Question

A random sample of 100 students at a high school was asked whether they would ask their father or mother for help with a homework assignment in science. A second sample of 100 different students was asked the same question for an assignment in history. Forty-three students in the first sample and 47 students in the second sample replied that they turned to their mother rather than their father for help. Construct a 98% confidence interval for p 1 - p 2.

Answer 1

98% confidence interval is given as [-0.204, 0.124]

Explanation:

In this question we have given

number of student in sample-1,
`N_(1)=100`

number of student in sample-2,
`N_(2)=100`

43 students in the first sample and 47 students in the second sample replied that they turned to their mother rather than their father for help

Therefore,

`p_(1)=(43)/(100)`

`p_(1)=0.43`

and

`p_(2)=(47)/(100)`

`p_(2)=0.47`

`p_(1)-p_(2)=0.43-0.47\\p_(1)-p_(2)= -0.04`

and we can determine p by using following formula,

`p = (n_(1)p_(1)+n_(2)p_(2) )/(n_(1)+n_(2))`..............(1)

put values of
`n_(1),n_(2),p_(1)` and
`p_(2)` in equation (1)

`p = (100* .43+100* .47)/(100+100)`

`p = (43+47)/(200)`

`p = (90)/(200)`

`p =.45`

Now we can determine q by using following formula

`q=1-p`................(2)

put value of p in equation 2

`q =1-.45`

`q =.55`

Now we can determine Standard Error of the difference between population proportions by using following formula

Standard error

=
`\sqrt{pq((1)/(n_(1))+(1)/(n_(2) ))`.............(3)

Standard error

=
`\sqrt{.45* .55((1)/(100)+(1)/(100))`

Standard error

=
`\sqrt.2475* .02`

standard error=0.07035

standard error=0.0704

z- score for 98% confidence is 2.3263

Now we can determine lower limit of confidence interval by using following formula

=
`(p1 - p2) - 2.3263*` standard error

lower limit of confidence interval =
`(.43-0.47) - 2.3263*0.0704`

lower limit of confidence interval =
`-0.04 - 2.3263*0.0704` lower limit of confidence interval=-0.2038

Similarly we can determine upper limit of confidence interval by using following formula

=
`(p1 - p2) +2.3263*` standard error

Therefore,

Upper limit of confidence interval =
`(.43-0.47) + 2.3263*0.0704`

Upper limit of confidence interval =
`-0.04 +2.3263*0.0704`

Upper limit of confidence interval= 0.1238

Therefore,

98% confidence interval is given as [-0.204, 0.124]