Question

finding 5 number summaries
(d) 3, 9, 14, 7, 12, 10, 13, 18, 20
5-number summary
min = ____
q1 = ____
median = ____
q3 = ____
max = ____

(e) 5, 7, 6, 12, 14, 10, 2, 3, 16
5-number summary
min = ____
q1 = ____
median = ____
q3 = ____
max = ____

(f) 20, 17, 3, 9, 5, 12, 16, 19, 10
5-number summary
min = ____
q1 = ____
median = ____
q3 = ____
max = ____

1, 4, 12, 15, 8, 10, 11, 17, 20
5-number summary
min = ____
q1 = ____
median = ____
q3 = ____
max = ____

19, 15, 7, 12, 5, 2, 17, 20, 3
5-number summary
min = ____
q1 = ____
median = ____
q3 = ____
max = ____

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Explanation:

Part (d)

Step1: Sort the data

Sort the data set \( 3, 9, 14, 7, 12, 10, 13, 18, 20 \) in ascending order: \( 3, 7, 9, 10, 12, 13, 14, 18, 20 \)

Step2: Find Min and Max

Min is the smallest value, so Min \( = 3 \). Max is the largest value, so Max \( = 20 \)

Step3: Find Median (Q2)

The number of data points \( n = 9 \) (odd). The median is the middle value, at position \( \frac{n + 1}{2}=\frac{9+ 1}{2}=5 \). The 5th value is \( 12 \), so Median \( = 12 \)

Step4: Find Q1 (first quartile)

Q1 is the median of the lower half (data before the median). Lower half: \( 3, 7, 9, 10 \) ( \( n = 4 \), even). Median of lower half: \( \frac{7 + 9}{2}=\frac{16}{2}=8 \), so Q1 \( = 8 \) (Wait, correction: Wait, the lower half for \( n = 9 \) is the first 4 values? Wait, no. When \( n \) is odd, the median is excluded from the lower and upper halves. So lower half: \( 3, 7, 9, 10 \) (4 values), upper half: \( 13, 14, 18, 20 \) (4 values). So Q1 is the median of \( 3, 7, 9, 10 \): average of 2nd and 3rd? Wait, no, for even number of values, median is average of middle two. So \( \frac{7+9}{2}=8 \)? Wait, no, 4 values: positions 1 - 4: 3 (1), 7 (2), 9 (3), 10 (4). The median of these 4 is the average of 2nd and 3rd: \( \frac{7 + 9}{2}=8 \). Correct.

Step5: Find Q3 (third quartile)

Q3 is the median of the upper half: \( 13, 14, 18, 20 \). Median of these 4 values: average of 2nd and 3rd: \( \frac{14+18}{2}=\frac{32}{2}=16 \), so Q3 \( = 16 \)

Step1: Sort the data

Sort \( 5, 7, 6, 12, 14, 10, 2, 3, 16 \) in ascending order: \( 2, 3, 5, 6, 7, 10, 12, 14, 16 \)

Step2: Find Min and Max

Min \( = 2 \), Max \( = 16 \)

Step3: Find Median

\( n = 9 \) (odd). Median at position \( \frac{9 + 1}{2}=5 \). 5th value is \( 7 \), so Median \( = 7 \)

Step4: Find Q1

Lower half (excluding median): \( 2, 3, 5, 6 \) (4 values). Median of lower half: \( \frac{3+5}{2}=\frac{8}{2}=4 \), so Q1 \( = 4 \)

Step5: Find Q3

Upper half (excluding median): \( 10, 12, 14, 16 \) (4 values). Median of upper half: \( \frac{12 + 14}{2}=\frac{26}{2}=13 \), so Q3 \( = 13 \)

Step1: Sort the data

Sort \( 20, 17, 3, 9, 5, 12, 16, 19, 10 \) in ascending order: \( 3, 5, 9, 10, 12, 16, 17, 19, 20 \)

Step2: Find Min and Max

Min \( = 3 \), Max \( = 20 \)

Step3: Find Median

\( n = 9 \) (odd). Median at position \( \frac{9+1}{2}=5 \). 5th value is \( 12 \), so Median \( = 12 \)

Step4: Find Q1

Lower half (excluding median): \( 3, 5, 9, 10 \) (4 values). Median of lower half: \( \frac{5 + 9}{2}=\frac{14}{2}=7 \), so Q1 \( = 7 \)

Step5: Find Q3

Upper half (excluding median): \( 16, 17, 19, 20 \) (4 values). Median of upper half: \( \frac{17+19}{2}=\frac{36}{2}=18 \), so Q3 \( = 18 \)

Answer:

Min \( = 3 \), Q1 \( = 8 \), Median \( = 12 \), Q3 \( = 16 \), Max \( = 20 \)

Part (e)