Chapter 22:

Comparing Two Proportions

Yet Another Standard Deviation (YASD)

Standard deviation of the sampling distribution

The variance of the sum or difference of two independent random variables is the sum of their variances

and are independent.

SD X Y Var X Var Y

X Y

Standard Deviation of the Difference Between Two

ProportionsProportions observed in independent random samples are independent

Sample proportions:

1 1 2 21 2

1 2

p q p qSD p p

n n

1 2 and p p

1 1 2 21 2

1 2

p q p qSE p p

n n

Assumptions & ConditionsIndependence Assumption:

Within each group, the data should be based on results for independent individuals

Randomization:The data in each group should be drawn independently and at random from a homogeneous population or generated by a randomized comparative study

Assumptions & Conditions10% condition:

When the data are sampled without replacement, the sample should not exceed 10% of the population.

Independent Samples condition:The two groups that are being compared must be independent of one another.

Success/failure:Both samples are big enough that at least 10 successes and and least 10 failures have been observed.

The Sampling DistributionThe sampling distribution model for a difference between two independent proportions:

Provided that the sampled values are independent, the samples are independent, and the sample sizes are large enough

1 2

1 2

1 1 2 21 2

1 2

the sampling distribution of is

modeled by a Normal model with mean

and standard deviation

p p

p p

p q p qSD p p

n n

A Two-proportion z-interval

When the conditions are met, find the confidence interval:

The critical value depends on the confidence level, C, that you specify.

1 1 2 21 2

1 2

p q p qSE p p

n n

1 2 1 2interval: *p p z SE p p

Intelligence – An ExampleIdentify the parameter and choose a confidence level

: proportion of men who think that men

can be described as intelligent

: proportion of women who think that men

can be described as intelligent

Parameter: Confidence: 95

M

F

M F

p

p

p p C %

Intelligence – An ExampleCheck the conditions.

Randomization: Gallup drew a random sample of U.S. adults10%: the sample size for each groups was certainly less than the U.S. population of men and womenIndependent Samples: the sample of women and the sample of men are independent of each other

Intelligence – An ExampleCheck the conditions.

Success/Failure:

Both samples exceed the minimum size.

506 28 0 142 10

506 72 0 364 10

520 14 0 73 10

506 86 0 447 10

. %

. %

. %

. %

n p men

nq men

n p women

nq women

Intelligence – An ExampleState the sampling distribution model:

Under these conditions, the sampling distribution of the difference between the sample proportions is approximately Normal with a mean of pM – pF, the true difference between the population proportions.

Find a two-proportion z-interval.

Intelligence – An ExampleCalculator

Intelligence – An ExampleInterpret the results:

We are 95% confident that the proportion of American men that the attribute of “intelligent” applies more to men than to women is between 9% and 19% more than American women who think that,

PoolingPooling combines the counts to get an overall proportionWhen we have counts for each group:

When we have only proportions:

Round to whole numbers

1 2

1 2pooled

Success Successp

n n

1 21 1 2 2 and Success n p Success n p

Two-proportion z-testThe conditions for the two-proportion z-test are the same as for the two-proportion z-intervalTest the hypothesisBecause we hypothesize that the proportions are equal, pool to find

1 2:OH p p

1 2

1 2pooled

Success Successp

n n

Two-proportion z-test

Standard Error:

Find the test statistic:

1 21 2

pooled pooled pooled pooledpooled

p q p qSE p p

n n

1 2

1 2pooled

p pzSE p p

Snoring Rates – An Example

Hypothesis:

HO:There is no difference in snoring rates

between those who are 18 – 29 years old and those who are 30 years old.

HA: The rates are different.

1 2 0p p

1 2 0p p

Snoring Rates – An Example

Check the conditions:Randomization: the patients were randomly selected and stratified by sex and region10%: the number of adults surveyed is certainly less than 10% of the population.Independent samples: the two groups are independent of each otherSuccess/failure:

Younger group: 48 snored, 136 didn’tOlder group: 318 snored, 493 didn’t

Snoring Rates – An Example

State the null model and choose your method:

Because the conditions are met, we can model the sampling distribution of the difference in proportions with a Normal modelPerform a two-proportion z-test

Snoring Rates – An Example

Calculator:

Snoring Rates – An Example

State your conclusion:The P-value of 0.0008 says that if there really were no difference in snoring rates between the two age groups, then the difference observed would only happen 8 out of 10,000 times. This is rare enough for us to reject the null hypothesis of no difference and conclude that there is a difference between older and younger adults. It appears that older adults are more likely to snore.

What Can Go Wrong???Don’t use two-sample proportion methods when the samples aren’t independent

Make sure there is no relationship between the two groupsWhen the assumption of independence is violated, this method gives wrong answers

Don’t apply inference methods when you don’t have random samplesDon’t interpret a significant difference in proportions causally.