Interpret or analyse given back-to-back stem-and-leaf

3 Two machines, \(A\) and \(B\), produce metal rods of a certain type. The lengths, in metres, of 19 rods produced by machine \(A\) and 19 rods produced by machine \(B\) are shown in the following back-to-back stem-and-leaf diagram. \begin{table}[h] \end{table}

1. Find the median and the interquartile range for machine \(A\).
   It is given that for machine \(B\) the median is 0.232 m , the lower quartile is 0.224 m and the upper quartile is 0.243 m .
2. Draw box-and-whisker plots for \(A\) and \(B\).
   \includegraphics[max width=\textwidth, alt={}, center]{a3b3ebd1-db9e-4552-9abe-bfdeba786d02-05_812_1205_616_511}
3. Hence make two comparisons between the lengths of the rods produced by machine \(A\) and those produced by machine \(B\).

3 The following back-to-back stem-and-leaf diagram represents the monthly salaries, in dollars, of 27 employees at each of two companies, \(A\) and \(B\). Key: 1 |27| 6 means \(
) 2710\( for company \)A\( and \)\\( 2760\) for company \(B\)

1. Find the median and the interquartile range of the monthly salaries of employees in company \(A\).
   The lower quartile, median and upper quartile for company \(B\) are \(
   ) 2600 , \\( 2690\) and \(
   ) 2780\( respectively.
2. Draw two box-and-whisker plots in a single diagram to represent the information for the salaries of employees at companies \)A\( and \)B$.
   \includegraphics[max width=\textwidth, alt={}, center]{f2666d82-4711-499a-98c0-3421e4c228fb-07_810_1406_573_411}
3. Comment on whether the mean would be a more appropriate measure than the median for comparing the given information for the two companies.

4 The back-to-back stem-and-leaf diagram shows the annual salaries of 19 employees at each of two companies, Petral and Ravon. Key: 2 | 31 | 5 means \\(31 200 for a Petral employee and
)31500 for a Ravon employee.

1. Find the median and the interquartile range of the salaries of the Petral employees.
   The median salary of the Ravon employees is \(
   ) 33800\(, the lower quartile is \)\\( 32000\) and the upper quartile is \(
   ) 34400$.
2. Represent the data shown in the back-to-back stem-and-leaf diagram by a pair of box-and-whisker plots in a single diagram.
   \includegraphics[max width=\textwidth, alt={}, center]{f979a442-da05-410b-84dc-3da3286514a0-07_707_1395_477_335}
3. Comment on whether the mean or the median would be a better representation of the data for the employees at Petral.

6 The lengths of some insects of the same type from two countries, \(X\) and \(Y\), were measured. The stem-and-leaf diagram shows the results. Key: 5 | 81 | 3 means an insect from country \(X\) has length 0.815 cm and an insect from country \(Y\) has length 0.813 cm .

1. Find the median and interquartile range of the lengths of the insects from country \(X\).
2. The interquartile range of the lengths of the insects from country \(Y\) is 0.028 cm . Find the values of \(q\) and \(r\).
3. Represent the data by means of a pair of box-and-whisker plots in a single diagram on graph paper.
4. Compare the lengths of the insects from the two countries.

4 The back-to-back stem-and-leaf diagram shows the values taken by two variables \(A\) and \(B\). Key: \(4 | 16 | 7\) means \(A = 0.164\) and \(B = 0.167\).

1. Find the median and the interquartile range for variable \(A\).
2. You are given that, for variable \(B\), the median is 0.171 , the upper quartile is 0.179 and the lower quartile is 0.164 . Draw box-and-whisker plots for \(A\) and \(B\) in a single diagram on graph paper.

3 The following back-to-back stem-and-leaf diagram shows the annual salaries of a group of 39 females and 39 males. Key: 2 | 20 | 3 means \\(20200 for females and
)20300 for males.

1. Find the median and the quartiles of the females' salaries. You are given that the median salary of the males is \(
   ) 24000\(, the lower quartile is \)\\( 22600\) and the upper quartile is \(
   ) 25300$.
2. Represent the data by means of a pair of box-and-whisker plots in a single diagram on graph paper.

7 The weights in kilograms of two groups of 17-year-old males from country \(P\) and country \(Q\) are displayed in the following back-to-back stem-and-leaf diagram. In the third row of the diagram, ... \(4 | 7 | 1 \ldots\) denotes weights of 74 kg for a male in country \(P\) and 71 kg for a male in country \(Q\).

1. Find the median and quartile weights for country \(Q\).
2. You are given that the lower quartile, median and upper quartile for country \(P\) are 84,94 and 98 kg respectively. On a single diagram on graph paper, draw two box-and-whisker plots of the data.
3. Make two comments on the weights of the two groups.

2 The following back-to-back stem-and-leaf diagram shows the reaction times in seconds in an experiment involving two groups of people, \(A\) and \(B\). Key: 5 | 22 | 6 means a reaction time of 0.225 seconds for \(A\) and 0.226 seconds for \(B\)

1. Find the median and the interquartile range for group \(A\).
   The median value for group \(B\) is 0.235 seconds, the lower quartile is 0.217 seconds and the upper quartile is 0.245 seconds.
2. Draw box-and-whisker plots for groups \(A\) and \(B\) on the grid.
   \includegraphics[max width=\textwidth, alt={}, center]{62812433-baee-490a-bad4-b6b0f917c234-03_805_1495_1729_365}

1. A sports teacher recorded the number of press-ups done by his students in two minutes. He recorded this information for a Year 7 class and for a Year 11 class.

The back-to-back stem and leaf diagram shows this information. Key: \(2 | 4 | 0\) means 42 press-ups for a Year 7 student and 40 press-ups for a Year 11 student

1. Find the median number of press-ups for each class. For the Year 11 class, the lower quartile is 38 and the upper quartile is 59
2. Find the lower quartile and the upper quartile for the Year 7 class.
3. Use the medians and quartiles to describe the skewness of each of the two distributions.
4. Give two reasons why the normal distribution should not be used to model the number of press-ups done by the Year 11 class.

2. The age in years of the residents of two hotels are shown in the back to back stem and leaf diagram below. Abbey Hotel \(8 | 5 | 0\) means 58 years in Abbey hotel and 50 years in Balmoral hotel Balmoral Hotel For the Balmoral Hotel,

1. write down the mode of the age of the residents,
2. find the values of the lower quartile, the median and the upper quartile.
3. 1. Find the mean, \(\bar { x }\), of the age of the residents.
   2. Given that \(\sum x ^ { 2 } = 81213\) find the standard deviation of the age of the residents. One measure of skewness is found using $$\frac { \text { mean - mode } } { \text { standard deviation } }$$
4. Evaluate this measure for the Balmoral Hotel. For the Abbey Hotel, the mode is 39 , the mean is 33.2 , the standard deviation is 12.7 and the measure of skewness is - 0.454
5. Compare the two age distributions of the residents of each hotel.

1. A random sample of 35 homeowners was taken from each of the villages Greenslax and Penville and their ages were recorded. The results are summarised in the back-to-back stem and leaf diagram below.

Key: 7 | 3 | 1 means 37 years for Greenslax and 31 years for Penville
Some of the quartiles for these two distributions are given in the table below.

1. Find the value of \(a\) and the value of \(b\). An outlier is a value that falls either $$\begin{aligned} & \text { more than } 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right) \text { above } Q _ { 3 }
   & \text { or more than } 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right) \text { below } Q _ { 1 } \end{aligned}$$
2. On the graph paper opposite draw a box plot to represent the data from Penville. Show clearly any outliers.
3. State the skewness of each distribution. Justify your answers.
   \includegraphics[max width=\textwidth, alt={}, center]{8270bcae-494c-4248-8229-a72e9e84eab0-03_930_1237_1800_367}

1. As part of their job, taxi drivers record the number of miles they travel each day. A random sample of the mileages recorded by taxi drivers Keith and Asif are summarised in the back-toback stem and leaf diagram below.

Key: 0184 means 180 for Keith and 184 for Asif
The quartiles for these two distributions are summarised in the table below.

1. Find the values of \(a , b\) and \(c\). Outliers are values that lie outside the limits $$Q _ { 1 } - 1.5 \left( Q _ { 3 } - Q _ { 1 } \right) \text { and } Q _ { 3 } + 1.5 \left( Q _ { 3 } - Q _ { 1 } \right) .$$
2. On graph paper, and showing your scale clearly, draw a box plot to represent Keith's data.
3. Comment on the skewness of the two distributions.

4. A College offers evening classes in GCSE Mathematics and English. In order to assess which age groups were reluctant to use the classes, the College collected data on the age in completed years of those currently attending each course. The results are shown in this back-to-back stem and leaf diagram. Key: 1|3|2 means age 31 doing Mathematics and age 32 doing English

1. Find the median and quartiles of the age in completed years of those attending the Mathematics classes.
   (4 marks)
2. On graph paper, draw a box plot representing the data for the Mathematics class. The median and quartiles of the age in completed years of those attending the English classes are 25,41 and 57 years respectively.
3. Draw a box plot representing the data for the English class using the same scale as for the data from the Mathematics class.
   (3 marks)
4. Using your box plots, compare and contrast the ages of those taking each class.