Solution

(i) Back-to-back stem-and-leaf diagram:

Key: \( 1 \mid 9 \mid 4 \) means \(91\ \text{kg}\) (Team A) and \(94\ \text{kg}\) (Team B).

(ii) Interquartile range for Team A:

Ordered weights: 82, 84, 84, 91, 96, 97, 98, 99, 100, 104, 107, 109, 115, 116, 122.

\( Q_1 = 91,\quad Q_3 = 109 \)

\( \text{IQR} = Q_3 - Q_1 = 109 - 91 = 18\ \text{kg}. \)

(iii) New player in Team B:

\( \sum_{15} = 1399 \)

\( \dfrac{1399 + x}{16} = 93.9 \)

\( 1399 + x = 16 \times 93.9 = 1502.4 \)

\( x = 1502.4 - 1399 = 103.4 \ \text{kg} \approx 103\ \text{kg}. \)

To create a stem-and-leaf diagram, separate each number into a stem (the leading digit) and a leaf (the trailing digit). For example, the number 23 is split into a stem of 2 and a leaf of 3.

Step 1: Organize the data

• Stems: 1, 2, 3, 4
• Leaves for stem 1: 4, 5, 7, 8, 9, 9
• Leaves for stem 2: 1, 2, 2, 3, 4, 5, 6, 6, 8, 8
• Leaves for stem 3: 0, 2, 6, 8
• Leaves for stem 4: 1, 2, 5, 6, 7

Step 2: Construct the diagram

Step 3: Key

Key: \( 1 \mid 4 \) represents \( 14 \) glasses of water.

Summary:

• The lowest number of glasses is 14 and the highest is 47.
• Most values are clustered in the 20s, indicating a typical weekly consumption between 20–30 glasses.
• The stem-and-leaf diagram provides a clear visualization of the distribution and helps identify patterns quickly.

Solution

(i) The back-to-back stem-and-leaf diagram is constructed as follows:

Key: \( 3 \mid 5 \mid 4 \) represents \( 53 \) seconds for adults and \( 54 \) seconds for children.

(ii) Comparisons:

• Children's estimates are more spread out than adults'.
• Adults' estimates are generally lower than children's.
• Adults' estimates are roughly symmetrical, whereas children's estimates are skewed to the right.

Solution

(i) Construct the stem-and-leaf diagram:

Key: \(1 \mid 4\) represents \$140.

(ii) Find the median and interquartile range:

Ordered data: 10, 40, 60, 80, 100, 130, 140, 140, 140, 150, 150, 150, 160, 160, 160, 160, 170, 180, 180, 200, 210, 250, 270, 280, 310, 450, 570

\(\text{Median} = 160\) (14th value)

\(\text{LQ} = 140\) (7th value), \(\text{UQ} = 210\) (21st value)

\(\text{IQR} = \text{UQ} - \text{LQ} = 210 - 140 = 70\)

(iii) Identify outliers:

\(1.5 \times \text{IQR} = 1.5 \times 70 = 105\)

\(\text{Lower limit} = 140 - 105 = 35\)

\(\text{Upper limit} = 210 + 105 = 315\)

Outliers: 10, 450, 570

(i) To create a stem-and-leaf diagram, separate the thousands and hundreds (stem) from the tens and units (leaf) for each price.

Sort the prices: 14 300, 15 300, 15 400, 15 500, 16 100, 16 400, 16 800, 16 800, 17 300, 17 700, 18 500.

Stem-and-leaf diagram:

Key: \(14 \mid 3\) represents \(14300\) dollars.

(ii) To find the lower quartile (Q1), arrange the prices in ascending order and find the median of the first half.

Sorted prices: 14 300, 15 300, 15 400, 15 500, 16 100, 16 400, 16 800, 16 800, 17 300, 17 700, 18 500.

The first half: 14 300, 15 300, 15 400, 15 500, 16 100.

The median of the first half (lower quartile) is 15 400.