Discuss the Null Hypothesis and the Alternate Hypothesis. How does one know when to use the One-Tailed Test or the Two-Tailed Test? What is a Type I error? How does one determine the test statistic and p-value? How does one determine the critical value and draw a conclusion?

There is a multi-step procedure for hypothesis testing that can be applied to determine the type of test, test statistic, critical value(s), and the appropriate response.

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Lets describe a five step method for hypothesis testing:

(1) Describe the hypotheses and the claim. There will be a null-hypothesis `H_0` which typically says that the parameter under study is as reported. Thus, the null-hypothesis for a mean is that the population mean is the mean reported. The null-hypothesis for linear regression is that the variables have no linear relationship.

The alternative hypothesis `H_1` is that there is some statistically significant difference in the true parameter. In the case of testing for the mean, we can say that the mean is less than or greater than the reported mean. (These are one-tailed tests— they go in one direction.) Or we can say that the true mean differs from the reported mean which is a two-tailed test. (It could be greater than or less than.)

One of the hypotheses is said to be the claim. Note that we can never prove the null-hypothesis (only present enough evidence to doubt it), nor can we disprove the alternative hypothesis.

(2) In the traditional method, we then calculate the critical value(s). (One value for a single tail test, two for a two-tailed test.) This value depends on `alpha` or the level of confidence. `alpha` essentially dictates how likely it is that we will make a type I error; that is, to reject a true null hypothesis. (In the US justice system, we are presumed innocent [the null-hypothesis]. A type I error is convicting an innocent person. A type II error, here releasing someone who is guilty, is harder to control.)

(3) We determine the test statistic. If we are testing a mean, we either use a z-test or t-test, typically. We use z if we know the population standard deviation and use t if all we have is the sample standard deviation. For population proportions we use z, and for standard deviation and variance we use `chi^2` .

(4) Using the traditional method, we check the test statistic against the critical values. If the test statistic is as extreme or more extreme than the critical value, we reject the null hypothesis; otherwise, we do not reject the null hypothesis. If using the p-value method, we compare p to `alpha` ; if `p<alpha` we reject the null hypothesis, as the probability of getting such an extreme sample by chance is small.

(The p-value is the probability of getting a statistic like the one in the sample, or more extreme, by chance alone.)

(5) We draw our conclusion based on the claim and whether we rejected the null hypothesis or not.

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