Question

In a recent poll of 1000 randomly selected​ adults, a​ president's approval rating stood at 52%.

​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 52​% of the country approved of the way he was handling his job at that time.

Answer 1

a)

Let p denotes the population proportion. Let o denotes the sample proportion and n denotes the sample size. Here, 0.520 * (1 – 0.520) P = 0.520, n = 1000, standard error = = 0.015799 0.0158 1000 At 95 percent confidence level, Margin of error is p* (1 - ) ME= * 2 1-0.95 0.520 * (1 – 0.520) 1000 * 20.025 0.520 * (1 – 0.520) * 1.959964 = 0.030965 ~ 0.031 1000 n a 95 percent confidence interval for the population proportion p is * (1-) == * 2 1-0.95 n = 0.5200 + 0.520 * (1 – 0.520) 1000 * 20.025 = 0.5200 + 0.520 * (1 – 0.520) 1000 * 1.959964 = 0.5200 + 0.030965 = (0.489035, 0.550965) ~ (0.489,0.551)

b) We fail to reject the null hypothesis as 0.52 lies within the confidence interval.

We can conclude that 52​% of the country approved of the way he was handling his job at that time.