# Likelyhood ratio test hypotheses and normal distribution

1. Feb 19, 2008

### Hummingbird25

1. The problem statement, all variables and given/known data

Given the normal distribution

$$X_{ij} \sim N(\mu_i, \omega^2)$$ where i = 1,2 and j = 1,..........,n

deduce that $$H_{0\mu}: \mu_1 = \mu _2$$

3. The attempt at a solution

Do I take in the Likelyhood function here?

and use it to analyse the case?

Sincerely Hummingbird

p.s. I have reading in Wiki that the Null hypo is rejected by the likehood ratio test, could be what I am expected to show here?

Last edited: Feb 19, 2008
2. Feb 19, 2008

### EnumaElish

You are taking 2 samples from 2 different normal distributions, where sample size is n for each sample. You are supposed to calculate the sample averages then test the null hyp. using a z-test, assuming their common variance $\omega^2$ is known (given).

If you don't know the variance you'll need to estimate it from the combined sample, then use a t-test for equality of means (assuming equal variances and equal sample sizes).

Last edited: Feb 19, 2008
3. Feb 19, 2008

### Hummingbird25

The sample average of the two norm distributions is that

$$\overline{x} = \frac{\sum_{i=1}^{2}f_i}{n}$$??

Sincerely
Hummingbird

Last edited: Feb 19, 2008
4. Feb 19, 2008

### EnumaElish

No.

$$\overline{x_1} = \frac{\sum_{j=1}^{n}x_{1j}}{n}$$

Same for i = 2.

5. Feb 19, 2008

### Hummingbird25

Hello again EnomaElish and thank you,

$$\overline{x_1} = \frac{\sum_{j=1}^{n}x_{1j}}{n}$$

$$\overline{x_2} = \frac{\sum_{j=1}^{n}x_{2j}}{n}$$

Then I say by the z-test then the null hypotheses is rejected if the variance isn't given since the samples aren't drawn from the same population.

But the null hypotheses is accepted if the means are equal which can be tested using the student t-test.

Is this it?

Sincerely Hummingbird

6. Feb 20, 2008

### EnumaElish

Is the variance given, or assumed known?