15
+ Chapter 10: Comparing Two Populations or Groups Section 10.1a Comparing Two Proportions

+ Chapter 10: Comparing Two Populations or Groups Section 10.1a Comparing Two Proportions

Embed Size (px)

Citation preview

Page 1: + Chapter 10: Comparing Two Populations or Groups Section 10.1a Comparing Two Proportions

+

Chapter 10: Comparing Two Populations or GroupsSection 10.1a

Comparing Two Proportions

Page 2: + Chapter 10: Comparing Two Populations or Groups Section 10.1a Comparing Two Proportions

+Chapter 10Comparing Two Populations or Groups

10.1a Comparing Two Proportions

Hw: pg 622: 7 – 13 odd

Page 3: + Chapter 10: Comparing Two Populations or Groups Section 10.1a Comparing Two Proportions

+ Section 10.1aComparing Two Proportions

After this section, you should be able to…

I can DETERMINE whether the conditions for performing inference are met.

I can CONSTRUCT and INTERPRET a confidence interval to compare two proportions.

Learning Objectives

Page 4: + Chapter 10: Comparing Two Populations or Groups Section 10.1a Comparing Two Proportions

+Co

mp

arin

g T

wo

Pro

portion

s Introduction

Suppose we want to compare the proportions of individuals with a certain characteristic in Population 1 and Population 2. Let’s call these parameters of interest p 1 and p 2. The ideal strategy is to take a separate random sample from each population and to compare the sample proportions with that characteristic.

What if we want to compare the effectiveness of Treatment 1 and Treatment 2 in a completely randomized experiment? This time, the parameters p 1 and p 2 that we want to compare are the true proportions of successful outcomes for each treatment. We use the proportions of successes in the two treatment groups to make the comparison. Here’s a table that summarizes these two situations.

Page 5: + Chapter 10: Comparing Two Populations or Groups Section 10.1a Comparing Two Proportions

+

The Sampling Distribution of a Difference Between Two Proportions

In Chapter 7, we saw that the sampling distribution of a sample proportion has the following properties:

Co

mp

arin

g T

wo

Pro

portion

s

Shape Approximately Normal if np ≥ 10 and n(1 - p) ≥ 10

Center ˆ p p

Spread ˆ p p(1 p)

n if the sample is no more than 10% of the population

To explore the sampling distribution of the difference between two proportions, let’s start with two populations having a known proportion of successes.

At School 1, 70% of students did their homework last night At School 2, 50% of students did their homework last night.

Suppose the counselor at School 1 takes an SRS of 100 students and records the sample proportion that did their homework.

School 2’s counselor takes an SRS of 200 students and records the sample proportion that did their homework.

1 2 ˆ ?ˆ p pWhat can we say about the difference in the sample proportions

Page 6: + Chapter 10: Comparing Two Populations or Groups Section 10.1a Comparing Two Proportions

+ The Sampling Distribution of a Difference Between Two Proportions

Using Fathom software, we generated an SRS of 100 students from School 1 and a separate SRS of 200 students from School 2. The difference in sample proportions was then calculated and plotted. We repeated this process 1000 times. The results are below:

Co

mp

arin

g T

wo

Pro

portion

s

What do you notice about the shape, center, and spreadof the sampling distribution of ˆ p 1 ˆ p 2 ?

Page 7: + Chapter 10: Comparing Two Populations or Groups Section 10.1a Comparing Two Proportions

+ The Sampling Distribution of a Difference Between Two Proportions C

om

pa

ring

Tw

o P

rop

ortions

Both ˆ p 1 and ˆ p 2 are random variables. The statistic ̂ p 1 ˆ p 2 is the differenceof these two random variables. In Chapter 6, we learned that for any two independent random variables X and Y,X Y X Y and X Y

2 X2 Y

2

1 21 2 ˆ 1 2ˆ ˆp̂ p p p p p Center :

1 22 11 2 ˆ ˆ 2 2 2 1 1 2 2ˆ ˆ ˆ ˆ

1 2

11 2

1 22

1 2

(1 ) (1 );

ˆ ˆ of the statistic : ˆ ˆ ˆ ˆ(1 ) (1 )

p pp p p p

p p p p

n n

p p p p

n np p

Use the

Spread:

When calculating CI: standard error

1 1 1 1 2 2 2 2

1 2

at least 10 When , (1 ), and (1 ) are all , the

ˆ ˆsampling distribution of is approximately Normal.

n p n p n p n p

p p

Shape

Page 8: + Chapter 10: Comparing Two Populations or Groups Section 10.1a Comparing Two Proportions

+

Example: Who Does More Homework?

Suppose that there are two large high schools, each with more than 2000 students, in a certain town. 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 an SRS of 100 students and records the proportion that did homework. School 2’s counselor takes an SRS of 200 students and records the proportion that did homework. School 1’s counselor and School 2’s counselor meet to discuss the results of their homework surveys. After the meeting, they both report to their principals that

Co

mp

arin

g T

wo

Pro

portion

s

1 2ˆ ˆ =0.10.p p

a) Describe the shape, center, and spread of the sampling distribution of ˆ p 1 ˆ p 2.

2 2

1

1

2

1 1

2 2

1 =200(0.5)=100

(1 ) 200(0

Because ,

and are all at least 10, the sampl

=100(0.7)=70, (1 ) 100(0

ing distribution

ˆ ˆof approximately

.

No

30

i

) 30

rmal.

.5 100

s

)

p

n p p n p

n p

p

n

Shape :

1 2mea: Its in 0.70 0.50 0.20s .p p Center

Its standard deviation i

0.7(0.3) 0.5(0.5)0.058.

100 20

s

0

Spread :

Page 9: + Chapter 10: Comparing Two Populations or Groups Section 10.1a Comparing Two Proportions

+ Example: Who Does More Homework?C

om

pa

ring

Tw

o P

rop

ortions

b) Find the probability of getting a difference in sample proportions ˆ p 1 ˆ p 2 of 0.10 or less from the two surveys.

1 2ˆ ˆStandardize: When 0.10,

p p

z

c) Does the result in part (b) give us reason to doubt the counselors' reported value?

There is only about a

as small as or smaller than the value of 0.10

4% chance of getting a difference in sample p

reported by the counselors.

This does seem su

ropor

spici

tions

ous!

Use normcdf: The area to the left of

under the standard Normal curve is

z

0.10 0.201.72

0.058

1.72

0.042 7.

Page 10: + Chapter 10: Comparing Two Populations or Groups Section 10.1a Comparing Two Proportions

+ Confidence Intervals for p 1 – p 2 Co

mp

arin

g

When data come from two random samples or two groups in a randomized

experiment, the statistic ˆ p 1 ˆ p 2 is our best guess for the value of p1 p2 . We

can use our familiar formula to calculate a confidence interval for p1 p2 :

1 1 2 21 2

1 2

CI: statistic (critical value) (standard deviation of statistic)

ˆ ˆ ˆ ˆ(1 ) (1 )ˆ ˆ: ( ) *

p p p pCI p p z

n n

Page 11: + Chapter 10: Comparing Two Populations or Groups Section 10.1a Comparing Two Proportions

+ Two-Sample z Interval for p 1 – p 2C

om

pa

ring

Tw

o P

rop

ortions

Two-Sample z Interval for a Difference Between Proportions

1

1 2

2

The data are produced by

a random sample and

a random sampl

or

of size from Pop

by two group

e of

of siz

ulation 1

size f

e and s in

rom Population 2

a randomized experime .nt

n

n n

n

Random

1 2

1 1 2 21 2

1 2

When the Random, Normal, and Independent conditions are met, an

ˆ ˆapproximate level C confidence interva

ˆ ˆ ˆ ˆ(1 ) (1 )ˆ

l for ( ) is

where

ˆ ) *

*

(p p p p

p p z

p

n

p

z

n

is the critical value for the standard Normal curve with area C

between * and *.z z

Page 12: + Chapter 10: Comparing Two Populations or Groups Section 10.1a Comparing Two Proportions

+ Two-Sample z Interval for p 1 – p 2C

om

pa

ring

Tw

o P

rop

ortions

Two-Sample z Interval for a Difference Between Proportions

1 1 1 1

2 2 2 2

The counts of "successes" and "failures"

in each sample or

ˆ ˆ, (1 ) 10,

ˆ ˆ

g

, (1 ) 10

roup --

.

n p n p

n p n p

Normal

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 ar at leaste

Independent

as the corresponding samples (t

10 t

he 10

imes

as large % condition).

Page 13: + Chapter 10: Comparing Two Populations or Groups Section 10.1a Comparing Two Proportions

+

Plan: We should use a two-sample z interval for p1 – p2 if the conditions are satisfied. Random The data come from a random sample of 800 U.S. teens and a separate random sample of 2253 U.S. adults. Normal We check the counts of “successes” and “failures” and note the Normal condition is met since they are all at least 10:

Independent We clearly have two independent samples—one of teens and one of adults. Individual responses in the two samples also have to be independent. The researchers are sampling without replacement, so we check the 10% condition: there are at least 10(800) = 8000 U.S. teens and at least 10(2253) = 22,530 U.S. adults.

Example: Teens and Adults on Social Networks

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.

Co

mp

arin

g T

wo

Pro

portion

s

1 1 1 1

2 2 2 2

ˆ ˆ= (1 )

ˆ ˆ= (1 )

n p n p

n p n p

State: Our parameters of interest are p1 = the proportion of all U.S. teens who use social networking sites and p2 = the proportion of all U.S. adults who use social-networking sites. We want to estimate the difference p1 – p2 at a 95% confidence level.

1 1 1 1

2 2 2 2

ˆ ˆ=800(0.73)=584 (1 ) 800(1 0.73) 216

ˆ ˆ=2253(0.47)=1058.91 1059 (1 ) 2253(1 0.47) 1194.09 1194

n p n p

n p n p

Page 14: + Chapter 10: Comparing Two Populations or Groups Section 10.1a Comparing Two Proportions

+ Example: Teens and Adults on Social NetworksC

om

pa

ring

Tw

o P

rop

ortions

Do: Since the conditions are satisfied, we can construct a two-sample z interval for the difference p1 – p2.

Conclude: We are 95% confident that the interval from 0.223 to 0.297 captures the true difference in the proportion of all U.S. teens and adults who use social-networking sites. This interval suggests that more teens than adults in the United States engage in social networking by between 22.3 and 29.7 percentage points.

1 1 2 21 2

1 2

ˆ ˆ ˆ ˆ(1 ) (1 )ˆ ˆ( ) *

p p p pp p z

n n

0.26 0.037

(0.223, 0.297)

0.73(0.27) 0.47(0.53)(0.73 0.47) 1.96

800 2253

Page 15: + Chapter 10: Comparing Two Populations or Groups Section 10.1a Comparing Two Proportions

+ Example: Teens and Adults on Social NetworksC

om

pa

ring

Tw

o P

rop

ortions

Do: Since the conditions are satisfied, we can construct a two-sample z interval for the difference p1 – p2.

Try with calculator. STAT:TEST:2-PropZInt x1:584 How did we get these?n1: 800x2:1059n2: 2253

Read pg. 608 - 611

1 1 2 21 2

1 2

ˆ ˆ ˆ ˆ(1 ) (1 )ˆ ˆ( ) *

p p p pp p z

n n

0.26 0.037

(0.223, 0.297)

0.73(0.27) 0.47(0.53)(0.73 0.47) 1.96

800 2253