To calculate the 5-number summary of the dataset \(5, 6, 6, 7, 7, 8, 8, 8, 9, 9, 14, 16\), we need to find the minimum, first quartile (Q1), median (Q2), third quartile (Q3), and maximum.
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Minimum: The smallest value in the dataset is \(5\).
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Maximum: The largest value in the dataset is \(16\).
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Median (Q2): Since there are \(12\) values (an even number), the median is the average of the \(6^{th}\) and \(7^{th}\) values.
- The \(6^{th}\) value is \(8\) and the \(7^{th}\) value is also \(8\).
- Median = \(\frac{8 + 8}{2} = 8\).
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First Quartile (Q1): This is the median of the first half of the data (the first 6 values: \(5, 6, 6, 7, 7, 8\)).
- The \(3^{rd}\) and \(4^{th}\) values are \(6\) and \(7\).
- Q1 = \(\frac{6 + 7}{2} = 6.5\).
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Third Quartile (Q3): This is the median of the second half of the data (the last 6 values: \(8, 8, 9, 9, 14, 16\)).
- The \(3^{rd}\) and \(4^{th}\) values are \(9\) and \(9\).
- Q3 = \(\frac{9 + 9}{2} = 9\).
Now we summarize:
- Minimum: \(5\)
- Q1: \(6.5\)
- Median: \(8\)
- Q3: \(9\)
- Maximum: \(16\)
Thus, the third quartile (Q3) is \(9\).
From the choices provided, the correct answer for quartile Q3 is: 9.