To determine the true statements, we need to calculate the means and standard deviations for both data sets.
Data set A: 4, 5, 6, 12, 13
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Calculating the mean: \[ \text{Mean}_A = \frac{4 + 5 + 6 + 12 + 13}{5} = \frac{40}{5} = 8 \]
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Calculating the standard deviation:
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First, find the deviations from the mean:
- \( 4 - 8 = -4 \)
- \( 5 - 8 = -3 \)
- \( 6 - 8 = -2 \)
- \( 12 - 8 = 4 \)
- \( 13 - 8 = 5 \)
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Square the deviations:
- \( (-4)^2 = 16 \)
- \( (-3)^2 = 9 \)
- \( (-2)^2 = 4 \)
- \( (4)^2 = 16 \)
- \( (5)^2 = 25 \)
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Sum of squared deviations: \[ 16 + 9 + 4 + 16 + 25 = 70 \]
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Variance: \[ \text{Variance}_A = \frac{70}{5} = 14 \]
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Standard deviation: \[ \text{Standard deviation}_A = \sqrt{14} \approx 3.74 \]
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Data set B: 4, 5, 7, 9, 10
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Calculating the mean: \[ \text{Mean}_B = \frac{4 + 5 + 7 + 9 + 10}{5} = \frac{35}{5} = 7 \]
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Calculating the standard deviation:
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First, find the deviations from the mean:
- \( 4 - 7 = -3 \)
- \( 5 - 7 = -2 \)
- \( 7 - 7 = 0 \)
- \( 9 - 7 = 2 \)
- \( 10 - 7 = 3 \)
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Square the deviations:
- \( (-3)^2 = 9 \)
- \( (-2)^2 = 4 \)
- \( (0)^2 = 0 \)
- \( (2)^2 = 4 \)
- \( (3)^2 = 9 \)
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Sum of squared deviations: \[ 9 + 4 + 0 + 4 + 9 = 26 \]
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Variance: \[ \text{Variance}_B = \frac{26}{5} = 5.2 \]
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Standard deviation: \[ \text{Standard deviation}_B = \sqrt{5.2} \approx 2.28 \]
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Comparison of means and standard deviations:
- Mean of Data set A: 8
- Mean of Data set B: 7
- Standard deviation of Data set A: approximately 3.74
- Standard deviation of Data set B: approximately 2.28
True Statements:
- Data set A has a larger mean than Data set B. (True: 8 > 7)
- Data set A has a larger standard deviation than Data set B. (True: approximately 3.74 > approximately 2.28)
The false statements are:
- Data set A has a smaller standard deviation than Data set B. (False)
- Data set A has the same mean as Data set B. (False)
So, the correct responses are:
- Data set A has a larger standard deviation than Data set B.
- Data set A has a larger mean than data set B.