let two stochastically independent random variables y1 and y2 with the distribution b(n1,p1) and b(n2,p2) respectively,how find a confidence interval for p1-p2 ?
To find a confidence interval for p1 - p2, we can use a method called the Wald interval. Here are the steps to calculate it:
1. Let's assume we have the following information:
- n1: the number of trials for the first random variable (y1) following the distribution B(n1, p1).
- n2: the number of trials for the second random variable (y2) following the distribution B(n2, p2).
- p1: the probability of success for the first random variable (y1).
- p2: the probability of success for the second random variable (y2).
- α: the significance level, which determines the width of the confidence interval.
2. The first step is to calculate the sample proportions for each random variable:
- Let x1 be the number of successful trials for y1.
- Let x2 be the number of successful trials for y2.
The sample proportions can be calculated as follows:
- p̂1 = x1 / n1
- p̂2 = x2 / n2
3. Calculate the standard error of the difference between the sample proportions:
- SE(p1 - p2) = sqrt((p̂1*(1-p̂1)) / n1 + (p̂2*(1-p̂2)) / n2)
4. Calculate the margin of error (MoE):
- MoE = z * SE(p1 - p2)
- For a 95% confidence interval, α = 0.05, and z ≈ 1.96.
5. Calculate the lower and upper bounds of the confidence interval:
- Lower Bound = (p̂1 - p̂2) - MoE
- Upper Bound = (p̂1 - p̂2) + MoE
6. Finally, the confidence interval for p1 - p2 will be the range between the lower and upper bounds.
Remember that this method assumes that the two random variables, y1 and y2, are stochastically independent.