Consider the random sample of response times in Exercise 2....

Consider the random sample of response times in Exercise 2. Use the Kolmogorov-Smirnov and $A^2$ tests at the $5 \%$ level to test the hypothesis that the response time is normally distributed. Assume the mean and variance are not known.

Key Concepts

Parameter Estimation in Normal Distribution

When testing if data are normally distributed and the parameters, mean and variance, are unknown, they must be estimated from the sample. This estimation is critical because the goodness-of-fit tests like the KS and Anderson-Darling tests require a fully specified distribution to compare against the sample data. The parameter estimation process ensures that the tests are conducted under the most appropriate normal distribution model, even when the exact parameters are not known a priori.

Hypothesis Testing

Hypothesis testing is a framework used to decide whether to reject a null hypothesis based on sample data. In the context of normality tests, the null hypothesis typically states that the sample comes from a normal distribution. The tests yield test statistics and p-values that are compared against predetermined significance levels (e.g., 5%) to decide whether the evidence is sufficient to reject the null hypothesis.

Anderson-Darling (A^2) Test

The Anderson-Darling test is a statistical test of whether a given sample comes from a specified distribution, placing more emphasis on the tails of the distribution compared to other goodness-of-fit tests. It evaluates the squared differences between the observed and expected cumulative distributions, weighted to give more importance to the ends of the distribution. This sensitivity makes it particularly effective in detecting deviations from normality, especially in the tails.

Kolmogorov-Smirnov (KS) Test

The KS test is a non-parametric goodness-of-fit test used to determine if a sample comes from a specified distribution. It compares the empirical cumulative distribution function (ECDF) of the sample with the cumulative distribution function (CDF) of the proposed distribution, computing the maximum absolute difference. The test is sensitive to discrepancies in both location and shape of the distribution, making it useful for assessing distributional assumptions such as normality.

Goodness-of-Fit Testing

Goodness-of-fit testing involves statistical procedures used to determine how well a statistical model fits a set of observations. In the context of testing for normality, these tests assess whether the sample data conform to a normal distribution by comparing observed frequencies or characteristics with those expected under the hypothesized distribution. Both the KS and Anderson-Darling tests are examples of such goodness-of-fit tests.

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