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

Answer:

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

Step-by-step 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,

$p_{1}=\frac{43}{100}$

$p_{1}=0.43$

and

$p_{2}=\frac{47}{100}$

$p_{2}=0.47$

$p_{1}-p_{2}=0.43-0.47\\p_{1}-p_{2}= -0.04$

and we can determine p by using following formula,

$p = \frac{n_{1}p_{1}+n_{2}p_{2} }{n_{1}+n_{2}}$..............(1)

put values of $n_{1},n_{2},p_{1}$ and $p_{2}$ in equation (1)

$p = \frac{100\times .43+100\times .47}{100+100}$

$p = \frac{43+47}{200}$

$p = \frac{90}{200}$

$p =.45$

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

= $\sqrt{pq(\frac{1}{n_{1}}+\frac{1}{n_{2} })$.............(3)

Standard error

= $\sqrt{.45\times .55(\frac{1}{100}+\frac{1}{100})$

Standard error

= $\sqrt.2475\times .02$

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

=$(p1 - p2) - 2.3263\times$ standard error

lower limit of confidence interval =$(.43-0.47) - 2.3263\times0.0704$

lower limit of confidence interval =$-0.04 - 2.3263\times0.0704$ lower limit of confidence interval=-0.2038

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

=$(p1 - p2) +2.3263\times$ standard error

Therefore,

Upper limit of confidence interval =$(.43-0.47) + 2.3263\times0.0704$

Upper limit of confidence interval =$-0.04 +2.3263\times0.0704$

Upper limit of confidence interval= 0.1238

Therefore,

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