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## Principles of Hypothesis Testing

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Hypothesis (goodness-of-fit) testing is a common method that uses statistical evidence from a sample to draw a conclusion about a population. In hypothesis testing, you assert a particular statement (a null hypothesis) and try to find evidence to support or reject that statement. A null hypothesis is an assumption about a population that you would like to test. It is “null” in the sense that it often represents a status-quo belief, such as the absence of a characteristic or the lack of an effect. You can formalize it by asserting that a population parameter, or a combination of population parameters, has a certain value. MuPAD® enables you to test the following null hypotheses:

• The data has the distribution function `f`. If `f` is a cumulative distribution function (CDF), you can use the classical chi-square goodness-of-fit test or the Kolmogorov-Smirnov test. If `f` is probability density function (PDF), a discrete probability function (PF), or an arbitrary distribution function, use the classical chi-square goodness-of-fit test.

• The data has a normal distribution function with a particular mean and a particular variance. For cumulative distribution functions, use the classical chi-square goodness-of-fit test or the Kolmogorov-Smirnov test. For other distribution functions, use the classical chi-square goodness-of-fit test.

• The data has a normal distribution function (with unknown mean and variance). Use the Shapiro-Wilk goodness-of-fit test for this hypothesis.

• The mean of the data is larger than some particular value. Use the t-Test for this hypothesis.

The main result returned by the hypothesis tests is the p-value (`PValue`). The p-value of a test indicates the probability, under the null hypothesis, of obtaining a value of the test statistic as extreme or more extreme than the value computed from the sample. If the p-value is larger than the significance level (stated and agreed upon before the test), the null hypothesis passes the test. A typical value of a significance level is 0.05. P-values below a significance level provide strong evidence for rejecting the null hypothesis.

#### Mathematical Modeling with Symbolic Math Toolbox

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