2.02b Histogram: area represents frequency

5 The times taken by 200 players to solve a computer puzzle are summarised in the following table.

1. Draw a histogram to represent this information. \includegraphics[max width=\textwidth, alt={}, center]{1a27e2ca-9be5-48a0-a1aa-01844573f4d4-08_1397_1198_808_516}
2. Calculate an estimate of the mean time taken by these 200 players.
3. Find the greatest possible value of the interquartile range of these times.

3 The times taken to travel to college by 2500 students are summarised in the table.

1. Draw a histogram to represent this information. \includegraphics[max width=\textwidth, alt={}, center]{d69f6a47-7c88-46b3-9e8f-07727106e987-04_1201_1198_1050_516} From the data, the estimate of the mean value of \(t\) is 31.44 .
2. Calculate an estimate of the standard deviation of the times taken to travel to college.
3. In which class interval does the upper quartile lie?
   It was later discovered that the times taken to travel to college by two students were incorrectly recorded. One student's time was recorded as 15 instead of 5 and the other's time was recorded as 65 instead of 75 .
4. Without doing any further calculations, state with a reason whether the estimate of the standard deviation in part (b) would be increased, decreased or stay the same.

5 The populations of 150 villages in the UK, to the nearest hundred, are summarised in the table.

1. Draw a histogram to represent this information. \includegraphics[max width=\textwidth, alt={}, center]{a7157882-d87e-4efb-abc5-9c9f58197012-08_1395_1195_1043_516}
2. Write down the class interval which contains the median for this information.
3. Find the greatest possible value of the interquartile range for the populations of the 150 villages.

3 The heights, in cm, of 200 adults in Barimba are summarised in the following table.

1. Draw a histogram to represent this information. \includegraphics[max width=\textwidth, alt={}, center]{a909cef1-8a22-4cef-b0b7-c48316304c0c-04_1397_1495_762_287}
2. The interquartile range is \(R \mathrm {~cm}\). Show that \(R\) is not greater than 15 .

3 At a summer camp an arithmetic test is taken by 250 children. The times taken, to the nearest minute, to complete the test were recorded. The results are summarised in the table.

1. Draw a histogram to represent this information. \includegraphics[max width=\textwidth, alt={}, center]{c1bc5ac2-6b0e-48c7-92e9-9b8b56b57d90-05_1000_1198_785_516}
2. State which class interval contains the median.
3. Given that an estimate of the mean time is 61.05 minutes, state what feature of the distribution accounts for the median and the mean being different.

3 The times taken, in minutes, by 150 students to complete a puzzle are summarised in the table.

1. Draw a histogram to represent this information. \includegraphics[max width=\textwidth, alt={}, center]{d1a3524c-a3b5-45fe-86a7-5cbda087efcd-06_1193_1489_886_328}
2. Calculate an estimate for the mean time for these students to complete the puzzle.
3. In which class interval does the lower quartile of the times lie?

7 A particular piece of music was played by 91 pianists and for each pianist, the number of incorrect notes was recorded. The results are summarised in the table.

1. Draw a histogram to represent this information. \includegraphics[max width=\textwidth, alt={}, center]{9f0f0e3c-7baf-42eb-a4fb-9ce61922c3cd-10_1488_1493_785_365}
2. State which class interval contains the lower quartile and which class interval contains the upper quartile. Hence find the greatest possible value of the interquartile range.
3. Calculate an estimate for the mean number of incorrect notes.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

3 The times taken, in minutes, by 360 employees at a large company to travel from home to work are summarised in the following table.

1. Draw a histogram to represent this information. \includegraphics[max width=\textwidth, alt={}, center]{217c5a58-2966-4b86-b3b6-9d1676d2979c-04_1198_1200_836_516}
2. Calculate an estimate of the mean time taken by an employee to travel to work.

4 The times taken, in minutes, to complete a word processing task by 250 employees at a particular company are summarised in the table.

1. Draw a histogram to represent this information. \includegraphics[max width=\textwidth, alt={}, center]{3e74785d-5981-480c-a0fd-f43d5d227f2d-06_1201_1198_1050_516} From the data, the estimate of the mean time taken by these 250 employees is 43.2 minutes.
2. Calculate an estimate for the standard deviation of these times.

4 The times, to the nearest minute, of 150 athletes taking part in a charity run are recorded. The results are summarised in the table.

1. Draw a histogram to represent this information. \includegraphics[max width=\textwidth, alt={}, center]{e8c2b51e-d788-4917-829e-1b056a24f520-08_1493_1397_936_415}
2. Calculate estimates for the mean and standard deviation of the times taken by the athletes.

7 A random sample of 97 people who own mobile phones was used to collect data on the amount of time they spent per day on their phones. The results are displayed in the table below.

1. Calculate estimates of the mean and standard deviation of the time spent per day on these mobile phones.
2. On graph paper, draw a fully labelled histogram to represent the data.

7 The numbers of chocolate bars sold per day in a cinema over a period of 100 days are summarised in the following table.

1. Draw a histogram to represent this information. \includegraphics[max width=\textwidth, alt={}, center]{3ada18de-c4f7-4049-9032-46b796be83c3-12_1203_1399_833_415}
2. What is the greatest possible value of the interquartile range for the data?
3. Calculate estimates of the mean and standard deviation of the number of chocolate bars sold.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

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.

5 As part of a data collection exercise, members of a certain school year group were asked how long they spent on their Mathematics homework during one particular week. The times are given to the nearest 0.1 hour. The results are displayed in the following table.

1. Draw, on graph paper, a histogram to illustrate this information.
2. Calculate an estimate of the mean time spent on their Mathematics homework by members of this year group.

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 \(A\).

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.

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.

4 The times taken, \(t\) seconds, by 1140 people to solve a puzzle are summarised in the table.

1. On the grid, draw a histogram to illustrate this information. \includegraphics[max width=\textwidth, alt={}, center]{7652f36c-59b5-4fcd-b17b-d796dc82aec0-05_812_1406_804_411}
2. Calculate an estimate of the mean of \(t\).

7 The following histogram represents the lengths of worms in a garden. \includegraphics[max width=\textwidth, alt={}, center]{67412184-38f6-4b37-afe3-4a149a2e0586-10_789_1195_301_466}

1. Calculate the frequencies represented by each of the four histogram columns.
2. On the grid on the next page, draw a cumulative frequency graph to represent the lengths of worms in the garden. \includegraphics[max width=\textwidth, alt={}, center]{67412184-38f6-4b37-afe3-4a149a2e0586-11_1111_1409_251_408}
3. Use your graph to estimate the median and interquartile range of the lengths of worms in the garden.
4. Calculate an estimate of the mean length of worms in the garden.
   {www.cie.org.uk} after the live examination series. }

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}