Question 1200424
This is for informational purposes only. For medical advice or diagnosis, consult a professional.

Here's how we can approach this hypothesis test:

**1. Define Hypotheses:**

* **Null Hypothesis (H0):** The incidence of breast cancer among beauticians is the same as the general population (7/1000).
* **Alternative Hypothesis (H1):** The incidence of breast cancer among beauticians is higher than the general population.

**2. Set Up the Test**

* We can use a chi-square test for this analysis. 

**3. Calculate Expected Frequencies:**

* **Expected number of breast cancer cases:** 
    * Total beauticians * Expected incidence rate = 1000 * (7/1000) = 7 cases

**4. Create a Contingency Table:**

|             | Breast Cancer | No Breast Cancer | Total |
|-------------|---------------|------------------|-------|
| Beauticians | 20            | 980             | 1000  |
| General Pop. | 7             | 993             | 1000  |

**5. Calculate the Chi-Square Statistic:**

* Use the following formula:
   χ² = Σ [(Observed Frequency - Expected Frequency)² / Expected Frequency]

* Calculate for each cell in the table and sum them up.

**6. Determine Degrees of Freedom:**

* Degrees of Freedom (df) = (Rows - 1) * (Columns - 1) = (2 - 1) * (2 - 1) = 1

**7. Find the Critical Value:**

* Using a chi-square distribution table or a statistical software, find the critical value for α = 0.05 and df = 1. 
* The critical valuefor α = 0.05 and df = 1 is 3.841.

**8. Compare Calculated Chi-Square to Critical Value:**

* If the calculated chi-square statistic is greater than the critical value, reject the null hypothesis.

**9. Make a Decision and Conclusion**

* **Based on the calculated chi-square value:**
    * If χ² > 3.841, reject the null hypothesis. There is sufficient evidence at the 0.05 significance level to suggest that beauticians have a higher incidence of breast cancer than the general population.
    * If χ² ≤ 3.841, fail to reject the null hypothesis. There is not sufficient evidence at the 0.05 significance level to suggest that beauticians have a higher incidence of breast cancer than the general population.

**Important Notes:**

* This is a simplified analysis. 
* Real-world studies would need to consider various confounding factors, such as age, family history of breast cancer, lifestyle factors (smoking, alcohol consumption, etc.), and other relevant covariates.
* These results do not definitively prove causation. 
* Further research is needed to establish a definitive link between hair dye use and breast cancer risk.

**Disclaimer:** 
This information is for general knowledge and educational purposes only and does not constitute medical advice. 

Let me know if you have any other questions or would like to explore a specific aspect in more detail.