Question

In a recent poll of 1100 randomly selected​ adults, a​ president's approval rating stood at 49%.

a) Make a 90% Confidence interval for his approval rating by all adults in the country.

​b) Based on the confidence​ interval, test the null hypothesis that 49% of the country approved of the way he was handling his job at that time.

Answer 1

Part a

Confidence interval for Population Proportion is given as below:

Confidence Interval = P ± Z* sqrt(P*(1 – P)/n)

Where, P is the sample proportion, Z is critical value, and n is sample size.

We are given

n = 1100

P = x/n = 0.49

Confidence level = 90%

Critical Z value = 1.6449

(by using z-table)

Confidence Interval = P ± Z* sqrt(P*(1 – P)/n)

Confidence Interval = 0.49 ± 1.6449* sqrt(0.49*(1 – 0.49)/1100)

Confidence Interval = 0.49 ± 1.6449* 0.0151

Confidence Interval = 0.49 ± 0.0248

Lower limit = 0.49 - 0.0248 = 0.4652

Upper limit = 0.49 + 0.0248 = 0.5148

Confidence interval = (0.4652, 0.5148)

Part b

Based on the above confidence interval, we do not reject the null hypothesis because the value 0.49 is included in the above confidence interval. There is sufficient evidence to conclude that 49% of the country approved of the way he was handling his job at that time.