SOLUTION: 5.Summary is obtained from two independent Normal samples Sample1 Sample2 Sample size 20

Question 1200870: 5.Summary is obtained from two independent Normal samples

Sample1 Sample2
Sample size 20 40
Sample mean 28 30
Sample standard deviation 8 10

a) Test whether one can assume equal variances.
b) With a suitable test procedure, test for equality of mean
Answer by asinus(45) (Show Source): You can put this solution on YOUR website!
**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.

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