Question

Use the information Give the rejection region for a chi-square test of specified probabilities if the experiment involves \(k\) categories. $$ k=10, \alpha=.01 $$

Short answer

Answer: The rejection region for a chi-square test with 10 categories and a significance level of 0.01 is all values of the test statistic greater than 21.666.

Step-by-step solution

Identify the degrees of freedom

In a chi-square test, the degrees of freedom are given by the number of categories minus 1, or \(df=k-1\). In this case, we have \(k=10\) categories, so \(df=10-1=9\).

Find the critical value for the chi-square distribution

We need to find the critical value of the chi-square distribution with 9 degrees of freedom, corresponding to a significance level of \(\alpha=0.01\). We can find this value using a chi-square table or a statistical calculator. The critical value for this case will be denoted as \(\chi^2_{0.99,9}\). Using a chi-square table or calculator, we find that the critical value is approximately \(\chi^2_{0.99,9}=21.666\).

Determine the rejection region

The rejection region for the chi-square test is given by all values of the test statistic greater than the critical value. In this case, the rejection region is for all values of the test statistic greater than \(\chi^2_{0.99,9}=21.666\).

Write the final answer

The rejection region for a chi-square test involving \(k=10\) categories and a significance level of \(\alpha=0.01\) is all values of the test statistic greater than \(\chi^2_{0.99,9}=21.666\).

Key concepts

Degrees of Freedom

The concept of \textbf{degrees of freedom} (df) is pivotal in statistical analyses, particularly when working with the chi-square test. Simply put, degrees of freedom refer to the number of values in a calculation that are free to vary. It’s much like having a certain number of choices, with one choice inevitably being determined by the others. In a chi-square test, we calculate the degrees of freedom by subtracting one from the number of categories, or levels, in the categorical variable being tested.

For instance, consider an experiment with 10 categories. To calculate the degrees of freedom, we would use the formula:
\[ df = k - 1 \]
where \(k\) is the number of categories. Hence, for our 10-category experiment, the degrees of freedom would be:
\[ df = 10 - 1 = 9 \]

This calculation helps set the stage for understanding the chi-square distribution, which depends heavily on the correct number of degrees of freedom to accurately test hypotheses about categorical data.

Critical Value

In statistical tests, the \textbf{critical value} is the threshold to which we compare our test statistic to determine whether to accept or reject the null hypothesis. The critical value acts as a cutoff point, separating the likely sample outcomes, under the assumption that the null hypothesis is true, from the unlikely ones. When you’re working with the chi-square test, the critical value is specifically denoted for a certain level of significance, \(\alpha\), and a certain amount of degrees of freedom, \(df\), which we previously calculated.

To find the critical value for a chi-square test, we use the chi-square distribution table or statistical software with our \(df\) and our chosen \(\alpha\). If we set \(\alpha = 0.01\) for a test with \(df = 9\), we're seeking the chi-square value which has only a 1% chance of being exceeded if the null hypothesis is true. Statistically, this value can be denoted as \(\chi^2_{0.99,9}\), which, in this case, is approximately \(21.666\). It marks the boundary beyond which we would regard the observed variation as too extreme to be attributable to chance alone.

Rejection Region

The \textbf{rejection region} is the range of values for which we reject the null hypothesis in hypothesis testing. It's effectively the 'danger zone' for the null hypothesis, where the observed data are considered too unusual to have occurred by random chance, given that the null hypothesis is true. When performing a chi-square test, once the critical value is determined, the rejection region is then defined as all test statistic values greater than this critical value.

In the example provided with \(k=10\) categories and a significance level of \(\alpha=0.01\), we already outlined the critical value as \(21.666\). Consequently, the rejection region is every chi-square value that surpasses this limit. If the calculated chi-square statistic from our experiment is greater than \(21.666\), we enter the rejection region and must reject the null hypothesis. Through this criterion, we're saying that the observed categorical data significantly deviate from what we would expect under the null hypothesis, suggesting an underlying pattern or relationship at work rather than random chance.