Answer: (0.5496, 0.5754)
Step-by-step explanation:
The confidence interval for population proportion (p) is given by :-
[tex]\hat{p}\pm z*\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}[/tex] , where [tex]\hat{p}[/tex]= Sample proportion , n= sample size , z*= Critical z-value.
Let p be the proportion of all college students who are in favor of banning Hawaiian shirt.
Given, A random sample of 4000 college students yielded 2250 who are in favor of banning Hawaiian shirts.
i.e. n=4000
[tex]\hat{p}=\dfrac{2250}{4000}=0.5625[/tex]
z-value for 90% confidence level is 1.645
Now , 90% confidence interval for p would be :
[tex]0.5625\pm (1.645)(\sqrt{\dfrac{0.5625(1-0.5625)}{4000}})[/tex]
[tex]=0.5625\pm (1.645)(\sqrt{0.0000615234375})\\\\=0.5625\pm (1.645)(0.00784368774876)\\\\\approx0.5625\pm0.0129\\\\=(0.5625-0.0129, \ 0.5625+0.0129)\\\\=(0.5496,\ 0.5754)[/tex]
Hence, the required 90% interval = (0.5496, 0.5754)
