Beginner to hypothesis testing. Have removed/replaced terminology, but these samples are from real-world populations doing a specific real world action where whichever option they picked splits them (binary choice). Samples are 1000 of each population, population for group 1 - 30100, population for group 2 - 75330.
Sample 1 mean 0.472, std 0.269, var 0.072
Sample 2 mean 0.418, std 0.243, var 0.059
$H_0$ : Sample 1 mean - sample 2 mean > 0
$H_1$ : Sample 1 mean - sample 2 mean <= 0
alpha=0.05, therefore norm.s.inv(0.95)=1.644
.
Observed sample means, $0.472-0.418=0.054$
$1.644 \cdot \sqrt{(0.072/1000)+(0.059/1000)} = 0.018$
$0.054$ is greater than $0.018$, therefore there is than 5% chance of getting this difference in sample means assuming the $H_0$, so we reject the null hypothesis.
However, depending who you ask, they'd switch the two hypothesis, so I tried that too, and it's basically exactly the same outcome.
$H_0$ : Sample 1 mean - sample 2 mean <= 0
$H_1$ : Sample 1 mean - sample 2 mean > 0 (note switched hypothesis)
alpha=0.05, therefore norm.s.inv(0.95)=1.644.
Observed sample means, 0.472-0.418=0.054
$1.644 \cdot \sqrt{(0.072/1000)+(0.059/1000)} = 0.018$
$0.054$ is greater than $0.018$, therefore there is than 5% chance of getting this difference in sample means assuming the $H_0$, so we reject the null hypothesis.
So I've rejected both null hypothesis here, which can't possibly be the case. What am I missing when it comes to interpretation of the hypothesis at the end?
Thanks.