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?


  • $\begingroup$ 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. $\endgroup$
    – AdamO
    Dec 4, 2018 at 21:07
  • $\begingroup$ 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 $\endgroup$ Dec 4, 2018 at 21:33
  • $\begingroup$ Scaling: multiplying by a scalar. e.g. a duration in years can be converted to days by scaling by 365.25 $\endgroup$
    – AdamO
    Dec 4, 2018 at 21:46

1 Answer 1



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

The alternative hypothesis would then point at which tail(s) to look at


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