Question

Use the information Give the rejection region for a chi-square test of specified probabilities if the experiment involves \(k\) categories. $$ k=5, \alpha=.05 $$

Short answer

Answer: The rejection region for a chi-square test with 5 categories and a significance level of 0.05 is when the chi-square test statistic is greater than or equal to 9.488.

Step-by-step solution

Identify the parameters

In this exercise, the given parameters are: Number of categories (k) = 5, Significance level (alpha) = 0.05.

Calculate the degrees of freedom

For a chi-square test, the degrees of freedom (df) is calculated as: df = k - 1. In this problem, df = 5 - 1 = 4.

Find the critical value

We need to find the chi-square critical value for the given significance level (alpha) and degrees of freedom (df). We can use a chi-square table or calculator for this. For df = 4 and alpha = 0.05, the critical value (and also the chi-square test statistic) is approximately 9.488.

Determine the rejection region

In a chi-square test, if the test statistic (computed value) is greater than or equal to the critical value, we reject the null hypothesis. Therefore, the rejection region is determined as: $$\chi^2 \geq 9.488$$

Conclusion

The rejection region for a chi-square test with specified probabilities, involving 5 categories and a significance level of 0.05, is when the chi-square test statistic is greater than or equal to 9.488.

Key concepts

Degrees of Freedom

In statistics, the term degrees of freedom (df) is crucial when determining the accuracy of certain test statistics, particularly in the chi-square test. Simply put, degrees of freedom are a measure of the amount of 'independence' present when estimating some parameter. In the context of chi-square tests—which are often used to compare observed data with data we would expect to obtain according to a specific hypothesis—the degrees of freedom are calculated by taking the number of categories (\(k\)) and subtracting one.

For example, if an experiment has 5 categories (\(k = 5\)), the degrees of freedom would be 4 (\(df = k - 1 = 5 - 1 = 4\)). This number is pivotal because it helps to determine the critical value from a chi-square distribution table, which in turn establishes the rejection region for the test.

Rejection Region

The rejection region defines the range of values for which the null hypothesis is not plausible and therefore should be rejected. It's essentially the 'danger zone' for the null hypothesis in any statistical test, where observed test statistics lead us to conclude that our initial assumption (the null hypothesis) may not hold true.

In a chi-square test, this rejection region is determined based on the critical value, which corresponds to the predetermined significance level, often denoted as \(\alpha\). If the test statistic calculated from the experiment is greater than or equal to this critical value, the data falls within the rejection region. For instance, given a critical value of 9.488 for a chi-square test with 4 degrees of freedom at a significance level of 0.05, the rejection region would be for any chi-square test statistic \(\chi^2\) that is \(\chi^2 \geq 9.488\).

Null Hypothesis

At the heart of any hypothesis-testing procedure is the null hypothesis, often represented as \(H_0\). This hypothesis posits that there is no effect or no difference, and it serves as a baseline assumption that is tested against experimental data. In chi-square tests, which are frequently used for goodness-of-fit tests, the null hypothesis typically asserts that the observed frequencies for a categorical variable match the expected frequencies.

The null hypothesis is either rejected or not rejected based on the test outcome. It is fundamental to understand that failing to reject the null hypothesis doesn't prove it's true; rather, it suggests that there is not enough evidence against it. The chi-square test provides a method for challenging the null hypothesis, by comparing the observed data with a chi-square distribution that corresponds to the degrees of freedom for the test. If our test statistic falls in the rejection region as previously described, we have grounds to reject the null hypothesis, suggesting that our observed data do not fit with what the null hypothesis would predict.