Answer:
When testing joint hypotheses for parameters, we look for appropriate critical values in the table of the chi-square distribution.
The chi-square distribution is commonly used in hypothesis testing when dealing with categorical data or testing the independence of variables. It is a probability distribution that takes on positive values only and is skewed to the right.
To understand the concept of critical values, let's consider an example. Suppose we want to test the hypothesis that the proportions of students who prefer different types of music are equal. We collect data from a sample of students and calculate the chi-square test statistic. To determine whether our test statistic is statistically significant, we compare it to the critical value from the chi-square distribution table.
The critical value represents the threshold beyond which we reject the null hypothesis. If our test statistic exceeds the critical value, we conclude that there is evidence to support the alternative hypothesis. On the other hand, if our test statistic is smaller than the critical value, we fail to reject the null hypothesis.
The chi-square distribution table provides critical values for different levels of significance (e.g., 0.05, 0.01). These values correspond to the areas in the tail of the chi-square distribution that define the rejection region for a given level of significance. By comparing our test statistic to the critical value in the table, we can determine whether our results are statistically significant.
In summary, when testing joint hypotheses for parameters, we use the chi-square distribution table to find the appropriate critical values. These critical values help us make decisions about rejecting or failing to reject the null hypothesis based on the test statistic calculated from our data.