An insurance company wants to know if the color of an automobile has a relationship with the number of moving violations. The following contingency table gives the results of data collected from police reports across the nation. The columns list the numbers of reported moving violations in a year. Use a level of significance of α=0.01α=0.01 to conduct a hypothesis test of association between automobile color and the number of moving violations. Observed Sample of Cars0–12–3More Than 3TotalWhite

52

33

30

115

Black

75

77

26

178

Red

25

22

21

68

Silver

39

41

18

98

Other

52

67

35

154

Total

243

240

130

613 Step 3 of 4 :  

Compute the value of the test statistic. Round your answer to three decimal places, if necessary. (**Note:** If calculating by hand, be sure to use at least six decimal places in numbers for all intermediate calculations to ensure accuracy of the result.)

To conduct a hypothesis test of association between automobile color and the number of moving violations, we can use the **Chi-Square Test of Independence**. We are provided with a contingency table, and we need to compute the value of the test statistic.

Here is a step-by-step approach:

### Step 1: Set up the observed values (O)
We already have the observed values from the contingency table.

| **Color/Violations** | **0–1** | **2–3** | **More than 3** | **Total** |
|----------------------|---------|---------|-----------------|-----------|
| **White**            | 52      | 33      | 30              | 115       |
| **Black**            | 75      | 77      | 26              | 178       |
| **Red**              | 25      | 22      | 21              | 68        |
| **Silver**           | 39      | 41      | 18              | 98        |
| **Other**            | 52      | 67      | 35              | 154       |
| **Total**            | 243     | 240     | 130             | 613       |

### Step 2: Compute the expected values (E)
The expected value for each cell is calculated using the formula:

\[
E = \frac{(\text{Row Total}) \times (\text{Column Total})}{\text{Grand Total}}
\]

We'll calculate the expected values for each cell:

\[
E_{ij} = \frac{(\text{Row Total for } i) \times (\text{Column Total for } j)}{613}
\]

For example, the expected value for the first cell (White, 0–1 violations) is:

\[
E_{White, 0-1} = \frac{115 \times 243}{613} = 45.585
\]

We'll compute all the expected values.

### Step 3: Compute the Chi-Square test statistic
The Chi-Square test statistic is given by:

\[
\chi^2 = \sum \frac{(O - E)^2}{E}
\]

Where:
- \( O \) is the observed value
- \( E \) is the expected value

Let me calculate this now.The value of the test statistic for the Chi-Square test is \( \chi^2 = 16.862 \), rounded to three decimal places.

Would you like to see the next steps or have any questions on the process so far?

Here are five related questions that might interest you:
1. How do we determine the critical value for the Chi-Square test?
2. What are the degrees of freedom for this test?
3. How do we interpret the results of the Chi-Square test?
4. What assumptions are made when using the Chi-Square test of independence?
5. How would the test result change if we used a different significance level?

**Tip:** Ensure your expected values are sufficiently large (at least 5) to meet the assumptions of the Chi-Square test for reliable results.

An insurance company wants to know if the color of an automobile has a relationship with the number of moving violations. The following contingency table gives the results of data collected from police reports across the nation. The columns list the numbers of reported moving violations in a year. Use a level of significance of α=0.01 to conduct a hypothesis test of association between automobile color and the number of moving violations. Observed Sample of Cars: 0–1, 2–3, More Than 3. Compute the value of the test statistic.

Conduct a hypothesis test of association between automobile color and the number of moving violations using the Chi-Square Test of Independence. The problem involves a contingency table with reported moving violations in different color groups. Learn how to compute expected values, the Chi-Square test statistic, and analyze the relationship between variables. Suitable for advanced high school or undergraduate students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation