2.02a Interpret single variable data: tables and diagrams

5 Each father in a random sample of fathers was asked how old he was when his first child was born. The following histogram represents the information. \includegraphics[max width=\textwidth, alt={}, center]{14e8a601-2180-4491-9336-cafd262f2596-3_789_1627_1468_260}

1. What is the modal age group?
2. How many fathers were between 25 and 30 years old when their first child was born?
3. How many fathers were in the sample?
4. Find the probability that a father, chosen at random from the group, was between 25 and 30 years old when his first child was born, given that he was older than 25 years. 632 teams enter for a knockout competition, in which each match results in one team winning and the other team losing. After each match the winning team goes on to the next round, and the losing team takes no further part in the competition. Thus 16 teams play in the second round, 8 teams play in the third round, and so on, until 2 teams play in the final round.

4 The lengths of time in minutes to swim a certain distance by the members of a class of twelve 9 -year-olds and by the members of a class of eight 16 -year-olds are shown below.

1. Draw a back-to-back stem-and-leaf diagram to represent the information above.
2. A new pupil joined the 16 -year-old class and swam the distance. The mean time for the class of nine pupils was now 13.6 minutes. Find the new pupil's time to swim the distance.

1 The stem-and-leaf diagram below represents data collected for the number of hits on an internet site on each day in March 2007. There is one missing value, denoted by \(x\). Key: 1 | 5 represents 15 hits

1. Find the median and lower quartile for the number of hits each day.
2. The interquartile range is 19 . Find the value of \(x\).

6 During January the numbers of people entering a store during the first hour after opening were as follows.

1. Find the values of \(a\) and \(b\).
2. Draw a cumulative frequency graph to represent this information. Take a scale of 2 cm for 10 minutes on the horizontal axis and 2 cm for 50 people on the vertical axis.
3. Use your graph to estimate the median time after opening that people entered the store.
4. Calculate estimates of the mean, \(m\) minutes, and standard deviation, \(s\) minutes, of the time after opening that people entered the store.
5. Use your graph to estimate the number of people entering the store between ( \(m - \frac { 1 } { 2 } s\) ) and \(\left( m + \frac { 1 } { 2 } s \right)\) minutes after opening.

2 The numbers of people travelling on a certain bus at different times of the day are as follows.

1. Draw a stem-and-leaf diagram to illustrate the information given above.
2. Find the median, the lower quartile, the upper quartile and the interquartile range.
3. State, in this case, which of the median and mode is preferable as a measure of central tendency, and why.

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.

6 There are 5000 schools in a certain country. The cumulative frequency table shows the number of pupils in a school and the corresponding number of schools.

1. Draw a cumulative frequency graph with a scale of 2 cm to 100 pupils on the horizontal axis and a scale of 2 cm to 1000 schools on the vertical axis. Use your graph to estimate the median number of pupils in a school.
2. \(80 \%\) of the schools have more than \(n\) pupils. Estimate the value of \(n\) correct to the nearest ten.
3. Find how many schools have between 201 and 250 (inclusive) pupils.
4. Calculate an estimate of the mean number of pupils per school.

5 A hotel has 90 rooms. The table summarises information about the number of rooms occupied each day for a period of 200 days.

1. Draw a cumulative frequency graph on graph paper to illustrate this information.
2. Estimate the number of days when over 30 rooms were occupied.
3. On \(75 \%\) of the days at most \(n\) rooms were occupied. Estimate the value of \(n\).

3 The following cumulative frequency table shows the examination marks for 300 candidates in country \(A\) and 300 candidates in country \(B\).

1. Without drawing a graph, show that the median for country \(B\) is higher than the median for country \(A\).
2. Find the number of candidates in country \(A\) who scored between 20 and 34 marks inclusive.
3. Calculate an estimate of the mean mark for candidates in country \(A\).

5 The lengths of the diagonals in metres of the 9 most popular flat screen TVs and the 9 most popular conventional TVs are shown below.

1. Represent this information on a back-to-back stem-and-leaf diagram.
2. Find the median and the interquartile range of the lengths of the diagonals of the 9 conventional TVs.
3. Find the mean and standard deviation of the lengths of the diagonals of the 9 flat screen TVs.

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.

5 The following are the annual amounts of money spent on clothes, to the nearest \(\\) 10$, by 27 people.

1. Construct a stem-and-leaf diagram for the data.
2. Find the median and the interquartile range of the data. An 'outlier' is defined as any data value which is more than 1.5 times the interquartile range above the upper quartile, or more than 1.5 times the interquartile range below the lower quartile.
3. List the outliers.

6 The weights, \(x\) kilograms, of 144 people were recorded. The results are summarised in the cumulative frequency table below.

1. On graph paper, draw a cumulative frequency graph to represent these results.
2. 64 people weigh more than \(c \mathrm {~kg}\). Use your graph to find the value of \(c\).
3. Calculate estimates of the mean and standard deviation of the weights.

7 A typing test is taken by 111 people. The numbers of typing errors they make in the test are summarised in the table below.

1. Draw a histogram on graph paper to represent this information.
2. Calculate an estimate of the mean number of typing errors for these 111 people.
3. State which class contains the lower quartile and which class contains the upper quartile. Hence find the least possible value of the interquartile range.

6 The times taken by 57 athletes to run 100 metres are summarised in the following cumulative frequency table.

1. State how many athletes ran 100 metres in a time between 10.5 and 11.0 seconds.
2. Draw a histogram on graph paper to represent the times taken by these athletes to run 100 metres.
3. Calculate estimates of the mean and variance of the times taken by these athletes.

1 Some adults and some children each tried to estimate, without using a watch, the number of seconds that had elapsed in a fixed time-interval. Their estimates are shown below.

1. Draw a back-to-back stem-and-leaf diagram to represent the data.
2. Make two comparisons between the estimates of the adults and the children.

2 The table summarises the lengths in centimetres of 104 dragonflies.

1. State which class contains the upper quartile.
2. Draw a histogram, on graph paper, to represent the data.

6 Seventy samples of fertiliser were collected and the nitrogen content was measured for each sample. The cumulative frequency distribution is shown in the table below.

1. On graph paper draw a cumulative frequency graph to represent the data.
2. Estimate the percentage of samples with a nitrogen content greater than 4.4.
3. Estimate the median.
4. Construct the frequency table for these results and draw a histogram on graph paper.

7 The amounts spent by 160 shoppers at a supermarket are summarised in the following table.

1. Draw a cumulative frequency graph of this distribution.
2. Estimate the median and the interquartile range of the amount spent.
3. Estimate the number of shoppers who spent more than \(\\) 115$.
4. Calculate an estimate of the mean amount spent.

5 The following are the maximum daily wind speeds in kilometres per hour for the first two weeks in April for two towns, Bronlea and Rogate.

1. Draw a back-to-back stem-and-leaf diagram to represent this information.
2. Write down the median of the maximum wind speeds for Bronlea and find the interquartile range for Rogate.
3. Use your diagram to make one comparison between the maximum wind speeds in the two towns.

2 A group of children played a computer game which measured their time in seconds to perform a certain task. A summary of the times taken by girls and boys in the group is shown below.

1. On graph paper, draw two box-and-whisker plots in a single diagram to illustrate the times taken by girls and boys to perform this task.
2. State two comparisons of the times taken by girls and boys.

2 Anabel measured the lengths, in centimetres, of 200 caterpillars. Her results are illustrated in the cumulative frequency graph below. \includegraphics[max width=\textwidth, alt={}, center]{184a04ac-4396-4a0f-8fa8-ab11a4b6df39-03_1173_1195_356_466}

1. Estimate the median and the interquartile range of the lengths.
2. Estimate how many caterpillars had a length of between 2 and 3.5 cm .
3. 6\% of caterpillars were of length \(l\) centimetres or more. Estimate \(l\).

5 The lengths, \(t\) minutes, of 242 phone calls made by a family over a period of 1 week are summarised in the frequency table below.

1. Find the value of \(a\).
2. Calculate an estimate of the mean length of these phone calls.
3. On the grid, draw a histogram to illustrate the data in the table. \includegraphics[max width=\textwidth, alt={}, center]{a813e127-d116-411c-88ec-2443fdbc9391-07_2002_1513_486_356}

1 The masses in kilograms of 50 children having a medical check-up were recorded correct to the nearest kilogram. The results are shown in the table.

1. Find which class interval contains the lower quartile.
2. On the grid, draw a histogram to illustrate the data in the table. \includegraphics[max width=\textwidth, alt={}, center]{dd75fa20-fead-48d6-aff4-c5e733769f9f-02_1397_1397_1187_415}