Question

In a recent poll of 1100 randomly selected​ adults, a​ president's approval rating stood at 46%.

​a) Make a 95​% confidence interval for his approval rating by all adults in the country.

​b) Based on the confidence​ interval, test the null hypothesis that 46​% of the country approved of the way he was handling his job at that time.

​a) Find the 95​% confidence interval as a true proportion. ​(_​,_)

​(Round to three decimal places as​ needed.)

Answer 1

a)
sample proportion, = 0.46
sample size, n = 1100
Standard error, SE = sqrt(pcap * (1 - pcap)/n)
SE = sqrt(0.46 * (1 - 0.46)/1100) = 0.015

Given CI level is 95%, hence α = 1 - 0.95 = 0.05
α/2 = 0.05/2 = 0.025, Zc = Z(α/2) = 1.96

CI = (pcap - z*SE, pcap + z*SE)
CI = (0.46 - 1.96 * 0.015 , 0.46 + 1.96 * 0.015)
CI = (0.431 , 0.489)

b)
As 0.46 is included in the CI, fail to reject H0.
There is evidence to conclude that the 46​% of the country approved of the way he was handling his job at that time.