Question

Give the rejection region for a chi-square test of independence if the contingency table involves \(r\) rows and \(c\) columns. $$r=2, c=4, \alpha=.05$$

Short answer

Answer: The rejection region for this chi-square test of independence is when the test statistic is greater than 7.815 (χ² > 7.815).

Step-by-step solution

1. Determine the degrees of freedom

To find the degrees of freedom for the chi-square test of independence, use the formula: $$df = (r-1)(c-1)$$ where \(r\) is the number of rows and \(c\) is the number of columns in the contingency table. In this case, \(r=2\) and \(c=4\). Substitute the values of \(r\) and \(c\) into the formula: $$df = (2-1)(4-1)$$

2. Calculate the degrees of freedom

Perform the calculations inside the parentheses first, and then multiply the results: $$df = (1)(3) = 3$$ So, the degrees of freedom for this chi-square test of independence is 3.

3. Find the critical value from the chi-square distribution table

To find the critical value for the chi-square test that corresponds to an alpha level of 0.05, look up the value in a chi-square distribution table using the degrees of freedom (3) and the alpha level (0.05). From the table, the critical value is approximately 7.815.

4. Determine the rejection region

The rejection region for the chi-square test of independence is defined as the area of the chi-square distribution where the test statistic would fall if the null hypothesis were to be rejected. For an alpha level of 0.05, the rejection region is: $$\chi^2 > 7.815$$ So, if the calculated chi-square test statistic is greater than 7.815, we would reject the null hypothesis.

Key concepts

Contingency Table

A contingency table, also known as a cross-tabulation or crosstab, is a type of table that displays the frequency distribution of variables to help determine the relationship between them.
In terms of the chi-square test of independence, such a table summarises the observed frequencies of data according to two categorical variables. For instance, in an exercise requiring a contingency table with 2 rows and 4 columns, this implies that we're examining the relationship between two variables where one has 2 categories and the other has 4 categories.

Understanding Rows and Columns

Rows in a contingency table represent the categories of one variable, while columns represent the categories of another. The cells of the table show how many data points fit into each cross-category setting, allowing us to discern patterns or associations in a visually intuitive and structured manner.

Degrees of Freedom

Degrees of freedom (df) are a concept critical to various statistical tests, including the chi-square test of independence. They represent the number of values that can vary in the calculation while estimating statistical parameters.

For chi-square tests involving contingency tables, degrees of freedom are calculated using the formula:
\[df = (r-1)(c-1)\]
where 'r' is the number of rows and 'c' is the number of columns in the table. The degrees of freedom provide a sense of how much 'room' there is for variability in the data and is crucial for determining the critical value from the chi-square distribution table.

Calculation Example

In practice, for a table with 2 rows and 4 columns, the degrees of freedom would be calculated as \(df = (2-1)(4-1) = 3\), meaning three independent values can vary.

Critical Value

The critical value in statistics is a threshold that determines the boundary between the rejection and non-rejection regions of a statistical test. For a chi-square test of independence, the critical value depends on the desired significance level (usually denoted as alpha or \( \alpha \)) and the degrees of freedom.

For example, at a significance level of 0.05 and 3 degrees of freedom, the critical value can be found using a chi-square distribution table. This value helps to decide whether to reject the null hypothesis.

Relevance to Hypothesis Testing

The critical value is pivotal as it anchors the conclusion of the hypothesis test. If the calculated test statistic is greater than the critical value, the implication is that the null hypothesis may not hold, suggesting an association between the variables under study.

Rejection Region

The rejection region is the range of values for which the null hypothesis is rejected in a statistical test. In the context of a chi-square test of independence, the rejection region is found at the tail end of the chi-square distribution, beyond the critical value.

For example, with a critical value of 7.815, the rejection region for a significance level of 0.05 would be characterized by the set of all chi-square values larger than this critical value.

Determining Rejection or Non-Rejection

The actual computation in a chi-square test yields a test statistic. If this statistic falls within the rejection region (\( \chi^2 > 7.815 \) in the example), this suggests that there's a low probability that the observed data would occur if the null hypothesis were true, leading to the rejection of the null hypothesis and a conclusion that there is a statistically significant association between the variables.