## Alpha = 0.05. Decision: p-value reject the null hypothesis.

### Please write out your hypothesis test before you look at mine.

Earlier, I said that your significance level α is thechance of being wrong that you can live with. Now I can be a littlemore precise. α is not the chance of *any* error;**α is the chance of a Type I error that you can live with.**If one Type I error in 20 hypothesis tests is unacceptable, use alower significance level — but then you make aType II error more likely. If *that’s*unacceptable, increase your sample size.

### With AGW, the problem is that the Hypotheses are written like this:

Comment: Even though you already have the sample data in theproblem, **when you write the hypotheses, ignore the sample.**In principle, you write the hypotheses, then plan the study and gatherdata. If you use any of the sample data in the hypotheses, somethingis wrong.

The above problems represent a comparison of a target or population variance with an observed sample variance, a comparison between several sample variances, or a comparison between frequency proportions. The standardized test statistic is called the Chi Square (χ^{2})test. Population variances are distributed according to the chi square distribution. Therefore, inferences about a single population variance will be based on chi square. The chi square test is widely used in two applications.

Case I. Comparing variances when the variance of the population is known.

Case ll. Comparing observed and expected frequencies of test outcomes when there is no defined population variance (attribute data).

When the population follows a normal distribution, the hypothesis tests for comparing a population variance, 0:, with a fixed value, 0:, are given by the following:

## The null and alternative hypotheses are:

In order to test a null hypothesis, a test calculation must be made from sample information. This calculated value is called a test statistic and is compared to an appropriate critical value. A decision can then be made to reject or not reject the null hypothesis.

## The test statistic calculation is:

The procedure employed in testing a hypothesis is strikingly similar to a court trial. The hypothesis is that the defendant is presumed not guilty until proven guilty. However, the term innocent does not apply to a null hypothesis. A null hypothesis can only be rejected, or fail to be rejected, it cannot be accepted because of a lack of evidence to reject it. If the means of two populations are different, the null hypothesis of equality can be rejected if enough data is collected. When rejecting the null hypothesis, the alternate hypothesis must be accepted.

## Therefore, the test statistic Z =

If a null hypothesis is established to test whether a sample value is smaller or larger than a population value, then the entire or risk is placed on one end of a distribution curve. This constitutes a one-tail test.

## Now it’s time to compute the test statistic (t) and thep-value.

Remember, in both cases here the Null Hypothesis doesn’t have to be *proved*. We just have to get to a point where we can no longer *fail to reject* it. Just like our Defendant in Example 2 above, we have to get to a point beyond reasonable doubt that we can *reject H0.* The burden on the scientist is **always** to *reject his own H1*.

## 3.0 - Hypothesis Testing | Statistics

The problem for many people comes in that when *B* happens, they also see *A* happening. What we don’t know, though, is 1) whether or not A *causes* B, B *causes* A, or neither (A and B may happen completely independent of one another). That’s why we write the default hypothesis the way that we do.