To evaluate the statements based on the given data sets, we'll first calculate the mean and standard deviation for both Data Set A and Data Set B.
Data Set A: 4, 5, 6, 12, 13
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Mean of Data Set A: \[ \text{Mean} = \frac{4 + 5 + 6 + 12 + 13}{5} = \frac{40}{5} = 8 \]
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Standard Deviation of Data Set A: \[ \text{Variance} = \frac{(4-8)^2 + (5-8)^2 + (6-8)^2 + (12-8)^2 + (13-8)^2}{5} \] \[ = \frac{16 + 9 + 4 + 16 + 25}{5} = \frac{70}{5} = 14 \] So, the standard deviation \( \sigma_A = \sqrt{14} \approx 3.74 \).
Data Set B: 4, 5, 7, 9, 10
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Mean of Data Set B: \[ \text{Mean} = \frac{4 + 5 + 7 + 9 + 10}{5} = \frac{35}{5} = 7 \]
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Standard Deviation of Data Set B: \[ \text{Variance} = \frac{(4-7)^2 + (5-7)^2 + (7-7)^2 + (9-7)^2 + (10-7)^2}{5} \] \[ = \frac{9 + 4 + 0 + 4 + 9}{5} = \frac{26}{5} = 5.2 \] So, the standard deviation \( \sigma_B = \sqrt{5.2} \approx 2.28 \).
Now, let's evaluate the statements:
- Data set A has the same mean as Data set B. - False (Mean of A is 8 and Mean of B is 7).
- Data set A has a larger standard deviation than Data set B. - True (Standard deviation of A is approximately 3.74 and B is approximately 2.28).
- Data set A has a smaller standard deviation than Data set B. - False (as just calculated).
- Data set A has a larger mean than data set B. - True (Mean of A is 8 and Mean of B is 7).
Thus, the two true statements are:
- Data set A has a larger standard deviation than Data set B.
- Data set A has a larger mean than data set B.