In Exercises 5–8, (a) find the expected frequency for each cell in the contingency table, (b) identify the claim and state H0 and Ha, (c) determine the degrees of freedom, find the critical value, and identify the rejection region, (d) find the chi-square test statistic, (e) decide whether to reject or fail to reject the null hypothesis, and (f) interpret the decision in the context of the original claim.


The contingency table shows the distribution of a random sample of fatal pedestrian and bicyclist motor vehicle collisions by time of day in a recent year. At α=0.10, can you conclude that the type of crash victim and the time of day are related? (Adapted from National Highway Traffic Safety Administration)

Verified step by step guidance

1. Step 1: Calculate the expected frequency for each cell in the contingency table. Use the formula for expected frequency: $E\; =\; (R\_i\; \times \; C\_j)\; /\; N$, where $R\_i$ is the row total, $C\_j$ is the column total, and $N$ is the grand total.
2. Step 2: Identify the claim and state the null hypothesis ($H\_0$) and alternative hypothesis ($H\_a$). The null hypothesis is that the type of crash victim and the time of day are independent, while the alternative hypothesis is that they are related.
3. Step 3: Determine the degrees of freedom using the formula: $df\; =\; (r\; -\; 1)(c\; -\; 1)$, where $r$ is the number of rows and $c$ is the number of columns. Then, find the critical value for the chi-square distribution at $\alpha \; =\; 0.10$ and identify the rejection region.
4. Step 4: Compute the chi-square test statistic using the formula: $\chi ^2\; =\; \Sigma ((O\; -\; E)^2\; /\; E)$, where $O$ is the observed frequency and $E$ is the expected frequency for each cell.
5. Step 5: Compare the chi-square test statistic to the critical value. If the test statistic falls in the rejection region, reject the null hypothesis; otherwise, fail to reject it. Finally, interpret the decision in the context of the original claim about the relationship between the type of crash victim and the time of day.

Key Concepts

Contingency Table

A contingency table is a type of data representation that displays the frequency distribution of variables. In this case, it shows the counts of fatal collisions categorized by the type of victim (pedestrian or bicyclist) and the time of day. This format allows for the analysis of the relationship between two categorical variables, which is essential for conducting chi-square tests.

Chi-Square Test

The chi-square test is a statistical method used to determine if there is a significant association between two categorical variables. It compares the observed frequencies in each category of a contingency table to the expected frequencies, which are calculated under the assumption that the variables are independent. The test results in a chi-square statistic that can be compared to a critical value to decide whether to reject the null hypothesis.

Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis (H0) represents a statement of no effect or no relationship, while the alternative hypothesis (Ha) suggests that there is an effect or a relationship. For this question, H0 would state that the type of crash victim and the time of day are independent, while Ha would claim that they are related. Formulating these hypotheses is crucial for guiding the analysis and interpreting the results of the chi-square test.