# Likelyhood ratio test hypotheses and normal distribution

• Hummingbird25
In summary, the conversation discusses using the normal distribution and likelihood function to test the null hypothesis that two sample means are equal. If the common variance is known, the z-test for equality of means should be used. If the common variance is not known, it should be calculated from the combined sample and then the t-test for equality of means should be used.
Hummingbird25

## Homework Statement

Given the normal distribution

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

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

## 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:
Hummingbird25 said:
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?
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:
EnumaElish said:
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).

The sample average of the two norm distributions is that

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

Sincerely
Hummingbird

Last edited:
No.

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

Same for i = 2.

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

Is the variance given, or assumed known?

Yes, the variance is given (or the problem assumes it is known).
What you need to do: use the z test for equality of means to determine whether or not the two means are equal. (You are not supposed to use the t test in this case.)

No, the variance is not given (nor does the problem assume the variance is known).
What you need to do: calculate the common variance from the combined sample. Then use the t test for equality of means to determine whether or not the two means are equal. (You are not supposed to use the z test in this case.)

## What is a likelihood ratio test?

A likelihood ratio test is a statistical method used to compare two or more statistical models. It is used to determine whether one model fits the data significantly better than another.

## How is a likelihood ratio test used to test hypotheses?

A likelihood ratio test is used to compare the likelihood of the data under the null hypothesis (H0) to the likelihood of the data under the alternative hypothesis (HA). If the likelihood under HA is significantly greater than under H0, then the null hypothesis is rejected, indicating that the alternative hypothesis is a better fit for the data.

## What is a normal distribution?

A normal distribution, also known as a Gaussian distribution, is a probability distribution that is symmetric and bell-shaped. It is commonly used to model continuous data in many fields, and is characterized by its mean and standard deviation.

## How is a normal distribution related to a likelihood ratio test?

A likelihood ratio test assumes that the data follows a normal distribution, meaning that the data is continuous and symmetric. This assumption is necessary for the test to accurately compare the likelihood of the data under different statistical models.

## What are the assumptions of a likelihood ratio test?

The main assumptions of a likelihood ratio test are that the data follows a normal distribution, and that the models being compared are nested (i.e. one model is a special case of the other). Additionally, the sample size should be large enough for the test to be valid, and the observations should be independent of each other.

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