# Hypothesis Testing For Two Sample Mean Proportion

We've two samples of size n_1 and n_2 from two different normally distributed populations. We don't know their population means but we know their population variances as (σ_1)^2 and (σ_2)^2.

Null hypothesis is,

H_0: μ_1 = μ_2 / 2


and alternative hypothesis is,

H_1: μ_1 < μ_2 / 2


How should we proceed in this hypothesis testing procedure? Which statistic model we should use?

Thanks!

• Just scale all the observations in the second sample by 0.5 and run the usual one-sided, 2 sample T-test with unequal variance. – AdamO Dec 4 '18 at 21:07
• What exactly do you mean by scaling the observations? Also, since we know population variances, can't we use 2 sample Z-test, why do we need to use T-test? @AdamO – Mert Akozcan Dec 4 '18 at 21:33
• Scaling: multiplying by a scalar. e.g. a duration in years can be converted to days by scaling by 365.25 – AdamO Dec 4 '18 at 21:46

If everything is independent, you can construct a sensible linear combination of the data which is normally distributed with known population variance and which would have population mean $$0$$ when the null hypothesis is true