Question 1200870
**a) Test for Equal Variances (Levene's Test)**

* **Hypothesis:**
    * **H0:** σ₁² = σ₂² (Variances are equal)
    * **H1:** σ₁² ≠ σ₂² (Variances are not equal)

* **Levene's Test:** 
    * Levene's Test is used to assess the equality of variances between two groups. 
    * You would typically use statistical software (like R, Python with libraries like SciPy, or software like SPSS or Minitab) to perform this test. 
    * The software will output a test statistic (often an F-statistic) and a p-value.

* **Decision:**
    * If the p-value from Levene's Test is greater than your chosen significance level (e.g., α = 0.05), you **fail to reject the null hypothesis**. This suggests that there is no significant evidence to conclude that the population variances are different. 
    * If the p-value is less than your chosen significance level, you **reject the null hypothesis**. This suggests that the population variances are likely different.

**b) Test for Equality of Means**

* **Based on the outcome of Levene's Test:**

    * **If variances are equal (Levene's Test not significant):**
        * Use the **t-test for independent samples assuming equal variances**. 
        * This test assumes that the populations have equal variances.

    * **If variances are not equal (Levene's Test significant):**
        * Use the **Welch's t-test for independent samples with unequal variances**. 
        * This test does not assume equal variances between the two groups.

* **Hypothesis:**
    * **H0:** μ₁ = μ₂ (Population means are equal)
    * **H1:** μ₁ ≠ μ₂ (Population means are not equal) 

* **Perform the t-test:**
    * Use the appropriate t-test (equal or unequal variances) based on the results of Levene's Test.
    * The software will provide a t-statistic, degrees of freedom, and a p-value.

* **Decision:**
    * If the p-value from the t-test is less than your chosen significance level (e.g., α = 0.05), you **reject the null hypothesis**. This suggests that there is a statistically significant difference between the population means.
    * If the p-value is greater than or equal to the significance level, you **fail to reject the null hypothesis**. This suggests that there is not enough evidence to conclude that the population means are different.

**Important Considerations:**

* **Software Implementation:** Use statistical software (like R, Python, SPSS, Minitab) to perform the Levene's Test and the appropriate t-test. These software packages will provide the necessary calculations and statistical output.
* **Assumptions:** Remember to check the assumptions of normality for both groups before conducting the t-tests. You can use graphical methods (histograms, Q-Q plots) or statistical tests (Shapiro-Wilk test) to assess normality.

By following these steps, you can determine whether there is a significant difference in the means of the two independent samples while appropriately accounting for the equality of variances.