Question

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

Short answer

Answer: The rejection region for this test is when the chi-square test statistic is greater than or equal to 3.841.

Step-by-step solution

Calculate the degrees of freedom (df)

To calculate the degrees of freedom for this chi-square test, we can use the formula: $$df = (r - 1) \times (c - 1)$$ Plugging in the given values \(r=2\) and \(c=2\), we have: $$df = (2 - 1) \times (2 - 1) = 1$$

Determine the critical chi-square value using a significance level of α=0.05

To find the critical chi-square value for a significance level of α=0.05 and 1 degree of freedom, we can use a chi-square distribution table or an online calculator. From the chi-square distribution table, we get the critical value as: $$\chi^2_{critical} = 3.841$$

Interpret the rejection region

The rejection region for a chi-square test of independence is the region where the observed chi-square test statistic would lead us to reject the null hypothesis (no association between the row and column variables). With the calculated critical chi-square value and α=0.05, the rejection region is: $$\chi^2 \ge 3.841$$ This means that if the calculated chi-square test statistic is greater than or equal to 3.841, we reject the null hypothesis suggesting that there is a significant association between the row and column variables.

Key concepts

Degrees of Freedom

In statistics, the concept of degrees of freedom is pivotal when working with various statistical tests, including the chi-square test of independence. Degrees of freedom, often abbreviated as 'df', can be thought of as the number of values in a calculation that are free to vary. To put it simply, it's a way of understanding how much 'freedom' there is within the data to change, given certain constraints applied to the dataset.

In the context of the chi-square test, which is often used to examine the relationship between categorical variables, the degrees of freedom are typically calculated based on the number of categories within each variable. For a contingency table with rows and columns, the formula to determine degrees of freedom is $$ df = (r - 1) \times (c - 1) $$where 'r' is the number of rows and 'c' is the number of columns. In the given exercise, with a 2x2 table, we have one degree of freedom, as calculated by $$ df = (2 - 1) \times (2 - 1) = 1 $$Understanding the degrees of freedom is critical because it directly influences the critical value we choose for determining the rejection region in hypothesis testing.

Critical Chi-Square Value

The critical chi-square value is a key component when utilizing a chi-square test of independence. This value acts as a threshold to decide whether the observed data is significantly different from the expected data under the null hypothesis. To select the proper critical value, we factor in the significance level of the test, often denoted by alpha (α), and the degrees of freedom in our data.

The significance level reflects the probability of rejecting the null hypothesis when it's actually true—commonly set at 0.05 (or 5%). This level demonstrates a balance between being too lenient and too strict when deciding if our results are due to mere chance or not. For the exercise's scenario with a significance level of 0.05 and 1 degree of freedom, the critical value obtained from a chi-square distribution table is approximately 3.841. Thus,
$$ chi^2_{critical} = 3.841 $$
To enhance understanding, it can be helpful to visualize the chi-square distribution and note how an increase in degrees of freedom affects the shape of the curve and the placement of the critical value.

Rejection Region

The rejection region is the set of all possible values of the test statistic that would lead to a rejection of the null hypothesis. Essentially, it's where we decide that our observed statistic is too extreme to be compatible with the assumption that there is no effect or no association—known as the null hypothesis. In the context of the chi-square test, our test statistic compares observed data with a theoretical distribution.

When we talk about the rejection region, we're looking at values that fall into the 'tails' of the distribution—often on the right side for a chi-square test because the distribution is not symmetrical and is bounded at zero on the left. For our given 2x2 table problem, with a critical value of 3.841, any chi-square test statistic that we calculate and find to be higher than 3.841 would land in the rejection region. Formally, the rejection region can be expressed as:
$$ chi^2 \text{ test statistic} eq 3.841 $$
This means that an observed chi-square statistic greater than 3.841 suggests a significant association between the variables in our table, leading us to reject the null hypothesis.