Workers and​ senior-level bosses were asked if it was seriously unethical to monitor employee​ e-mail. The results are summarized in the table to the right. Use a 0.05 significance level to test the claim that the response is independent of whether the subject is a worker or a boss.
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Yes
No
 
Workers
198
242
Bosses
40
87


To test the claim that the response (whether monitoring employee email is unethical) is independent of whether the subject is a worker or a boss, we can use a **Chi-Square Test of Independence**. Here's how to perform the test:

### Step 1: State the Hypotheses
- **Null Hypothesis (H₀):** The response is independent of whether the subject is a worker or a boss.
- **Alternative Hypothesis (H₁):** The response is dependent on whether the subject is a worker or a boss.

### Step 2: Construct the Contingency Table

|                  | Yes | No  | Total |
|------------------|-----|-----|-------|
| **Workers**      | 198 | 242 |  440  |
| **Bosses**       |  40 |  87 |  127  |
| **Total**        | 238 | 329 |  567  |

### Step 3: Calculate the Expected Frequencies
For each cell in the table, the expected frequency is calculated using the formula:

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

**Calculating expected frequencies:**

1. **For Workers, Yes:**
\[
\text{Expected Frequency} = \frac{(440 \times 238)}{567} \approx 184.7
\]

2. **For Workers, No:**
\[
\text{Expected Frequency} = \frac{(440 \times 329)}{567} \approx 255.3
\]

3. **For Bosses, Yes:**
\[
\text{Expected Frequency} = \frac{(127 \times 238)}{567} \approx 53.3
\]

4. **For Bosses, No:**
\[
\text{Expected Frequency} = \frac{(127 \times 329)}{567} \approx 73.7
\]

### Step 4: Calculate the Chi-Square Test Statistic
The test statistic is calculated using the formula:

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

Where:
- \( O \) = Observed frequency
- \( E \) = Expected frequency

**Calculating each term:**

1. **For Workers, Yes:**
\[
\frac{(198 - 184.7)^2}{184.7} \approx 0.96
\]

2. **For Workers, No:**
\[
\frac{(242 - 255.3)^2}{255.3} \approx 0.67
\]

3. **For Bosses, Yes:**
\[
\frac{(40 - 53.3)^2}{53.3} \approx 3.26
\]

4. **For Bosses, No:**
\[
\frac{(87 - 73.7)^2}{73.7} \approx 2.53
\]

**Total Chi-Square Statistic:**
\[
\chi^2 \approx 0.96 + 0.67 + 3.26 + 2.53 = 7.42
\]

### Step 5: Determine the Critical Value and Make a Decision
- **Degrees of Freedom (df):** 
  \[
  \text{df} = (r - 1) \times (c - 1) = (2 - 1) \times (2 - 1) = 1
  \]

- **Critical Value for \(\chi^2\) at \( \alpha = 0.05 \) and 1 df:** From the chi-squared table, the critical value is 3.841.

### Step 6: Conclusion
- If the calculated \(\chi^2\) is greater than the critical value, reject the null hypothesis.
- Since \( \chi^2 = 7.42 \) > 3.841, we reject the null hypothesis.

### **Conclusion:**
There is sufficient evidence to conclude that the response to whether monitoring employee email is unethical is dependent on whether the subject is a worker or a boss.

---

Would you like more details on any part of this process, or do you have any other questions?

### Follow-up Questions:
1. How do you interpret the chi-square statistic in other contexts?
2. What happens if the significance level is changed from 0.05 to 0.01?
3. How would you conduct this test if you had more categories for the responses?
4. What are some common applications of the chi-square test in real-world scenarios?
5. How does the chi-square test relate to other statistical tests for independence?

### Tip:
Always ensure your expected frequencies are large enough (generally at least 5) to validate the use of the chi-square test.

Explore the Chi-Square Test of Independence to determine if monitoring employee email is perceived as unethical based on worker and boss responses. Includes step-by-step calculations, contingency table construction, and interpretation of results at a significance level of 0.05.

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