Likelyhood ratio test hypotheses and normal distributionby Hummingbird25 Tags: distribution, hypotheses, likelyhood, normal, ratio, test 

#1
Feb1908, 07:02 AM

P: 86

1. The problem statement, all variables and given/known data
Given the normal distribution [tex]X_{ij} \sim N(\mu_i, \omega^2)[/tex] where i = 1,2 and j = 1,..........,n deduce that [tex]H_{0\mu}: \mu_1 = \mu _2[/tex] 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? 



#2
Feb1908, 10:28 AM

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P: 2,483

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



#3
Feb1908, 12:00 PM

P: 86

[tex]\overline{x} = \frac{\sum_{i=1}^{2}f_i}{n}[/tex]?? Sincerely Hummingbird 



#4
Feb1908, 03:37 PM

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P: 2,483

Likelyhood ratio test hypotheses and normal distribution
No.
[tex]\overline{x_1} = \frac{\sum_{j=1}^{n}x_{1j}}{n}[/tex] Same for i = 2. 



#5
Feb1908, 03:50 PM

P: 86

Hello again EnomaElish and thank you,
[tex]\overline{x_1} = \frac{\sum_{j=1}^{n}x_{1j}}{n}[/tex] [tex]\overline{x_2} = \frac{\sum_{j=1}^{n}x_{2j}}{n}[/tex] Then I say by the ztest 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 ttest. Is this it? Sincerely Hummingbird 



#6
Feb2008, 08:48 AM

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P: 2,483

Is the variance given, or assumed known?
Possible answer 1: 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.) Possible answer 2: 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.) 


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