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Theoretical Issues About Testing of Population Mean

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How do hypothesis-testing procedures differ for the population mean when the population standard deviation is known or unknown? What is the relationship between two-sided hypothesis tests for means and confidence intervals?

Why would the population variability and the sample size affects the power of a test? How do the level of significance and the power of a test relate to each other?
Hint: We are dealing with hypothesis testing issues.

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Solution Summary

This solution deals with certain theoretical issues about testing of population mean. The test construction when population standard deviation is known (normal test) and when population standard deviation is unknown (students t test) are discussed. The relation between power and significance level, influence of population variance and sample size on the power of the test etc. is also discussed. This solution is 420 words and provided as an attached Word document.

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Consider a sample of size n taken from a normal population with mean  and standard deviation  i.e., from . Let be the sample mean and be the sample variance. It has been proved that
(1)
(Chi square distribution with n-1 d.f) (2)
Hence it is clear that
(3)
and

(Students t with n - 1 d.f) (4)

Q1. Consider the testing of population mean i.e., against
If the population standard deviation  is known one can use the z statistic given by equation (3) as the test statistic so that the critical values can be obtained from the standard ...

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