I have two samples. $S_1$ has mean of $83.17$, with standard deviation of $15.50$. $S_2$ has a mean of $84.00$ and standard deviation of $23.95$. How do you calculate the probability that $S_1$ is greater than $S_2$ by some value $x$ or more?
How to calculate the probability that the value of $S_1$ is $x$ amount higher than vaule of $S_2$
1 Answers
We need some assumptions which have not been stated. First we need the assumption of independence. We will also assume that our random variables have normal distribution. The sample sizes are not specified. We will assume they are large. If they are quite small (and specified) the problem can be solved, but is more complicated.
Let $S_1$ and $S_2$ be independent normally distributed random variables, with means $\mu_1$ and $\mu_2$, and variances $\sigma_1^2$ and $\sigma_2^2$ respectively. Let $W=S_1-S_2$. Then $W$ has normal distribution, with mean $\mu_1-\mu_2$, and variance $\sigma_1^2+\sigma_2^2$ (the plus sign here is correct).
Calculate the mean and variance of $W$. Then find the standard deviation of $W$. Since $W$ has normal distribution, you have arrived at a familiar kind of problem.
Remark: The standard deviations seem awfully high.