To evaluate the statements based on Data Set A (4, 5, 6, 12, 13) and Data Set B (4, 5, 7, 9, 10), we need to calculate the means and the standard deviations of both data sets.
Step 1: Calculate the means.
For Data Set A:
- Mean (A) = (4 + 5 + 6 + 12 + 13) / 5 = 40 / 5 = 8
For Data Set B:
- Mean (B) = (4 + 5 + 7 + 9 + 10) / 5 = 35 / 5 = 7
Step 2: Compare the means.
- Mean of A: 8
- Mean of B: 7
So, Data Set A has a larger mean than Data Set B.
Step 3: Calculate the standard deviations.
For Data Set A:
- Find the variance:
- Mean = 8
- Variance = [(4-8)² + (5-8)² + (6-8)² + (12-8)² + (13-8)²] / 5
- = [16 + 9 + 4 + 16 + 25] / 5
- = 70 / 5 = 14
- Standard deviation = √14 ≈ 3.74
For Data Set B:
- Find the variance:
- Mean = 7
- Variance = [(4-7)² + (5-7)² + (7-7)² + (9-7)² + (10-7)²] / 5
- = [9 + 4 + 0 + 4 + 9] / 5
- = 26 / 5 = 5.2
- Standard deviation = √5.2 ≈ 2.28
Step 4: Compare the standard deviations.
- Standard deviation of A: ≈ 3.74
- Standard deviation of B: ≈ 2.28
Data set A has a larger standard deviation than Data set B.
Conclusion: Based on our calculations:
- Statement A (Data set A has a smaller standard deviation than Data set B) is false.
- Statement B (Data set A has a larger mean than data set B) is true.
- Statement C (Data set A has a larger standard deviation than Data set B) is true.
- Statement D (Data set A has the same mean as Data set B) is false.
Therefore, the two true statements are:
- B. Data set A has a larger mean than data set B.
- C. Data set A has a larger standard deviation than Data set B.