ch 10 comparing two proportions

Ch 10Ch 10Comparing Two Comparing Two ProportionsProportions

Target Goal: I can determine the significance Target Goal: I can determine the significance of a two sample proportion.of a two sample proportion.

10.1b10.1bh.w: pg 623: 15, 17, 21, 23h.w: pg 623: 15, 17, 21, 23

a bag of M&M's for day 1 chapter 11.Bring

If we want to If we want to comparecompare two populationstwo populations or compare the responses to two or compare the responses to two treatments from treatments from independent samples independent samples which involve the which involve the mean of a mean of a quantitative variablequantitative variable, we look at a , we look at a two-two-sample method.sample method.

Comparing Proportions of Successes in Comparing Proportions of Successes in Two GroupsTwo Groups

The null hypothesis is that there is The null hypothesis is that there is no no differencedifference between the two parameters. between the two parameters.

0 2 1 21 0 or : 0 : H pp p pH

The Alternative Hypothesis could The Alternative Hypothesis could be thatbe that

(two-sided)(two-sided)

(one-sided)(one-sided)

(one-sided)(one-sided)

The alternative says The alternative says what kind of what kind of difference we expect.difference we expect.

1 2:aH p p

2 1 21 or : 0:a aH pH p p p

2 1 21 or : 0:a aH pH p p p

Pooled Sample ProportionPooled Sample Proportion

When the observations come from a When the observations come from a single single populationpopulation; instead of estimating; instead of estimating p p11 and and pp22

separately, we separately, we pool the two samples pool the two samples (combined)(combined) and and use the overall sample use the overall sample proportionproportion the the estimate the single estimate the single parameter pparameter p..

= = Count of Count of successessuccesses in in both samples combinedboth samples combined

Count of Count of observationsobservations in in both samples combinedboth samples combined

ˆCp

Check your assumptions about a Check your assumptions about a proportion!proportion!

1)1) Random:Random: Both samples formed from a random sample or Both samples formed from a random sample or by two groups from a randomized experiment.by two groups from a randomized experiment.

2)2) Independently chosen samples. Independently chosen samples.

(A randomized sample taken covers independence)(A randomized sample taken covers independence)

Population Population 10n; population is at least 10 times as large 10n; population is at least 10 times as large as the sample.as the sample.

3.3. Normal:Normal:

10 10 for a for a 2 sample hypothesis 2 sample hypothesis testtest

10 10 for for 2 sample a C.I.2 sample a C.I.

1 1 1 1, 2 2 2 2ˆ ˆ ˆ ˆ, , n p n q n p n q

1 1 1 1, 2 2 2 2ˆ ˆ ˆ ˆ, , n p n q n p n q

If these assumptions hold, then If these assumptions hold, then the the difference in sample proportionsdifference in sample proportions is an is an unbiased estimator of the difference in unbiased estimator of the difference in population proportions,population proportions,

The The meanmean of is equal to . of is equal to .

1 2ˆ ˆThe Sampling Distribution of .p p

1 2ˆ ˆp p 1 2p p

The variance of The variance of

is the is the sum of the variancessum of the variances of of

Recall that the Recall that the variances variances addadd but the but the standard deviations do not.standard deviations do not.

1 2ˆ ˆp p

1 2ˆ ˆ .p and p

112 21 2ˆ1 1 2 2

ˆ ˆ

1 2

ˆp p p p

p q p q

n n

When sample sizes are large, the When sample sizes are large, the distribution of is approximately normal.distribution of is approximately normal.1 2ˆ ˆp p

Many comparative studies start with just one sample and then divide into two groups based on the data gathered from the subjects. Use the two sample z procedures.

Replace the population proportions Replace the population proportions with the sample proportions to find SEwith the sample proportions to find SE..

1 2Recall: Confidence Intervals for .p p

*estimate z SE

1 1 2 2

1 2

ˆ ˆ ˆ ˆp q p

n nS

qE

1 1 2 21 2

1 2

ˆ ˆ ˆ ˆˆ ˆ( ) *

p q p q

nI p

nC p z

Significance Tests for Significance Tests for Comparing Two ProportionsComparing Two Proportions

Significance testsSignificance tests help us to decide if help us to decide if the the effecteffect we see in the samples is we see in the samples is really there really there in the proportions.in the proportions.

The null hypothesis is that there is The null hypothesis is that there is no no differencedifference between the two parameters. between the two parameters.

10 02 1 2: or : 0H Hp p p p

In order to In order to standardizestandardize , subtract , subtract the mean and divide by the standard error the mean and divide by the standard error (SE):(SE):

Use this test statistic to carry out a test of Use this test statistic to carry out a test of significance.significance.

Note: most 2-sample problems are pooled.Note: most 2-sample problems are pooled.

1 2ˆ ˆp p

1 2 1 2

1 2

ˆ ˆ

1 1ˆ ˆ

p p p p

pqn n

z

pooled

Ex: Ex: Cholesterol and Heart AttacksCholesterol and Heart Attacks

High levels of cholesterol in the blood are High levels of cholesterol in the blood are associated with higher risk of heart attacks. associated with higher risk of heart attacks. Study: Middle age men were assigned at Study: Middle age men were assigned at random to one of two treatments: 2051 men took random to one of two treatments: 2051 men took the drug gemfibrozil to reduce their cholesterol the drug gemfibrozil to reduce their cholesterol levels, and a control group of 2030 men took a levels, and a control group of 2030 men took a placebo. placebo.

During the next five years, 56 men in the During the next five years, 56 men in the gemfibrozil group and 84 men in the placebo gemfibrozil group and 84 men in the placebo group had heart attacks.group had heart attacks.

The sample proportion that had The sample proportion that had heart attacks are:heart attacks are:

(gemfibrozil group)(gemfibrozil group)

(placebo group)(placebo group)

Is the apparent benefit of gemfibrozil Is the apparent benefit of gemfibrozil statistically significant?statistically significant?

1p̂ 560.0273

2051

2p̂84

0.04142030

Step 1.State Step 1.State Identify the population of interest and the Identify the population of interest and the parameter you want to draw a conclusion about.parameter you want to draw a conclusion about.

We hope to show that gemfibrozil We hope to show that gemfibrozil reduces reduces heart attacksheart attacks by by comparing two comparing two proportions of middle age men.proportions of middle age men.

: : There is There is no no difference in the reduction of heart difference in the reduction of heart attacks attacks for middle age men taking for middle age men taking gemfibrozil compared to the placebo.gemfibrozil compared to the placebo.

0 1 2:H p p

There There is a reduction in the number of heart is a reduction in the number of heart attacksattacks of middle age men taking of middle age men taking gemfibrozil gemfibrozil compared tocompared to the middle age the middle age men taking the placebo.men taking the placebo.

1 2:aH p p

Step 2. Step 2. Plan Plan If the conditions are met, If the conditions are met, we should perform a two-we should perform a two-sample z test for .sample z test for .

Since both samples come from a Since both samples come from a single single populationpopulation, we will , we will pool the samples.pool the samples.

The The pooled proportionpooled proportion of heart attacks for of heart attacks for the two groups in the Helinski Heart study the two groups in the Helinski Heart study is: is: = = Count of heart attacksCount of heart attacks in both samples combinedin both samples combined

Count of subjectsCount of subjects in both samples combinedin both samples combinedp̂

1 2p p

56 84

2051 2030p̂

140ˆ 0.0343

4081p

Verify the conditions.Verify the conditions.

Random: Random: subjects randomly assignedsubjects randomly assigned Independent:Independent: Yes, Yes, due to random assignmentdue to random assignment, ,

these groups these groups can be viewed as independent.can be viewed as independent.

Individual observations are also independent. Individual observations are also independent. Knowing whether one subject has a heart attack Knowing whether one subject has a heart attack gives no information gives no information on another subject.on another subject.

Verify the conditions.Verify the conditions.

Normal:Normal:

; ; 10.10.

1 1ˆ (0.0273)2051 55.99n p 1 1̂, (.9727) 1995.012051n q

2 1ˆ (0.0414)2030 84.042n p 2 2ˆ, (0.9586) 1945.90 6203n q

pooled

Step 3Step 3. . Carry out the inference Carry out the inference procedureprocedure

The The z z statistic isstatistic is 1 2 1 2

1 2

ˆ ˆ

1ˆ ˆ

1z

p

p p p p

nqn

(0.0343)(0.9657)

0.0273 0.0414 0

1 12051 2030

z

2.47z

Determine the p-valueDetermine the p-value

normalcdf(-E99,-2.47) = 0.0068normalcdf(-E99,-2.47) = 0.0068

Step 4:Step 4: Conclude - Conclude - Interpret your results in Interpret your results in the context of the problem.the context of the problem.

Since P < 0.01, the results are significant Since P < 0.01, the results are significant at the at the α= 0.01α= 0.01 level. We level. We reject Hreject Hoo and and conclude that conclude that gemfibrozil reduced the rate gemfibrozil reduced the rate of heart attacks in middle age men.of heart attacks in middle age men.

Verify by calculator:Verify by calculator: On the TI-83+, use On the TI-83+, use 2-PropZInt and 2-2-PropZInt and 2-

PropZTestPropZTest to construct confidence to construct confidence intervals and perform significance tests. intervals and perform significance tests.

Stat:tests:2-propZTest:Stat:tests:2-propZTest:

In class FR: 2009B #3In class FR: 2009B #3 Read pg. 611 - 619Read pg. 611 - 619

Gives you z score using pooled proportions.