Answer:
The 99% confidence interval for the difference in two proportions is (0.0456, 0.1944).
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.
Western residents:
39% out of 640, so:
![p_1 = 0.39](https://tex.z-dn.net/?f=p_1%20%3D%200.39)
![s_1 = \sqrt{\frac{0.39*0.61}{640}} = 0.0193](https://tex.z-dn.net/?f=s_1%20%3D%20%5Csqrt%7B%5Cfrac%7B0.39%2A0.61%7D%7B640%7D%7D%20%3D%200.0193)
Eastern residents:
51% out of 540, so:
![p_2 = 0.51](https://tex.z-dn.net/?f=p_2%20%3D%200.51)
![s_2 = \sqrt{\frac{0.51*0.49}{540}} = 0.0215](https://tex.z-dn.net/?f=s_2%20%3D%20%5Csqrt%7B%5Cfrac%7B0.51%2A0.49%7D%7B540%7D%7D%20%3D%200.0215)
Distribution of the difference:
![p = p_2 - p_1 = 0.51 - 0.39 = 0.12](https://tex.z-dn.net/?f=p%20%3D%20p_2%20-%20p_1%20%3D%200.51%20-%200.39%20%3D%200.12)
![s = \sqrt{s_2^2+s_1^2} = \sqrt{0.0215^2+0.0193^2} = 0.0289](https://tex.z-dn.net/?f=s%20%3D%20%5Csqrt%7Bs_2%5E2%2Bs_1%5E2%7D%20%3D%20%5Csqrt%7B0.0215%5E2%2B0.0193%5E2%7D%20%3D%200.0289)
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
.
99% 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 = 0.12 - 2.575*0.0289 = 0.0456](https://tex.z-dn.net/?f=p%20-%20zs%20%3D%200.12%20-%202.575%2A0.0289%20%3D%200.0456)
The upper bound of the interval is:
![p + zs = 0.12 + 2.575*0.0289 = 0.1944](https://tex.z-dn.net/?f=p%20%2B%20zs%20%3D%200.12%20%2B%202.575%2A0.0289%20%3D%200.1944)
The 99% confidence interval for the difference in two proportions is (0.0456, 0.1944).