Question

In a study to estimate the proportion of residents in a certain city and its suburbs who favor the construction of a nuclear power plant, it is found that 210 of 300 urban residents favor the construction while only 180 of 200 suburban residents are in favor.

(a) Is there a significant difference between the proportion of urban and suburban residents who favor construction of the nuclear power plant? α=0.05.

(b) Construct a 95% confidence interval for the difference of favor proportions between urban residents and suburban residents.

(c) By using the confidence interval from (b) test if there is a significant difference between the proportion of urban and suburban residents who favor construction of the nuclear power plant. (You need to explain how you make the conclusion)  

Answer 1

a)

Below are the null and alternative Hypothesis,
Null Hypothesis, H0: p1 = p2
Alternate Hypothesis, Ha: p1 not = p2

Test statistic
z = (p1cap - p2cap)/sqrt(pcap * (1-pcap) * (1/N1 + 1/N2))
z = (0.7-0.9)/sqrt(0.78*(1-0.78)*(1/300 + 1/200))
z = -5.29

P-value Approach
P-value = 01
As P-value < 0.05, reject null hypothesis.

b)

Here, , n1 = 300 , n2 = 200
p1cap = 0.7 , p2cap = 0.9

Standard Error, sigma(p1cap - p2cap),
SE = sqrt(p1cap * (1-p1cap)/n1 + p2cap * (1-p2cap)/n2)
SE = sqrt(0.7 * (1-0.7)/300 + 0.9*(1-0.9)/200)
SE = 0.0339

For 0.95 CI, z-value = 1.96
Confidence Interval,
CI = (p1cap - p2cap - z*SE, p1cap - p2cap + z*SE)
CI = (0.7 - 0.9 - 1.96*0.0339, 0.7 - 0.9 + 1.96*0.0339)
CI = (-0.27 , -0.13)

c)

yes, because confidence interval does not contains 0