Question 1181427
Here's how to construct a 95% confidence interval for the difference between the population means, assuming equal variances:

**1. Calculate the Sample Means:**

*   **Drug A:** (3.5 + 5.7 + 3.4 + 6.9 + 17.8 + 3.8 + 3.0 + 6.4 + 6.8 + 3.6 + 6.9 + 5.7) / 12 = 6.275
*   **Drug B:** (4.5 + 11.7 + 10.8 + 4.5 + 6.3 + 3.8 + 6.2 + 6.6 + 7.1 + 6.4 + 4.5 + 5.1 + 3.2 + 4.7 + 4.5 + 3.0) / 16 = 5.8

**2. Calculate the Sample Standard Deviations:**

*   **Drug A:**  Use your calculator or statistical software. The sample standard deviation is approximately 4.22.
*   **Drug B:** Use your calculator or statistical software. The sample standard deviation is approximately 2.34.

**3. Calculate the Pooled Standard Deviation:**

Since we assume equal variances, we calculate a pooled standard deviation to get a better estimate of the common population standard deviation.

*   s_p = sqrt[((n_A - 1) * s_A^2 + (n_B - 1) * s_B^2) / (n_A + n_B - 2)]
*   s_p = sqrt[((12 - 1) * 4.22^2 + (16 - 1) * 2.34^2) / (12 + 16 - 2)]
*   s_p ≈ sqrt[(11 * 17.8084 + 15 * 5.4756) / 26]
*   s_p ≈ sqrt(277.6224 / 26)
*   s_p ≈ 3.27

**4. Determine the Critical t-Value:**

*   Degrees of freedom (df) = n_A + n_B - 2 = 12 + 16 - 2 = 26
*   For a 95% confidence interval and a two-tailed test, the alpha level is 1 - 0.95 = 0.05.  We then divide alpha by 2 since it is a two-tailed test, giving us 0.025.
*   Using a t-table or calculator, find the t-value associated with df = 26 and α/2 = 0.025. The critical t-value is approximately 2.056.

**5. Calculate the Margin of Error:**

*   Margin of Error = t * s_p * sqrt(1/n_A + 1/n_B)
*   Margin of Error = 2.056 * 3.27 * sqrt(1/12 + 1/16)
*   Margin of Error ≈ 2.056 * 3.27 * 0.354
*   Margin of Error ≈ 2.37

**6. Construct the Confidence Interval:**

*   (x̄_A - x̄_B) ± Margin of Error
*   (6.275 - 5.8) ± 2.37
*   0.475 ± 2.37

**7. Final Answer:**

The 95% confidence interval for the difference between the population means is approximately (-1.90, 2.85).