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

Key: 8 | 7 | 2 means 78 kg for Rebels and 72 kg for Sharks.

(b) To find the median for the Rebels, arrange the data in order: 75, 78, 79, 80, 82, 82, 83, 84, 85, 86, 89, 93, 95, 99, 102. The median is the 8th value, which is 84 kg.

To find the interquartile range (IQR), calculate the lower quartile (Q1) and upper quartile (Q3). Q1 is the 4th value: 80 kg. Q3 is the 12th value: 93 kg. Thus, IQR = Q3 - Q1 = 93 - 80 = 13 kg.

(c) Draw a box-and-whisker plot for the Rebels with endpoints at 75 and 102, and median and quartiles as found in part (b).

(d) One comparison is that the average weight of the Rebels is higher than the average weight of the Sharks.

(a) To find the median for machine A, arrange the data in ascending order: 0.220, 0.231, 0.231, 0.232, 0.233, 0.234, 0.235, 0.236, 0.237, 0.238, 0.239, 0.240, 0.241, 0.242, 0.243, 0.244, 0.245, 0.246, 0.247. The median is the 10th value, which is 0.238 m.

To find the interquartile range (IQR), calculate the lower quartile (LQ) and upper quartile (UQ). LQ is the 5th value: 0.231 m. UQ is the 15th value: 0.245 m. Thus, IQR = UQ - LQ = 0.245 - 0.231 = 0.014 m.

(b) For the box-and-whisker plot, use the following values for machine A: LQ = 0.231, Median = 0.238, UQ = 0.245, minimum = 0.220, maximum = 0.254. For machine B: LQ = 0.224, Median = 0.232, UQ = 0.243, minimum = 0.211, maximum = 0.256.

(c) Comparison 1: The median length of rods produced by machine A is greater than that of machine B (0.238 m vs. 0.232 m).

Comparison 2: The interquartile range for machine A is smaller than that for machine B (0.014 m vs. 0.019 m), indicating that the lengths of rods produced by machine A are less spread out.

(i) Advantage: A box-and-whisker plot provides a visual summary of data distribution, showing the median, quartiles, and potential outliers, which helps in understanding the spread and shape of the data.

Disadvantage: It does not provide detailed information about individual data points or the exact distribution of data within the quartiles.

(iii)(a) To draw the box-and-whisker plot:

• Identify the minimum value: 190
• Identify the maximum value: 375
• Calculate the median: 280
• Calculate the lower quartile (LQ): 256 or 256.5
• Calculate the upper quartile (UQ): 329

Draw the box from LQ to UQ with a line at the median. Extend whiskers from the minimum to LQ and from UQ to the maximum.

(iii)(b) The interquartile range (IQR) is calculated as:

\(IQR = UQ - LQ = 329 - 256 = 73\) or \(329 - 256.5 = 72.5\)

To draw the box-and-whisker plot, we first find the quartiles:

Lower quartile \(( Q_1 ) = 18\)

Median\( = 25\)

Upper quartile \(( Q_3 ) = 50\)

Interquartile range\( (IQR) = Q_3 - Q_1 = 50 - 18 = 32\)

For outliers, calculate 1.5 times the IQR:

\(1.5 × IQR = 1.5 × 32 = 48\)

Check for outliers:

Lower bound\( = Q_1 - 1.5 × IQR = 18 - 48 = -30 \)(no values below this)

Upper bound \(= Q_3 + 1.5 × IQR = 50 + 48 = 98 \)(no values above this)

Since all data values are within the bounds, there are no outliers.

First, arrange the data in ascending order:

2, 2, 5, 6, 9, 10, 10, 10, 12, 12, 13, 13, 14, 17, 17, 18, 18, 28, 28, 34, 35, 38, 44, 65, 82, 88, 104

There are \(27\) data points, so the median is the \(14^\text{th}\) value:

\(\text{Median} = 17\)

To find the lower quartile (LQ), take the median of the first \(13\) values (the \(7^\text{th}\) value):

\(\text{LQ} = 10\)

To find the upper quartile (UQ), take the median of the last \(13\) values (the \(7^\text{th}\) of that half, overall \(21^\text{st}\) value):

\(\text{UQ} = 35\)

Draw a box-and-whisker plot with these values:

• Minimum = 2
• LQ = 10
• Median = 17
• UQ = 35
• Maximum = 104