A steel company is considering the relocation of one of its manufacturing plants. The company’s executives have selected three areas that they believe are suitable locations. However, they want to determine if the average wages are significantly different in any of the locations, since this could have a major impact on the cost of production. A survey of hourly wages of similar workers in each of the three areas is performed with the following results. Do the data indicate a significant difference among the average hourly wages in the three areas? Hourly Wages ($)Area 1Area 2Area 3

19

24

11

17

25

9

21

18

16

14

15

25

18

23

11

25

23

15

22

16

10

17

17

21 ind the value of the test statistic to test for a difference in the areas. Round your answer to two decimal places, if necessary.

This problem is asking whether there is a significant difference in the average hourly wages across the three areas. Since we are comparing the means of three different groups (Area 1, Area 2, and Area 3), we can use **ANOVA** (Analysis of Variance) to test for significant differences among the means of the groups.

### Steps:
1. **Null Hypothesis ($H_0$)**: The average hourly wages in the three areas are the same (i.e., $ \mu_1 = \mu_2 = \mu_3 $).
2. **Alternative Hypothesis ($H_A$)**: At least one of the areas has a different average hourly wage.

### Given Data:

- **Area 1**: 19, 17, 21, 14, 18, 25, 22, 17
- **Area 2**: 24, 25, 18, 15, 23, 23, 16, 17
- **Area 3**: 11, 9, 16, 25, 11, 15, 10, 21

### Steps for calculating ANOVA:
1. Calculate the **grand mean** (overall average wage).
2. Compute the **sum of squares for treatment (SST)** — this measures the variation between the group means.
3. Compute the **sum of squares for error (SSE)** — this measures the variation within the groups.
4. Calculate the **mean square for treatment (MST)** and **mean square for error (MSE)**.
5. Finally, compute the **F-statistic**: 
   \[
   F = \frac{\text{MST}}{\text{MSE}}
   \]
6. Compare the computed F-statistic to the critical value from the F-distribution table or use a p-value to determine significance.

Let me calculate these values for you.The value of the test statistic (F-statistic) is approximately **3.23**.

This F-value can be compared to the critical value from the F-distribution table for a given significance level (typically $\alpha = 0.05$) or used to compute the p-value to determine if the difference in the means is statistically significant.

Would you like to proceed with calculating the p-value or have more details about the process?

Here are 5 related questions:
1. How do we interpret the F-statistic in ANOVA tests?
2. What is the significance of the p-value in ANOVA?
3. How do we check assumptions in an ANOVA test?
4. What are post-hoc tests, and when should they be used?
5. What happens if the assumptions of ANOVA are violated?

**Tip:** ANOVA assumes that the data in each group follows a normal distribution and that the variances across the groups are equal (homoscedasticity). Always check these assumptions before interpreting results.

A steel company is considering the relocation of one of its manufacturing plants. The company’s executives have selected three areas that they believe are suitable locations. However, they want to determine if the average wages are significantly different in any of the locations, since this could have a major impact on the cost of production. A survey of hourly wages of similar workers in each of the three areas is performed with the following results. Do the data indicate a significant difference among the average hourly wages in the three areas? Hourly Wages ($) Area 1, Area 2, Area 3: 19, 24, 11, 17, 25, 9, 21, 18, 16, 14, 15, 25, 18, 23, 11, 25, 23, 15, 22, 16, 10, 17, 17, 21. Find the value of the test statistic to test for a difference in the areas. Round your answer to two decimal places, if necessary.

Explore how to perform an ANOVA test to determine if there are significant differences in the average hourly wages across three areas. Using real-world data from a steel company, this step-by-step guide calculates the F-statistic and walks through the process of hypothesis testing in statistics. Suitable for college-level statistics 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