Answer: [tex](0.522,\ 0.658)[[/tex]
Step-by-step explanation:
The formula to calculate the confidence interval is given by :-
[tex]\hat{p}\pm z^*\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}[/tex]
, where n= sample size .
[tex]\hat{p}[/tex] = sample proportion.
z* = cRitical value .
Let p be the population proportion of students who receive some financial aid.
The dean randomly selects 200 students and finds that 118 of them are receiving financial aid.
i.e. n= 200
[tex]\hat{p}=\dfrac{118}{200}=0.59[/tex]
We know that the critical value for 95% confidence interval : z*= 1.96
Then, the 95% confidence interval for population proportion will be :-
[tex](0.59)\pm (1.96)\sqrt{\dfrac{0.59(1-0.59)}{200}}[/tex]
[tex]0.59\pm (1.96)\sqrt{0.0012095}[/tex]
[tex]0.59\pm (1.96)(0.0347778665246)[/tex]
[tex]0.59\pm0.0681646183882\approx0.59\pm0.068\\\\=(0.59-0.068,\ 0.59+0.068)\\\\=(0.522,\ 0.658)[/tex]
Hence, 95% confidence interval to estimate the proportion of students on financial aid = [tex](0.522,\ 0.658)[[/tex]
