# Hypothesis Testing? Probability

I'd be really grateful if someone can help me. I found this problem in the textbook Probability and Statistics by Wapole. This question look something similar to a Hypothesis test, but I have no clue how to solve it. I have tried re-reading parts of the book, but I still can't figure out how to do this.

For a) I figure that Z = (X-u)/(std/sqrt(n)) would give me the probability, so I know whether the amount is significant or low probability...But what do I then do for b)??

We have 30 samples, and the std of each sample is known. So the std of the mean is going to be $5 / \sqrt(30) \approx 1\%$ so, roughly applying the central limit theorem (be careful when applying the CLT ! it's full of traps ! learn about them), the distribution of the mean of the 30 samples is going to be a Gaussian, of std 1%, centered around $\mu_a$ Seeing a value that close to 65% for the realization of that variable, it's quite likely that the hypothesis that $\mu_a \geq 65$ is wrong
For the second part, we want to compute the probability of the difference of two Gaussians of std 1% to be bigger than 5.5. Very roughly, the difference of the two Gaussians will have std $\approx\sqrt(2)\approx1.5$ so that 5.5 is between 3 and 4 stds of deviation. That's a pretty good separation and is strong evidence for $\mu_a \neq \mu_b$
• If X and Y are two Gaussians of mean $\mu_x$ and $\mu_y$ and variance 1, then their difference is a Gaussian of mean the difference of the means $\mu_x - \mu_y$ and of variance the sum of the variance $1+1=2$, so the std is $\sqrt(2)$ Jul 3 '15 at 14:29