Answer:
The 90% confidence interval for the difference between the proportions of Democrats and Republicans who approve of the mayor's performance is (-0.1078, 0.1702).
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 estabilishes 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 ![s = \sqrt{\frac{p(1-p)}{n}}](https://tex.z-dn.net/?f=s%20%3D%20%5Csqrt%7B%5Cfrac%7Bp%281-p%29%7D%7Bn%7D%7D)
Subtraction of normal variables:
When two normal variables are subtracted, the mean is the subtraction of the means while the standard deviation is the square root of the sum of the variances.
In a random sample of 42 Democrats from one city, 10 approved of the mayor's performance.
This means that:
![p_D = \frac{10}{42} = 0.2381, s_D = \sqrt{\frac{0.2381*(1-0.2381)}{42}} = 0.0657](https://tex.z-dn.net/?f=p_D%20%3D%20%5Cfrac%7B10%7D%7B42%7D%20%3D%200.2381%2C%20s_D%20%3D%20%5Csqrt%7B%5Cfrac%7B0.2381%2A%281-0.2381%29%7D%7B42%7D%7D%20%3D%200.0657)
In a random sample of 58 Republicans from the city, 12 approved of the mayor's performance.
This means that:
![p_R = \frac{12}{58} = 0.2069, s_R = \sqrt{\frac{0.2069*(1-0.2069)}{58}} = 0.0532](https://tex.z-dn.net/?f=p_R%20%3D%20%5Cfrac%7B12%7D%7B58%7D%20%3D%200.2069%2C%20s_R%20%3D%20%5Csqrt%7B%5Cfrac%7B0.2069%2A%281-0.2069%29%7D%7B58%7D%7D%20%3D%200.0532)
Distribution of the difference:
![p = p_D - p_R = 0.2381 - 0.2069 = 0.0312](https://tex.z-dn.net/?f=p%20%3D%20p_D%20-%20p_R%20%3D%200.2381%20-%200.2069%20%3D%200.0312)
![s = \sqrt{0.0657^2+0.0532^2} = 0.0845](https://tex.z-dn.net/?f=s%20%3D%20%5Csqrt%7B0.0657%5E2%2B0.0532%5E2%7D%20%3D%200.0845)
Confidence interval:
We have that to find our
level, that is the subtraction of 1 by the confidence interval divided by 2. So:
![\alpha = \frac{1 - 0.9}{2} = 0.05](https://tex.z-dn.net/?f=%5Calpha%20%3D%20%5Cfrac%7B1%20-%200.9%7D%7B2%7D%20%3D%200.05)
Now, we have to find z in the Ztable as such z has a pvalue of
.
That is z with a pvalue of
, so Z = 1.645.
Now, find the margin of error M as such
![M = zs](https://tex.z-dn.net/?f=M%20%3D%20zs)
![M = 1.645*0.0845 = 0.139](https://tex.z-dn.net/?f=M%20%3D%201.645%2A0.0845%20%3D%200.139)
The lower end of the interval is the sample mean subtracted by M. So it is 0.0312 - 0.139 = -0.1078
The upper end of the interval is the sample mean added to M. So it is 0.0312 + 0.139 = 0.1702
The 90% confidence interval for the difference between the proportions of Democrats and Republicans who approve of the mayor's performance is (-0.1078, 0.1702).