Answer:
90% confidence interval for the difference in true proportion of the two groups is (-0.0717, 0.0517).
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
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.
First group: Sample of 163, 13% has a second episode.
This means that:
Second group: Sample of 160, 14% has a second episode
This means that:
Distribution of the difference:
Confidence interval:
The confidence interval is:
In which
z is the zscore that has a pvalue of .
90% confidence level
So , z is the value of Z that has a pvalue of , so .
Lower bound:
Upper bound:
90% confidence interval for the difference in true proportion of the two groups is (-0.0717, 0.0517).