How is the rejection region defined and how is that related to the z-score and t

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How is the rejection region defined and how is that related to the z-score and the p value? When do you reject or fail to reject the null hypothesis? Why do you think statisticians are asked to complete hypothesis testing? Can you think of examples in courts, in medicine, or in your area?

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The rejection region is the range of values that, under a specified confidence level, it is impossible for the null hypothesis to be a valid answser.The "rejection region" is whatever tail(s) of the normal probability distribution.For instance, if you are conducting a "one-tailed test" where the alternate 
hypothesis alleges that some average is (let's say) HIGHER than a particular 
value, and you want to test it at alpha = 0.05, then the "rejection region" is 
the highest 5% of the sample means that would result if the ACTUAL average 
were exactly the particular value mentioned in the null hypothesis. 

Using the z-score from the given data, you find the level of possibility (the p value) for a given value at H-zero, and if that p is less than the threshold specified in the problem (usually as α=0.05), then H-zero is rejected.

If p is greater than that line, then there is a possibility that H-zero can be possible, not proven true, but accepted as a possibility.
Statisticians find this kind of hypothesis testing as a key part of the scientific method. If a hypothesis can be proven false in the math on the drawing board, it can save a lot of money in the testing stage.
There was a classic case in California of a blonde woman with a ponytail and a black man with a beard and moustache who drove a yellow car who were arrested and convicted after a mugging. Prosecutors used the facts that these two were near, and their features that matched the description made it a surety that they were the ones who committed the crime. They used probabilities that a the car was yellow was 1 in 10, that the man had a beard was 1 in 4, and that she had a ponytail was 1 in 10, and so on to prove that the chances that it was not them to be more than 12 million to one, so since there are not 12 million people in San Pedro, these must be the ones. Under appeal, the board stated that "In any event, we think that under the circumstances the "trial by mathematics" so distorted the role of the jury and so disadvantaged counsel for the defense, as to constitute in itself a miscarriage of justice," and overturned the verdict.

 Another kind of example that comes to mind is the sort of tests that have to be submitted to FDA when seeking approval for a new drug. It has to be proven that using the drug makes a significant difference to patients' well-being.

Enovnannm A (570)
UT Austin

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