Question

1. the weights, in pounds, of the 24 members of a varsity football team are listed below. 144 192 142 150 160 181 131 164 141 151 140 149 144 155 156 147 168 178 159 130 159 148 171 163 a) fill in the 5 - number summary. b) create the box - and - whisker plot: c) in what range do we find the weights of players that are in the highest quarter of the data? d) what are the weights of players in the central 50% of the data? e) which quarter shows the greatest range (spread) of data? 2. the humanities division recorded the number of students signed up for the study abroad program each quarter. the results are listed below. 58 49 11 39 70 47 42 38 44 56 52 64 68 59 63 36 34 45 51 50 a) create a stem - and - leaf display of the data. be sure to include a key. b) what is the median of the data set? c) what is the range of the data?

Explanation:

Step1: Sort the football - team weight data

130, 131, 140, 141, 142, 144, 144, 147, 148, 149, 150, 151, 155, 156, 159, 159, 159, 160, 163, 164, 168, 171, 178, 181, 192

Step2: Find the lowest value

The lowest value is 130.

Step3: Calculate Q1

Since \(n = 24\), the position of Q1 is \(\frac{n + 1}{4}=\frac{24+1}{4}=6.25\). The 6th value is 144 and the 7th value is 144. So \(Q1=\frac{144 + 144}{2}=144\).

Step4: Calculate the median

The position of the median for \(n = 24\) is \(\frac{n}{2}=12\) and \(\frac{n}{2}+1 = 13\). The 12th value is 151 and the 13th value is 155. So the median \(M=\frac{151+155}{2}=153\).

Step5: Calculate Q3

The position of Q3 is \(\frac{3(n + 1)}{4}=\frac{3\times(24 + 1)}{4}=18.75\). The 18th value is 160 and the 19th value is 163. So \(Q3=\frac{160+163}{2}=161.5\).

Step6: Find the highest value

The highest value is 192.

a)

Statistic	Value
Q1	144
Median	153
Q3	161.5
Highest	192

b) To create a box - and - whisker plot:

• Draw a number line that includes the range from 130 to 192.
• Mark the lowest value (130) with a dot or a small circle.
• Mark Q1 (144) with a vertical line.
• Mark the median (153) with a vertical line.
• Mark Q3 (161.5) with a vertical line.
• Mark the highest value (192) with a dot or a small circle.
• Draw a box from Q1 to Q3 and draw whiskers from the box to the lowest and highest values.

c) The highest quarter of the data is from Q3 to the highest value. So the range is from 161.5 to 192.
d) The central 50% of the data is from Q1 to Q3. So the weights are in the range 144 to 161.5.
e)

• The range of the first quarter (from lowest to Q1): \(144-130 = 14\).
• The range of the second quarter (from Q1 to median): \(153 - 144=9\).
• The range of the third quarter (from median to Q3): \(161.5 - 153 = 8.5\).
• The range of the fourth quarter (from Q3 to highest): \(192-161.5 = 30.5\).

The fourth quarter shows the greatest range of data.

For the study - abroad data:

Step1: Sort the study - abroad data

11, 34, 36, 38, 39, 42, 44, 45, 47, 49, 50, 51, 52, 56, 58, 59, 63, 64, 68, 70

Step2: Create a stem - and - leaf display

Stem	Leaf
3	4 6 8 9
4	2 4 5 7 9
5	0 1 2 6 8 9
6	3 4 8
7	0

Key: \(3|4\) means 34.

Step3: Calculate the median

Since \(n = 20\), the position of the median is \(\frac{n}{2}=10\) and \(\frac{n}{2}+1 = 11\). The 10th value is 49 and the 11th value is 50. So the median \(M=\frac{49 + 50}{2}=49.5\).

Step4: Calculate the range

The highest value is 70 and the lowest value is 11. The range \(R=70 - 11=59\).

b) The median of the data set is 49.5.
c) The range of the data is 59.

Answer:

a)

Statistic	Value
Q1	144
Median	153
Q3	161.5
Highest	192

b) (Box - and - whisker plot description as above)
c) 161.5 to 192
d) 144 to 161.5
e) Fourth quarter
For study - abroad data:
a)

Stem	Leaf
3	4 6 8 9
4	2 4 5 7 9
5	0 1 2 6 8 9
6	3 4 8
7	0

Key: \(3|4\) means 34.
b) 49.5
c) 59