Answer: (0.1658, 0.2742)
Step-by-step explanation:
Formula to find the confidence interval for population proportion is given by :-
[tex]\hat{p}\pm z^*\sqrrt{\dfrac{\hat{p}(1-\hat{p})}{n}}[/tex]
, where n= sample size
z*= Critical value
[tex]\hat{p}[/tex] = sample proportion.
As per given , we have
Significance level : [tex]\alpha=1-0.99=0.01[/tex]
According to z-table, Critical value for 99% confidence interval : z*=2.576
Let p be the proportion of people in the country that have red hair.
n= 388
[tex]\hat{p}=0.22[/tex]
Now, required confidence interval for proportion of people in the country that have red hair will be :-
[tex]0.22\pm (2.576)\sqrt{\dfrac{0.22(1-0.22)}{388}}[/tex]
[tex]0.22\pm (2.576)\sqrt{0.000442268}[/tex]
[tex]0.22\pm (2.576)(0.02103)[/tex]
[tex](0.22-0.05417328,\ 0.22+0.05417328 )=(0.16582672,\ 0.27417328)\\\\\approx(0.1658,\ 0.2742)[/tex]
The 99% confidence interval for the proportion of people in the country that have red hair= (0.1658, 0.2742)
