Answer:
The point estimate that should be used in constructing the confidence interval is 0.11.
The 80% confidence interval for the difference in two proportions is (0.0856, 0.1344).
Step-by-step explanation:
Before building the confidence interval, we need to understand the central limit theorem and subtraction of normal variables.
Central Limit Theorem
The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean
and standard deviation
, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
and standard deviation
.
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
For a proportion p in a sample of size n, the sampling distribution of the sample proportion will be approximately normal with mean
and standard deviation
Subtraction between normal variables:
When two normal variables are subtracted, the mean is the difference of the means, while the standard deviation is the square root of the sum of the variances.
Midwest:
50% of 1380, so:
![p_M = 0.5](https://tex.z-dn.net/?f=p_M%20%3D%200.5)
![s_M = \sqrt{\frac{0.5*0.5}{1380}} = 0.0135](https://tex.z-dn.net/?f=s_M%20%3D%20%5Csqrt%7B%5Cfrac%7B0.5%2A0.5%7D%7B1380%7D%7D%20%3D%200.0135)
South:
39% of 1300, so:
![p_S = 0.39](https://tex.z-dn.net/?f=p_S%20%3D%200.39)
![s_S = \sqrt{\frac{0.39*0.61}{1300}} = 0.0135](https://tex.z-dn.net/?f=s_S%20%3D%20%5Csqrt%7B%5Cfrac%7B0.39%2A0.61%7D%7B1300%7D%7D%20%3D%200.0135)
Distribution of the difference:
![p = p_M - p_S = 0.5 - 0.39 = 0.11](https://tex.z-dn.net/?f=p%20%3D%20p_M%20-%20p_S%20%3D%200.5%20-%200.39%20%3D%200.11)
So the point estimate that should be used in constructing the confidence interval is 0.11.
![s = \sqrt{s_M^2+s_S^2} = \sqrt{0.0135^2+0.0135^2} = 0.0191](https://tex.z-dn.net/?f=s%20%3D%20%5Csqrt%7Bs_M%5E2%2Bs_S%5E2%7D%20%3D%20%5Csqrt%7B0.0135%5E2%2B0.0135%5E2%7D%20%3D%200.0191)
Confidence interval:
![p \pm zs](https://tex.z-dn.net/?f=p%20%5Cpm%20zs)
In which
z is the z-score that has a p-value of
.
80% confidence level
So
, z is the value of Z that has a p-value of
, so
.
The lower bound of the interval is:
![p - zs = 0.11 - 1.28*0.0191 = 0.0856](https://tex.z-dn.net/?f=p%20-%20zs%20%3D%200.11%20-%201.28%2A0.0191%20%3D%200.0856)
The upper bound of the interval is:
![p + zs = 0.11 + 1.28*0.0191 = 0.1344](https://tex.z-dn.net/?f=p%20%2B%20zs%20%3D%200.11%20%2B%201.28%2A0.0191%20%3D%200.1344)
The 80% confidence interval for the difference in two proportions is (0.0856, 0.1344).