Topic 21—Comparison of Proportions

Example

Test of Significance – Problem #1

• A test of significance for the difference of two proportions

• Requirements:– Indep. SRS or Random assign. to groups?• Randomly assigned to groups—OK

– .

– Results should be valid

5; (1 ) 5; 5; (1 ) 5

17 418 38 397435( ) 17;435( ) 418;435( ) 38;435( ) 397

435 435 435 435

A A P PA A P Pn p n p n p n p

• Ho: Pa = Pp The proportion of all HIV+ people who develop AIDS on AZT equals proportion of HIV+ people who develop AIDS on placebo

• Ha: Pa < Pp The proportion of all HIV+ people who develop AIDS on AZT is less than the proportion of HIV+ people who develop AIDS on placebo

• Z =

• P(z<-2.9256) = .0017

.039 .0872.9256

1 1 1 1( )(1 )( ) (.063)(.937)( )

435 435

A P

c cA P

p p

p pn n

• Because our p-value is less than 5%, we reject the null hypothesis. We have enough evidence to conclude that the proportion of HIV+ people who develop AIDS on AZT is less than the proportion of HIV+ people who develop AIDS on a placebo.

Confidence Interval – Problem #1

• 95% confidence interval to estimate the difference between 2 proportions

• Requirements– Same as above

• .

• .

* (1 ) (1 )( ) A A P P

A PA P

p p p pp p Z

n n

.039(.961) .087(.913)(.039 .087) 1.96

435 435

• (-.0805, -.0161)• I am 95% confident that the true difference

between the proportions who developed AIDS between the AZT and placebo groups is between -.0805 and -.0161

• ***AZT group developed AIDS between 1.6% and 8% less often.

Test of Significance – Problem #2

• A test of significance for the difference of two proportions

• Requirements:– Indep. SRS or Random assign. to groups?• Indep groups (urban vs. rural) but unsure how chosen-X

– .

– Results may be questionable

5; (1 ) 5; 5; (1 ) 5

52 13 30 2565( ) 52;65( ) 13;55( ) 30;55( ) 2565 65 55 55

A A P PA A P Pn p n p n p n p

• Ho: Purb = Prur The proportion of urban students who pass course

in chemical engineering at NC St. equals the proportion of rural students who pass the same course.

• Ha: Purb ≠ Prur The proportion of urban students who pass course in

chemical engineering at NC St. does not equal the proportion of rural students who pass the same course.

• Z =

• 2 x P(z>|2.987|) = .0028

52 3065 55 2.987

1 1 82 38 1 1( )(1 )( )120 120 65 55

u r

c cu r

p p

p pn n

• Because our p-value is less than 5%, we reject the null hypothesis. We have enough evidence to conclude the proportion of urban students who pass course in chemical engineering at NC St. does not equal the proportion of rural students who pass the same course .

Confidence Interval – Problem #2

• 90% confidence interval to estimate the difference between 2 proportions

• Requirements– Same as above

• .

• .

* (1 ) (1 )( ) u u r r

u ru r

p p p pp p Z

n n

52 13 30 2552 30 65 65 55 55( ) 1.64565 55 65 55

• (.11723, .39186)• I am 90% confident that the true difference

between the proportions of urban students who pass the chemical engineering class at NC St. and the proportion of rural students who pass the class is between .11723 and .39186.