To determine the true statements based on the given data sets, let's calculate the means and standard deviations of both Data Set 1 and Data Set 2.
Data Set 1: 5, 5, 6, 6, 7, 7
Mean of Data Set 1:
\[
\text{Mean} = \frac{5 + 5 + 6 + 6 + 7 + 7}{6} = \frac{36}{6} = 6
\]
Standard Deviation of Data Set 1:
- Calculate the variance: \[ \text{Variance} = \frac{(5-6)^2 + (5-6)^2 + (6-6)^2 + (6-6)^2 + (7-6)^2 + (7-6)^2}{6} \] \[ = \frac{(1) + (1) + (0) + (0) + (1) + (1)}{6} = \frac{4}{6} = \frac{2}{3} \]
- Now the standard deviation is the square root of variance: \[ \text{SD} = \sqrt{\frac{2}{3}} \approx 0.8165 \]
Data Set 2: 1, 3, 5, 7, 9, 11
Mean of Data Set 2:
\[
\text{Mean} = \frac{1 + 3 + 5 + 7 + 9 + 11}{6} = \frac{36}{6} = 6
\]
Standard Deviation of Data Set 2:
- Calculate the variance: \[ \text{Variance} = \frac{(1-6)^2 + (3-6)^2 + (5-6)^2 + (7-6)^2 + (9-6)^2 + (11-6)^2}{6} \] \[ = \frac{(25) + (9) + (1) + (1) + (9) + (25)}{6} = \frac{70}{6} \approx 11.6667 \]
- Now the standard deviation is the square root of variance: \[ \text{SD} = \sqrt{\frac{70}{6}} \approx 2.6833 \]
Now we compare the results:
-
Means:
- Data Set 1 Mean = 6
- Data Set 2 Mean = 6
-
Standard Deviations:
- Data Set 1 SD ≈ 0.8165
- Data Set 2 SD ≈ 2.6833
True Statements
- Data Set 2 has the same mean as Data Set 1. (True)
- Data Set 2 has a larger standard deviation than Data Set 1. (True)
Conclusion
The two true statements are:
- Data Set 2 has the same mean as Data Set 1.
- Data Set 2 has a larger standard deviation than Data Set 1.