Question 1192439: two independent samples of size n1 = 16 and n2 = 9 from two populations given
sum of X1 = 960 sum of x1 square + 38140, sum of X2 = 450 and sum of X2 square + 22700
find 90% confidence interval for difference betweenn two population means
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! **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.
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