Question

the data set 5, 6, 7, 8, 9, 9, 9, 10, 12, 14, 17, 17, 18, 19, 19 represents the number of hours spent on the internet in a week by students in a mathematics class. which box - and - whisker plot represents the data?

Explanation:

Step1: Find the minimum and maximum

The data set is \(5, 6, 7, 8, 9, 9, 9, 10, 12, 14, 17, 17, 18, 19, 19\). The minimum value (\( \text{min} \)) is \(5\) and the maximum value (\( \text{max} \)) is \(19\).

Step2: Find the median (second quartile, \(Q_2\))

There are \(n = 15\) data points. The median is the \(\frac{n + 1}{2} = 8^{\text{th}}\) term. The \(8^{\text{th}}\) term is \(10\), so \(Q_2 = 10\).

Step3: Find the first quartile (\(Q_1\))

The first quartile is the median of the lower half (excluding the median if \(n\) is odd). The lower half is \(5, 6, 7, 8, 9, 9, 9\) (7 terms). The median of this set is the \(4^{\text{th}}\) term, which is \(8\), so \(Q_1 = 8\).

Step4: Find the third quartile (\(Q_3\))

The third quartile is the median of the upper half (excluding the median if \(n\) is odd). The upper half is \(12, 14, 17, 17, 18, 19, 19\) (7 terms). The median of this set is the \(4^{\text{th}}\) term, which is \(17\), so \(Q_3 = 17\).

Now we analyze the box - and - whisker plots:

• The minimum should be \(5\), maximum \(19\), \(Q_1 = 8\), \(Q_2=10\), \(Q_3 = 17\).
• Looking at the plots, we check the positions of the minimum, \(Q_1\), \(Q_2\), \(Q_3\) and maximum. The correct plot should have the left whisker starting at \(5\), the box starting at \(8\) ( \(Q_1\) ), the median line at \(10\) ( \(Q_2\) ), the box ending at \(17\) ( \(Q_3\) ) and the right whisker ending at \(19\). By comparing with the given plots, we find that the correct plot is the one (let's assume the first plot is the correct one based on the calculations, but we need to match with the visual. Wait, maybe I made a mistake in the initial analysis. Wait, let's re - check:

Wait, the data set: \(n = 15\). The median is the 8th term. Let's list the data in order:

1:5, 2:6, 3:7, 4:8, 5:9, 6:9, 7:9, 8:10, 9:12, 10:14, 11:17, 12:17, 13:18, 14:19, 15:19

So lower half (first 7 terms: positions 1 - 7): 5,6,7,8,9,9,9. Median of lower half ( \(Q_1\) ) is the 4th term, which is 8.

Upper half (last 7 terms: positions 9 - 15): 12,14,17,17,18,19,19. Median of upper half ( \(Q_3\) ) is the 4th term, which is 17.

Minimum = 5, Maximum = 19, \(Q_1 = 8\), \(Q_2 = 10\), \(Q_3 = 17\).

Now, looking at the plots, we need to find the plot where the left whisker is at 5, the box starts at 8, the median is at 10, the box ends at 17 and the right whisker is at 19.

Assuming that plot 1 (the first one) has these values (since the x - axis is marked with 2,4,6,8,10,12,14,16,18,20). So the correct box - and - whisker plot should have minimum 5, \(Q_1 = 8\), \(Q_2 = 10\), \(Q_3 = 17\), maximum 19.

Answer:

The box - and - whisker plot with minimum 5, \(Q_1 = 8\), median ( \(Q_2\) ) = 10, \(Q_3 = 17\) and maximum 19 (the first plot, assuming the first plot in the options has these characteristics). If we consider the options as 1), 2), 3), 4), the correct one is 1) (assuming the first plot matches the calculated quartiles and min/max).