

n n

Find the P-value by calculating the probability of getting a z statistic this large or larger in the direction

specified by the alternative hypothesis Ha:


of 14

Conditions: Use this test when

Random The data are produced by a random sample of size n1 from Population 1 and a random sample

of size n2 from Population 2 or by two groups of size n1 and n2 in a randomized experiment.

Normal 1 1p̂ n , 1 1

ˆ1 p n , 2 2p̂ n , 2 2

ˆ1 p n are all at least 10.

Independent Both the samples or groups themselves and the individual observations in each sample or

group are independent. When sampling without replacement, check that the two populations are at least

10 times as large as the corresponding samples (the 10% condition).

2. A Vermont study published in December 2001 by the American Academy of

Pediatrics examined parental influence on teenagers decisions to smoke. A group

of students who had never smoked were questioned about their parent’s attitudes

toward smoking. These students were questioned again two years later to see if

they had started smoking. The researchers found that among the 284 students who

indicated that their parents disapproved of teenagers smoking, 54 had become

established smokers. Among the 41 students who initially said their parents were

lenient about smoking, 11 became established smokers. Do these data provide

strong evidence that parental attitude influences teenager’s decision about

smoking?

a. Is this an experiment or an observational study? If it is an experiment, diagram the experiment. If it is an

observational study is it a retrospective study or a prospective study?


of 14

b. Conduct an appropriate test of significance.


of 14

3. Consider again the Vermont study in question 2. Construct and interpret a 95% confidence interval for the

difference in proportion of children who may smoke and have approving parents and those who may smoke and

have disapproving parents. You do not need to recheck that the conditions have been met.


of 14

Significance Tests for 1 2p p

The National Sleep Foundation asked a random sample of 1010 US adults questions about their sleep habits.

The sample was selected in the fall of 2010 from random telephone numbers, stratified by region and sex, and

an equal number of men and women were interviewed.

One of the questions asked was about snoring. Of the 995 respondents, 37% of adults reported that they snored

at least a couple nights a week during the past year. Would you expect that percentage to be the same for all

age groups? When split into two age categories, 26% of the 184 people under 30 snored, compared with 39%

of the 811 in the over 30 group. Is this difference of 13% real, or simply due to natural variation in the

samples?

To answer this question we need to conduct a hypothesis test for the difference of two proportions.

1. a. State the inference test to be conducted including the null and alternative hypothesis both in symbols and

words.

In order to conduct this test we need to understand the sampling distribution for the difference of two

proportions 1 2ˆ ˆp p . But we did this already in section 10.1a when we constructed confidence intervals. In

that investigation we found that the standard error for the difference of the two proportions 1 2ˆ ˆp p was

1 1 2 2

1 2

ˆ ˆ ˆ ˆ1 1p p p pSE

n n

. This formula will be different for a hypothesis test. Recall that in a

hypothesis test we assume that the null hypothesis 1 2 0p p is true. This implies that the two population

proportions are equal. Therefore, assuming random samples from each population we can also assume that

1 2ˆ ˆp p . As a result, there should be a single value for p̂ that we could use in the formula for SE. The

question is, how do we find this single p̂ for our snoring problem? We will determine this in part b below.

http://www.sleepfoundation.org/


of 14

b. How many snorers are in the under 30 group?

How many where in the over 30 group?

What is the combined proportion of snorers in both groups?

We call this combined proportion from part b the pooled sample proportion and it’s the value we will use for

calculating the standard error SE. In general, the pooled proportion which we designate ˆCp . This combined

proportion is found using the following formula.

1 2

1 2

ˆC

Success Successp

n n

Where 1Success and 2Success are the number of success in groups 1 and 2 respectively. So ˆCp is the overall

proportion of success.

The z statistic then (assuming the conditions have been met) for a two-proportion z hypotheses test is as follows:

1 2

1 2

ˆ ˆ 0


p pz

p p p p

n n

Where ˆCp is the pooled sample proportion.

What does the 0 in the test statistic formula above represent?

Fortunately the conditions for a significance test for comparing two proportions are the same as the conditions

for constructing the confidence interval.


of 14

c. Verify that the conditions have been met and perform the two proportion z test for the snoring problem.

What would you conclude?


of 14

TECHNOLOGY

As you probably expected you can conduct a two proportion z test of significance on your graphing calculator.

From the TEST screen below left select option 6: 2-PropZTest and enter the appropriate data in the screen

below right.

Verify your results in the preceding problem. Use your graphing calculator from this point on to do the

necessary calculations. As always write the appropriate recipes.

Let’s summarize this test:

Two-Sample z Test for the Difference between Two Proportions

Suppose the Random, Normal, and Independent conditions are met. To test the hypothesis H0: p1 − p2 = 0, first

find the pooled proportion ˆCp of successes in both samples combined. Then compute the z statistic

1 2

1 2

ˆ ˆ 0


p pz

p p p p

n n

Find the P-value by calculating the probability of getting a z statistic this large or larger in the direction

specified by the alternative hypothesis Ha:


of 14

Conditions: Use this test when

Random The data are produced by a random sample of size n1 from Population 1 and a random sample

of size n2 from Population 2 or by two groups of size n1 and n2 in a randomized experiment.

Normal 1 1p̂ n , 1 1

ˆ1 p n , 2 2p̂ n , 2 2

ˆ1 p n are all at least 10.

Independent Both the samples or groups themselves and the individual observations in each sample or

group are independent. When sampling without replacement, check that the two populations are at least

10 times as large as the corresponding samples (the 10% condition).

2. A Vermont study published in December 2001 by the American Academy of

Pediatrics examined parental influence on teenagers decisions to smoke. A group

of students who had never smoked were questioned about their parent’s attitudes

toward smoking. These students were questioned again two years later to see if

they had started smoking. The researchers found that among the 284 students who

indicated that their parents disapproved of teenagers smoking, 54 had become

established smokers. Among the 41 students who initially said their parents were

lenient about smoking, 11 became established smokers. Do these data provide

strong evidence that parental attitude influences teenager’s decision about

smoking?

a. Is this an experiment or an observational study? If it is an experiment, diagram the experiment. If it is an

observational study is it a retrospective study or a prospective study?


of 14

b. Conduct an appropriate test of significance.


of 14

3. Consider again the Vermont study in question 2. Construct and interpret a 95% confidence interval for the

difference in proportion of children who may smoke and have approving parents and those who may smoke and

have disapproving parents. You do not need to recheck that the conditions have been met.

Section 10.1b Comparing Two Proportions-Significance …€¦ · · 2018-03-15Section 10.1b...

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