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Statistics / Mathematical Statistics and Data Analysis 3 / Chapter 11 / Problem 10

Mathematical Statistics and Data Analysis | 3rd Edition | ISBN: 9788131519547 | Authors: John A. Rice

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Verify that the two-sample t test at level of H0: X = Y

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QUESTION:

Verify that the two-sample t test at level \(\alpha\) of \(H_0: \mu_X=\mu_Y\) versus \(H_A: \mu_X \neq \mu_Y\) rejects if and only if the confidence interval for \(\mu_X-\mu_Y\) does not contain zero.

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QUESTION:

Verify that the two-sample t test at level \(\alpha\) of \(H_0: \mu_X=\mu_Y\) versus \(H_A: \mu_X \neq \mu_Y\) rejects if and only if the confidence interval for \(\mu_X-\mu_Y\) does not contain zero.

ANSWER:

Step 1 of 4

This should be clear intuitively?the confidence interval is the set of all values for which we are fairly certain they could be the true value of the population parameter. Therefore, if the difference between the two population means, , is our parameter of interest, and if the confidence interval for that parameter does not contain the value 0, then we can be quite sure that is not the real value of , i.e. is not equal to .

In other words, we would reject the hypothesis that the two population means are equal, and would conclude that

Formally, let

be the test statistic used for testing the stated hypotheses, where and are sample sizes of the first and second sample, respectively.

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