0
$\begingroup$

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!

$\endgroup$
  • $\begingroup$ 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 '18 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 $\endgroup$ – Mert Akozcan Dec 4 '18 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 '18 at 21:46
1
$\begingroup$

Hint:

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.