Testing that the mean of one group is greater than the other There are two independent normal variables:
$X\sim N(\mu_1,\sigma^2)$ and $Y\sim N(\mu_2,\sigma^2)$ with $\sigma^2$ unknown.
I want to test whether $\mu_1\gt\mu_2$ under the significance level of $10\%$.
I have two sets of data with the vector of (sample size, mean, standard deviation) respectively being $(101,75,14)$ and $(89,71,12.5)$
Here's my approach:
First calculate the pooled standard deviation:
$s^2_p=\frac{100*14*14+88*12.5*12.5}{101+89-2}=177.3936$
Secondly we calculate the t statistics: $t=\frac{75-71}{\sqrt{177.3936*\left(1/101+1/89\right)}}=2.06$
Then we compare the t statistics with $t_{188}(0.1)$, which is $1.286$.
Since $2.06\gt1.286$, we reject the null hypothesis that $\mu_1\gt\mu_2$
It feels kind of strange rejecting the null hypothesis when the sample mean of $X$ is greater than the sample mean of $Y$. But I don't know what I did wrong.
 A: 
It feels kind of strange rejecting the null hypothesis when the sample mean of $X$ is greater than the sample mean of $Y$. But I don't know what I did wrong.

What you are doing with this testresult is not rejecting the hypothesis $\mu_1 > \mu_2$  but instead the hypothesis $\mu_1 \leq \mu_2$.
So you found a result $\bar{x_1} > \bar{x_2}$ that corresponds with an effect. And you call it significant (measured with precision) because it is unlikely under the null (no effect) hypothesis.
The point of this hypothesis testing is to test whether some theory is correct by testing whether you find some unlikely deviations from the theory. But, often it is used when somebody already has an idea that there should be a deviation, and then the point of the hypothesis test is to be sure that the test to verify this predicted deviation is not occuring by mere chance.
The hypothesis test tells that the measurements are precise enough to be able to tell (statistically speaking) whether there is a difference. The result is statistically significant (whether the effect size is large enough to be called significant is another story).
