Question

In a random sample of 42 Democrats from one city, 10 approved of the mayor's performance. In a random sample of 58 Republicans from the city, 12 approved of the mayor's performance. Find a 90% confidence interval for the difference between the proportions of Democrats and Republicans who approve of the mayor's performance.

Answer 1

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).

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
`\mu` and standard deviation
`\sigma`, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
`\mu` and standard deviation
`s = (\sigma)/(√(n))`.

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
`\mu = p` and standard deviation
`s = \sqrt{(p(1-p))/(n)}`

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 = (10)/(42) = 0.2381, s_D = \sqrt{(0.2381*(1-0.2381))/(42)} = 0.0657`

In a random sample of 58 Republicans from the city, 12 approved of the mayor's performance.

This means that:

`p_R = (12)/(58) = 0.2069, s_R = \sqrt{(0.2069*(1-0.2069))/(58)} = 0.0532`

Distribution of the difference:

`p = p_D - p_R = 0.2381 - 0.2069 = 0.0312`

`s = √(0.0657^2+0.0532^2) = 0.0845`

Confidence interval:

We have that to find our
`\alpha` level, that is the subtraction of 1 by the confidence interval divided by 2. So:

`\alpha = (1 - 0.9)/(2) = 0.05`

Now, we have to find z in the Ztable as such z has a pvalue of
`1 - \alpha`.

That is z with a pvalue of
`1 - 0.05 = 0.95`, so Z = 1.645.

Now, find the margin of error M as such

`M = zs`

`M = 1.645*0.0845 = 0.139`

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).