Question

answer the statistical measures and create a box and whiskers plot for the following set of data. enter all values as whole numbers or decimals. 2,2,4,4,5,6,7,7,7,10,15,15,17,17,17 min: q1: med: q3: max: create the box plot by dragging the lines:

Explanation:

Step1: Find Minimum (Min)

The smallest value in the data set \(2,2,4,4,5,6,7,7,7,10,15,15,17,17,17\) is \(2\).

Step2: Find Median (Med)

There are \(n = 15\) data points. The median is the \(\frac{n + 1}{2}=\frac{15+ 1}{2}=8^{th}\) value when sorted. The sorted data: \(2,2,4,4,5,6,7,7,7,10,15,15,17,17,17\). The \(8^{th}\) value is \(7\).

Step3: Find Q1 (First Quartile)

The first quartile is the median of the lower half (excluding the median if \(n\) is odd). The lower half is \(2,2,4,4,5,6,7\) (7 values). The median of this set is the \(\frac{7 + 1}{2}=4^{th}\) value, which is \(4\).

Step4: Find Q3 (Third Quartile)

The third quartile is the median of the upper half (excluding the median if \(n\) is odd). The upper half is \(10,15,15,17,17,17\) (wait, no, original data after median: the upper half is \(10,15,15,17,17,17\)? Wait, no, for \(n = 15\), the lower half is first 7 values (\(2,2,4,4,5,6,7\)), upper half is last 7 values: \(10,15,15,17,17,17\)? Wait, no, correct upper half: after the \(8^{th}\) value (median \(7\)), the upper half is \(10,15,15,17,17,17\)? Wait, no, the data points are \(15\), so lower half: positions \(1 - 7\) (\(2,2,4,4,5,6,7\)), upper half: positions \(9 - 15\) (\(10,15,15,17,17,17\))? Wait, no, position \(8\) is median. So lower half: \(n_1=7\) values, median of lower half: \(\frac{7 + 1}{2}=4^{th}\) value in lower half. Lower half: \(2,2,4,4,5,6,7\), \(4^{th}\) value is \(4\). Upper half: \(10,15,15,17,17,17\) (wait, no, original data: after the \(8^{th}\) value (7), the remaining values are \(10,15,15,17,17,17\)? No, the data is \(2,2,4,4,5,6,7,7,7,10,15,15,17,17,17\). So upper half (values above median) are \(10,15,15,17,17,17\)? Wait, no, the median is at position 8 (value 7). The upper half is from position 9 to 15: \(10,15,15,17,17,17\) (6 values? Wait, no, \(n = 15\), so lower half: 7 values (positions 1 - 7), median at 8, upper half: 7 values (positions 9 - 15). Wait, position 9: 10, 10:15, 11:15, 12:17, 13:17, 14:17, 15:17? Wait, no, original data: let's count again: 1:2, 2:2, 3:4, 4:4, 5:5, 6:6, 7:7, 8:7, 9:7, 10:10, 11:15, 12:15, 13:17, 14:17, 15:17. Oh! I made a mistake earlier. The sorted data is \(2,2,4,4,5,6,7,7,7,10,15,15,17,17,17\). So \(n = 15\), median is at \(\frac{15 + 1}{2}=8^{th}\) value, which is \(7\) (correct). Lower half: values before the median (positions 1 - 7): \(2,2,4,4,5,6,7\) (7 values). Upper half: values after the median (positions 9 - 15): \(10,15,15,17,17,17\)? No, position 9 is \(7\)? Wait no, position 8 is \(7\), position 9 is \(7\), position 10 is \(10\). Wait, no, the data is: index 1:2, 2:2, 3:4, 4:4, 5:5, 6:6, 7:7, 8:7, 9:7, 10:10, 11:15, 12:15, 13:17, 14:17, 15:17. Ah! So the lower half is positions 1 - 7: \(2,2,4,4,5,6,7\) (7 values). The median of the lower half (Q1) is the \(\frac{7+1}{2}=4^{th}\) value, which is \(4\) (position 4: 4). The upper half is positions 9 - 15: \(7,10,15,15,17,17,17\)? Wait, no, position 8 is 7, position 9 is 7, position 10 is 10, position 11 is 15, position 12 is 15, position 13 is 17, position 14 is 17, position 15 is 17. So upper half (values above the median? No, quartiles: Q1 is median of lower 50%, Q3 is median of upper 50%. For \(n = 15\), lower 50% is first 7 values (since 15/2 = 7.5, so first 7), upper 50% is last 7 values. So upper half: positions 9 - 15: \(7,10,15,15,17,17,17\). The median of upper half (Q3) is the \(\frac{7 + 1}{2}=4^{th}\) value in upper half. Upper half data: \(7,10,15,15,17,17,17\), \(4^{th}\) value is \(15\).

Step5: Find Maximum (Max)

The largest value in the data set is \(17\).

Answer:

Min: \(2\), Q1: \(4\), Med: \(7\), Q3: \(15\), Max: \(17\)

For the box - plot:

• The left whisker starts at Min (\(2\)) and ends at Q1 (\(4\)).
• The box starts at Q1 (\(4\)), has a line at Med (\(7\)) and ends at Q3 (\(15\)).
• The right whisker starts at Q3 (\(15\)) and ends at Max (\(17\)).