Respuesta :
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, [tex]N_{1}=100[/tex]
number of student in sample-2, [tex]N_{2}=100[/tex]
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,
[tex]p_{1}=\frac{43}{100}[/tex]
[tex]p_{1}=0.43[/tex]
and
[tex]p_{2}=\frac{47}{100}[/tex]
[tex]p_{2}=0.47[/tex]
[tex]p_{1}-p_{2}=0.43-0.47\\p_{1}-p_{2}= -0.04[/tex]
and we can determine p by using following formula,
[tex]p = \frac{n_{1}p_{1}+n_{2}p_{2} }{n_{1}+n_{2}}[/tex]..............(1)
put values of [tex]n_{1},n_{2},p_{1}[/tex] and [tex]p_{2}[/tex] in equation (1)
[tex]p = \frac{100\times .43+100\times .47}{100+100}[/tex]
[tex]p = \frac{43+47}{200}[/tex]
[tex]p = \frac{90}{200}[/tex]
[tex]p =.45 [/tex]
Now we can determine q by using following formula
[tex]q=1-p[/tex]................(2)
put value of p in equation 2
[tex]q =1-.45 [/tex]
[tex]q =.55 [/tex]
Now we can determine Standard Error of the difference between population proportions by using following formula
Standard error
= [tex]\sqrt{pq(\frac{1}{n_{1}}+\frac{1}{n_{2} })[/tex].............(3)
Standard error
= [tex]\sqrt{.45\times .55(\frac{1}{100}+\frac{1}{100})[/tex]
Standard error
= [tex]\sqrt.2475\times .02[/tex]
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
=[tex](p1 - p2) - 2.3263\times[/tex] standard error
lower limit of confidence interval =[tex](.43-0.47) - 2.3263\times0.0704[/tex]
lower limit of confidence interval =[tex]-0.04 - 2.3263\times0.0704[/tex] lower limit of confidence interval=-0.2038
Similarly we can determine upper limit of confidence interval by using following formula
=[tex](p1 - p2) +2.3263\times[/tex] standard error
Therefore,
Upper limit of confidence interval =[tex](.43-0.47) + 2.3263\times0.0704[/tex]
Upper limit of confidence interval =[tex]-0.04 +2.3263\times0.0704[/tex]
Upper limit of confidence interval= 0.1238
Therefore,
98% confidence interval is given as [-0.204, 0.124]
