section 10.1a comparing two proportions€¦ · · 2018-03-1512 2 v ppˆˆ first then ... section...

Section 10.1a Comparing Two Proportions

of 20

Often in statistics we want to compare parameters from two different populations. For example we might want

to compare the opinions of men and women on abortion and know if the difference is significant. Comparing

parameters from two or more populations is an important topic in statistics and in this investigation we will look

at comparing the difference between two proportions from two different populations. As you might have

guessed we will need to investigate the sampling distribution for the difference between two proportions.

The sampling Distribution of a Difference between Two Proportions

To explore the sampling distribution of 1 2

ˆ ˆp p , let’s start with two populations having a known proportion of

successes. Suppose that there are two large high schools, each with over 2000 students, in a certain town.

1. At School 1, 70% of students did their homework last night. Only

50% of the students at School 2 did their homework last night. The

counselor at School 1 takes a SRS of 100 students and records the

sample proportion 1p̂ that did the homework. School 2’s counselor

takes a SRS of 200 students and records the sample proportion 2p̂ that

did the homework.

We want to look at the sampling distribution for the difference 1 2ˆ ˆp p

in the sample proportions.

Notice that we can consider 1p̂ and 2p̂ as random variables since the values will vary from random sample to

random sample.

a. What is the mean of the sampling distribution for 1 2ˆ ˆp p ? How do we know?

1 2ˆ ˆp p

b. What is the standard deviation of the sampling distribution for 1 2ˆ ˆp p ? Recall that variances for

independent random variables add, but standard deviations do not. So find the variance 1 2

2

ˆ ˆp p first then

determine the standard deviation.

1 2

2

ˆ ˆp p

1 2ˆ ˆp p

What must be true about the random variables in order to be able to add their variances?


of 20

c. We now have the mean and the standard deviation for the sampling distribution of 1 2

ˆ ˆp p . What about the

shape of the distribution?

The Fathom file sampdifferenceofprop shown below shows a partial sampling distribution (500 sample p̂ s

from samples of size n) for two different p̂ values from two different populations. The graph at the upper left is

a portion of the sampling distribution for 1p̂ and the graph at the upper right is a portion of the sampling

distribution for 2p̂ . The graph below these two graphs is a portion of the sampling distribution of the difference

1 2ˆ ˆp p .

The slider bars allow you to change the sample sizes n and the population proportion p for each sampling

distribution. Change the values of the proportions and the sample sizes to the values given in our school

problem for School 1 and School 2 and notice the shapes of the distributions. In particular what do you think

you can conclude about the shape of the distribution for the difference in the sample proportions?

Experiment with other values and observe the shape of the sampling distribution for the difference of these

proportions. What can you conclude?

What is the test to determine if the distribution for a sample proportions is normal?


of 20

If two random variables are approximately normally distributed, is the difference between them also normally

distributed?

Considering the two statements above, determine criteria for determining if the sampling distribution of

1 2ˆ ˆp p is approximately normal.

d. Assuming the criteria in part c are satisfied, sketch a graph of the sampling distribution of 1 2ˆ ˆp p .


of 20

Here is a summary of the facts for the sampling distribution of 1 2

ˆ ˆp p :

Choose an SRS of size n1 from Population 1 with proportion of successes p1 and an independent SRS of size n2

from Population 2 with proportion of successes p2.

Shape: When n1p1, n1(1 − p1), n2p2, and n2(1 − p2) are all at least 10, the sampling distribution of

1 2ˆ ˆp p is approximately Normal.

Center: The mean of the sampling distribution is p1 − p2. That is, the difference in sample proportions is

an unbiased estimator of the difference in population proportions.

Spread: The standard deviation of the sampling distribution 1 2

ˆ ˆp p is

1 1 2 2

1 2

1 1p p p p

n n

as long as each sample is no more than 10% of its population (the 10% condition) and we have independence of

the two samples.

The figure below displays the sampling distribution for 1 2ˆ ˆp p


of 20

e. Suppose that the two counselors from School 1 and School 2 meet to compare the proportions of students

who did their homework last night and find that 1 2

ˆ ˆp p is 0.10. Find the probably of getting this difference or

less.

f. Does the probability in part e give you a reason to doubt the counselors’ reported difference of 0.20? Explain.

CHECK YOUR UNDERSTANDING

Your teacher brings two bags of colored goldfish crackers to class. She tells you that Bag 1 has 25% red

crackers and Bag 2 has 35% red crackers. Each bag contains more than 500 crackers. Using a paper cup, your

teacher takes an SRS of 50 crackers from Bag 1 and a separate SRS of 40 crackers from Bag 2. Let 1 2ˆ ˆp p be

the difference in the sample proportions of red crackers.

a. What is the shape of the sampling distribution of 1 2ˆ ˆp p ? Why?

b. Find the mean and standard deviation of the sampling distribution. Show your work.


of 20

c. Draw a picture of the sampling distribution of 1 2

ˆ ˆp p and find the probability that 1 2

ˆ ˆp p is less than or

equal to −0.02. Show your work.

d. Based on your answer to Question 3, would you be surprised if the difference in the proportion of red

crackers in the two samples was 1 2ˆ ˆp p = −0.02? Explain.


of 20

Confidence Intervals for 1 2p p

2. Who do you think are more intelligent, men or women? To find out what

people think, the Gallup Poll selected a random sample of 520 women and 506

men. The pollsters showed them a list of personal attributes and asked them to

indicate whether each attribute was “generally more true of men than women”.

When asked about intelligence, 28% of the men thought men were generally

more intelligent, but only 13% of the women agreed. Is there a gender gap in

opinions about which sex is smarter or is the difference simply due to chance?

And if there is a gap, then what would we estimate the true size of that gap to be?

Comparing parameters from two different populations, like the difference in the

opinions of men and women on gender intelligence is a fundamental idea in

statistics and in this chapter we will explore this important idea.

We know that the difference between the two proportions (men – women) from our samples is 15%, but what is

the true difference for all men and women? Let’s construct a 95% confidence interval for this difference.

Recall that the general recipe for a confidence interval is:

statistic (critical value) (standard deviation of statistic)

Before we construct the confidence interval we should verify that we meet the conditions.

a. Verify that we meet the Random condition.

b. Independence. For this check we need to verify that the two groups are independent and that the

individuals within the groups are independent. If you are sampling without replacement you also need to check

the 10% condition for both groups. Check this condition.

c. Normality. Since the population parameter is unknown and is not assumed (as in a hypothesis test) we need

to use the statistics from both of our samples to check this condition. Check that this condition is met.


of 20

To construct the confidence interval we need the standard deviation

1 2

1 1 2 2

ˆ ˆ

1 2

1 1p p

p p p p

n n

. But

again since we do not know the population proportions 1p and 2p we will need to substitute the sample

proportions 1p̂ and 2p̂ to get what we call the standard error

1 1 2 2

1 2

ˆ ˆ ˆ ˆ1 1p p p pSE

n n

.

e. Based on what you now know, what is the formula for calculating a level C confidence interval for the

difference of two proportions?

g. Construct a 95% confidence interval for estimating the true difference in the proportions of men and women

with regard to their opinions on intelligence. Interpret your results in the context of the problem situation.


of 20

The confidence interval you constructed in 2g is called a Two-Sample z Interval for a Difference of Two

Proportions. The summary is as follows:

When the Random, Normal, and Independent conditions are met, a level C confidence interval for 1 2

ˆ ˆp p is

1 1 2 2

1 2

1 2

ˆ ˆ ˆ ˆ1 1ˆ ˆ *

p p p pp p z

n n

where z* is the critical value for the standard Normal curve with area C between −z* and z*.

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 1ˆ1 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).

TECHNOLOGY

Your calculator can do the calculations for you when performing inference on the difference of two proportions.

Select STAT then TESTS. For a confidence interval on the difference in proportions select option

B: 2-PropZInt… as shown in the screen below.

Verify your confidence interval from question 2g by entering the appropriate values in screen shown below and

having the calculator find the interval. Note that x1 and x2 must be whole numbers, so you may have to round

accordingly in some problems.


of 20


As part of the Pew Internet and American Life Project, researchers

conducted two surveys in late 2009. The first survey asked a random

sample of 800 U.S. teens about their use of social media and the

Internet. A second survey posed similar questions to a random

sample of 2253 U.S. adults. In these two studies, 73% of teens and

47% of adults said that they use social-networking sites. Use these

results to construct and interpret a 95% confidence interval for the

difference between the proportion of all U.S. teens and adults who

use social-networking sites. Follow all the appropriate steps for

construction a confidence interval.


of 20

Often in statistics we want to compare parameters from two different populations. For example we might want

to compare the opinions of men and women on abortion and know if the difference is significant. Comparing

parameters from two or more populations is an important topic in statistics and in this investigation we will look

at comparing the difference between two proportions from two different populations. As you might have

guessed we will need to investigate the sampling distribution for the difference between two proportions.

The sampling Distribution of a Difference between Two Proportions

To explore the sampling distribution of 1 2

ˆ ˆp p , let’s start with two populations having a known proportion of

successes. Suppose that there are two large high schools, each with over 2000 students, in a certain town.

1. At School 1, 70% of students did their homework last night. Only

50% of the students at School 2 did their homework last night. The

counselor at School 1 takes a SRS of 100 students and records the

sample proportion 1p̂ that did the homework. School 2’s counselor

takes a SRS of 200 students and records the sample proportion 2p̂ that

did the homework.

We want to look at the sampling distribution for the difference 1 2ˆ ˆp p

in the sample proportions.

Notice that we can consider 1p̂ and 2p̂ as random variables since the values will vary from random sample to

random sample.

a. What is the mean of the sampling distribution for 1 2ˆ ˆp p ? How do we know?

1 2ˆ ˆp p

b. What is the standard deviation of the sampling distribution for 1 2ˆ ˆp p ? Recall that variances for

independent random variables add, but standard deviations do not. So find the variance 1 2

2

ˆ ˆp p first then

determine the standard deviation.

1 2

2

ˆ ˆp p

1 2ˆ ˆp p

What must be true about the random variables in order to be able to add their variances?


of 20

c. We now have the mean and the standard deviation for the sampling distribution of 1 2

ˆ ˆp p . What about the

shape of the distribution?

The Fathom file sampdifferenceofprop shown below shows a partial sampling distribution (500 sample p̂ s

from samples of size n) for two different p̂ values from two different populations. The graph at the upper left is

a portion of the sampling distribution for 1p̂ and the graph at the upper right is a portion of the sampling

distribution for 2p̂ . The graph below these two graphs is a portion of the sampling distribution of the difference

1 2ˆ ˆp p .

The slider bars allow you to change the sample sizes n and the population proportion p for each sampling

distribution. Change the values of the proportions and the sample sizes to the values given in our school

problem for School 1 and School 2 and notice the shapes of the distributions. In particular what do you think

you can conclude about the shape of the distribution for the difference in the sample proportions?

Experiment with other values and observe the shape of the sampling distribution for the difference of these

proportions. What can you conclude?

What is the test to determine if the distribution for a sample proportions is normal?


of 20

If two random variables are approximately normally distributed, is the difference between them also normally

distributed?

Considering the two statements above, determine criteria for determining if the sampling distribution of

1 2ˆ ˆp p is approximately normal.

d. Assuming the criteria in part c are satisfied, sketch a graph of the sampling distribution of 1 2ˆ ˆp p .


of 20

Here is a summary of the facts for the sampling distribution of 1 2

ˆ ˆp p :

Choose an SRS of size n1 from Population 1 with proportion of successes p1 and an independent SRS of size n2

from Population 2 with proportion of successes p2.

Shape: When n1p1, n1(1 − p1), n2p2, and n2(1 − p2) are all at least 10, the sampling distribution of

1 2ˆ ˆp p is approximately Normal.

Center: The mean of the sampling distribution is p1 − p2. That is, the difference in sample proportions is

an unbiased estimator of the difference in population proportions.

Spread: The standard deviation of the sampling distribution 1 2

ˆ ˆp p is

1 1 2 2

1 2

1 1p p p p

n n

as long as each sample is no more than 10% of its population (the 10% condition) and we have independence of

the two samples.

The figure below displays the sampling distribution for 1 2ˆ ˆp p


of 20

e. Suppose that the two counselors from School 1 and School 2 meet to compare the proportions of students

who did their homework last night and find that 1 2

ˆ ˆp p is 0.10. Find the probably of getting this difference or

less.

f. Does the probability in part e give you a reason to doubt the counselors’ reported difference of 0.20? Explain.


Your teacher brings two bags of colored goldfish crackers to class. She tells you that Bag 1 has 25% red

crackers and Bag 2 has 35% red crackers. Each bag contains more than 500 crackers. Using a paper cup, your

teacher takes an SRS of 50 crackers from Bag 1 and a separate SRS of 40 crackers from Bag 2. Let 1 2ˆ ˆp p be

the difference in the sample proportions of red crackers.

a. What is the shape of the sampling distribution of 1 2ˆ ˆp p ? Why?

b. Find the mean and standard deviation of the sampling distribution. Show your work.


of 20

c. Draw a picture of the sampling distribution of 1 2

ˆ ˆp p and find the probability that 1 2

ˆ ˆp p is less than or

equal to −0.02. Show your work.

d. Based on your answer to Question 3, would you be surprised if the difference in the proportion of red

crackers in the two samples was 1 2ˆ ˆp p = −0.02? Explain.


of 20

Confidence Intervals for 1 2p p

2. Who do you think are more intelligent, men or women? To find out what

people think, the Gallup Poll selected a random sample of 520 women and 506

men. The pollsters showed them a list of personal attributes and asked them to

indicate whether each attribute was “generally more true of men than women”.

When asked about intelligence, 28% of the men thought men were generally

more intelligent, but only 13% of the women agreed. Is there a gender gap in

opinions about which sex is smarter or is the difference simply due to chance?

And if there is a gap, then what would we estimate the true size of that gap to be?

Comparing parameters from two different populations, like the difference in the

opinions of men and women on gender intelligence is a fundamental idea in

statistics and in this chapter we will explore this important idea.

We know that the difference between the two proportions (men – women) from our samples is 15%, but what is

the true difference for all men and women? Let’s construct a 95% confidence interval for this difference.

Recall that the general recipe for a confidence interval is:

statistic (critical value) (standard deviation of statistic)

Before we construct the confidence interval we should verify that we meet the conditions.

a. Verify that we meet the Random condition.

b. Independence. For this check we need to verify that the two groups are independent and that the

individuals within the groups are independent. If you are sampling without replacement you also need to check

the 10% condition for both groups. Check this condition.

c. Normality. Since the population parameter is unknown and is not assumed (as in a hypothesis test) we need

to use the statistics from both of our samples to check this condition. Check that this condition is met.


of 20

To construct the confidence interval we need the standard deviation

1 2

1 1 2 2

ˆ ˆ

1 2

1 1p p

p p p p

n n

. But

again since we do not know the population proportions 1p and 2p we will need to substitute the sample

proportions 1p̂ and 2p̂ to get what we call the standard error

1 1 2 2

1 2

ˆ ˆ ˆ ˆ1 1p p p pSE

n n

.

e. Based on what you now know, what is the formula for calculating a level C confidence interval for the

difference of two proportions?

g. Construct a 95% confidence interval for estimating the true difference in the proportions of men and women

with regard to their opinions on intelligence. Interpret your results in the context of the problem situation.


of 20

The confidence interval you constructed in 2g is called a Two-Sample z Interval for a Difference of Two

Proportions. The summary is as follows:

When the Random, Normal, and Independent conditions are met, a level C confidence interval for 1 2

ˆ ˆp p is

1 1 2 2

1 2

1 2

ˆ ˆ ˆ ˆ1 1ˆ ˆ *

p p p pp p z

n n

where z* is the critical value for the standard Normal curve with area C between −z* and z*.

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 1ˆ1 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).

TECHNOLOGY

Your calculator can do the calculations for you when performing inference on the difference of two proportions.

Select STAT then TESTS. For a confidence interval on the difference in proportions select option

B: 2-PropZInt… as shown in the screen below.

Verify your confidence interval from question 2g by entering the appropriate values in screen shown below and

having the calculator find the interval. Note that x1 and x2 must be whole numbers, so you may have to round

accordingly in some problems.


of 20


As part of the Pew Internet and American Life Project, researchers

conducted two surveys in late 2009. The first survey asked a random

sample of 800 U.S. teens about their use of social media and the

Internet. A second survey posed similar questions to a random

sample of 2253 U.S. adults. In these two studies, 73% of teens and

47% of adults said that they use social-networking sites. Use these

results to construct and interpret a 95% confidence interval for the

difference between the proportion of all U.S. teens and adults who

use social-networking sites. Follow all the appropriate steps for

construction a confidence interval.

section 10.1a comparing two proportions€¦ · · 2018-03-1512 2 v ppˆˆ first then ... section...

Documents