Answer:
The 90% confidence interval is (0.0131, 0.0845).
Explanation:
Before finding 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.
Old process:
125 out of 580, so:


New process:
130 out of 780. So


Distribution of the difference:


Confidence interval:

In which
z is the z-score that has a p-value of
.
90% confidence level
So
, z is the value of Z that has a p-value of
, so
.
The lower bound of the interval is:

The upper bound of the interval is:

The 90% confidence interval is (0.0131, 0.0845).