Question 1192439
**1. Calculate Sample Means and Standard Deviations**

* **Sample 1:**
    * Sample mean (x̄1) = sum(X1) / n1 = 960 / 16 = 60
    * Sample variance (s1²) = (sum(x1²) - (sum(X1))² / n1) / (n1 - 1) 
                        = (38140 - (960)² / 16) / (16 - 1) 
                        = (38140 - 57600) / 15 
                        = -129.33 
    * Sample standard deviation (s1) = √s1² = √(-129.33) 
        * Note: Since the sample variance is negative, there might be an error in the provided data. 

* **Sample 2:**
    * Sample mean (x̄2) = sum(X2) / n2 = 450 / 9 = 50
    * Sample variance (s2²) = (sum(X2²) - (sum(X2))² / n2) / (n2 - 1) 
                        = (22700 - (450)² / 9) / (9 - 1) 
                        = (22700 - 22500) / 8 
                        = 25
    * Sample standard deviation (s2) = √s2² = √25 = 5

**2. Calculate the Standard Error of the Difference**

* Standard Error (SE) = √[(s1²/n1) + (s2²/n2)] 
    * SE = √[(-129.33/16) + (25/9)] 
    * SE = √[-8.08 + 2.78] 
    * SE = √(-5.3) 
        * Note: The standard error is imaginary due to the negative sample variance in Sample 1. This indicates an issue with the provided data.

**3. Determine the Critical Value**

* For a 90% confidence interval, the critical value (Z-score) is 1.645.

**4. Calculate the Margin of Error**

* Margin of Error (ME) = Z-score * SE 
    * Since the standard error is imaginary, the margin of error cannot be calculated.

**5. Construct the Confidence Interval**

* Confidence Interval = (x̄1 - x̄2) ± ME 
    * Due to the issues with the data (negative sample variance), the confidence interval cannot be calculated.

**Conclusion**

* There appears to be an error in the provided data for Sample 1, as the calculated sample variance is negative. 
* Due to this error, the standard error and margin of error cannot be calculated, and therefore, the 90% confidence interval for the difference between the two population means cannot be determined.

**Recommendations**

* Double-check the provided data for Sample 1, especially the sum of squares of X1. 
* If the data is corrected, the calculations can be repeated to obtain the correct confidence interval.

**Important Note:** This analysis assumes that the two populations are normally distributed and independent.