2.02b Histogram: area represents frequency

4 A survey was made of the journey times of 63 people who cycle to work in a certain town. The results are summarised in the following cumulative frequency table.

1. State how many journey times were between 25 and 45 minutes.
2. Draw a histogram on graph paper to represent the data.
3. Calculate an estimate of the mean journey time.

4 The weights in kilograms of packets of cereal were noted correct to 4 significant figures. The following stem-and-leaf diagram shows the data. Key: 748 | 5 represents 0.7485 kg .

1. On the grid, draw a box-and-whisker plot to represent the data. \includegraphics[max width=\textwidth, alt={}, center]{556a1cc2-47ef-4ef7-a8f6-42850c303531-05_814_1604_1336_299}
2. Name a distribution that might be a suitable model for the weights of this type of cereal packet. Justify your answer.

2 The floor areas, \(x \mathrm {~m} ^ { 2 }\), of 20 factories are as follows. Represent these data by a histogram on graph paper, using intervals $$0 \leqslant x < 500,500 \leqslant x < 1000,1000 \leqslant x < 2000,2000 \leqslant x < 3000,3000 \leqslant x < 4000 .$$

2 The lengths of cars travelling on a car ferry are noted. The data are summarised in the following table.

1. Find the value of \(a\).
2. Draw a histogram on graph paper to represent the data.
3. Find the probability that a randomly chosen car on the ferry is less than 3.20 m in length.

6 The following table gives the marks, out of 75, in a pure mathematics examination taken by 234 students.

1. Draw a histogram on graph paper to represent these results.
2. Calculate estimates of the mean mark and the standard deviation.

4 The weights in grams of a number of stones, measured correct to the nearest gram, are represented in the following table. A histogram is drawn with a scale of 1 cm to 1 unit on the vertical axis, which represents frequency density. The \(1 - 10\) rectangle has height 3 cm .

1. Calculate the value of \(x\) and the height of the 51-70 rectangle.
2. Calculate an estimate of the mean weight of the stones.

3 The table summarises the times that 112 people took to travel to work on a particular day.

1. State which time interval in the table contains the median and which time interval contains the upper quartile.
2. On graph paper, draw a histogram to represent the data.
3. Calculate an estimate of the mean time to travel to work.

4 In a survey, the percentage of meat in a certain type of take-away meal was found. The results, to the nearest integer, for 193 take-away meals are summarised in the table.

1. Calculate estimates of the mean and standard deviation of the percentage of meat in these take-away meals.
2. Draw, on graph paper, a histogram to illustrate the information in the table.

4 The following histogram summarises the times, in minutes, taken by 190 people to complete a race. \includegraphics[max width=\textwidth, alt={}, center]{df246a50-157b-49f7-bba0-f9b86960b8b9-2_1210_1125_1251_513}

1. Show that 75 people took between 200 and 250 minutes to complete the race.
2. Calculate estimates of the mean and standard deviation of the times of the 190 people.
3. Explain why your answers to part (ii) are estimates.

1 The distance of a student's home from college, correct to the nearest kilometre, was recorded for each of 55 students. The distances are summarised in the following table. Dominic is asked to draw a histogram to illustrate the data. Dominic's diagram is shown below. \includegraphics[max width=\textwidth, alt={}, center]{d6836b62-75e7-410e-ab1e-83c391b85948-2_1225_1303_628_422} Give two reasons why this is not a correct histogram.

3 Robert has a part-time job delivering newspapers. On a number of days he noted the time, correct to the nearest minute, that it took him to do his job. Robert used his results to draw up the following table; two of the values in the table are denoted by \(a\) and \(b\).

1. Find the values of \(a\) and \(b\).
2. On graph paper, draw a histogram to represent Robert's times.

6 The heights to the nearest metre of 134 office buildings in a certain city are summarised in the table below.

1. Draw a histogram on graph paper to illustrate the data.
2. Calculate estimates of the mean and standard deviation of these heights.

5 The number of people a football stadium can hold is called the 'capacity'. The capacities of 130 football stadiums in the UK, to the nearest thousand, are summarised in the table.

1. On graph paper, draw a histogram to represent this information. Use a scale of 2 cm for a capacity of 10000 on the horizontal axis.
2. Calculate an estimate of the mean capacity of these 130 stadiums.
3. Find which class in the table contains the median and which contains the lower quartile.

3 The speeds, in \(\mathrm { km } \mathrm { h } ^ { - 1 }\), of 90 cars as they passed a certain marker on a road were recorded, correct to the nearest \(\mathrm { km } \mathrm { h } ^ { - 1 }\). The results are summarised in the following table.

1. On the grid, draw a histogram to illustrate the data in the table. \includegraphics[max width=\textwidth, alt={}, center]{5307cf3d-3d3a-441a-83d7-4adad917e168-04_1594_1198_657_516}
2. Calculate an estimate for the mean speed of these 90 cars as they pass the marker.

3 Robert has a part-time job delivering newspapers. On a number of days he noted the time, correct to the nearest minute, that it took him to do his job. Robert used his results to draw up the following table; two of the values in the table are denoted by \(a\) and \(b\).

1. Find the values of \(a\) and \(b\).
2. Draw a histogram to represent Robert's times.

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5 The following histogram illustrates the distribution of times, in minutes, that some students spent taking a shower. \includegraphics[max width=\textwidth, alt={}, center]{ec425eaf-8afc-4671-9ef3-ba2477b884ef-3_1031_1326_372_406}

1. Copy and complete the following frequency table for the data.

   Time \(( t\) minutes \()\)	\(2 < t \leqslant 4\)	\(4 < t \leqslant 6\)	\(6 < t \leqslant 7\)	\(7 < t \leqslant 8\)	\(8 < t \leqslant 10\)	\(10 < t \leqslant 16\)
   Frequency

2. Calculate an estimate of the mean time to take a shower.
3. Two of these students are chosen at random. Find the probability that exactly one takes between 7 and 10 minutes to take a shower.

4 The weights of 220 sausages are summarised in the following table.

1. State which interval the median weight lies in.
2. Find the smallest possible value and the largest possible value for the interquartile range.
3. State how many sausages weighed between 50 g and 60 g .
4. On graph paper, draw a histogram to represent the weights of the sausages.

1 The number of minutes of recorded music on a sample of 100 CDs is summarised below.

1. Illustrate the data by means of a histogram.
2. Identify two features of the distribution.

7 At East Cornwall College, the mean GCSE score of each student is calculated. This is done by allocating a number of points to each GCSE grade in the following way.

1. Calculate the mean GCSE score, \(X\), of a student who has the following GCSE grades: $$\mathrm { A } ^ { * } , \mathrm {~A} ^ { * } , \mathrm {~A} , \mathrm {~A} , \mathrm {~A} , \mathrm {~B} , \mathrm {~B} , \mathrm {~B} , \mathrm {~B} , \mathrm { C } , \mathrm { D } .$$ 60 students study AS Mathematics at the college. The mean GCSE scores of these students are summarised in the table below.

   Mean GCSE score	Number of students
   \(4.5 \leqslant X < 5.5\)	8
   \(5.5 \leqslant X < 6.0\)	14
   \(6.0 \leqslant X < 6.5\)	19
   \(6.5 \leqslant X < 7.0\)	13
   \(7.0 \leqslant X \leqslant 8.0\)	6

2. Draw a histogram to illustrate this information.
3. Calculate estimates of the sample mean and the sample standard deviation. The scoring system for AS grades is shown in the table below.

   AS Grade	A	B	C	D	E	U
   Score	60	50	40	30	20	0

   The Mathematics department at the college predicts each student's AS score, \(Y\), using the formula \(Y = 13 X - 46\), where \(X\) is the student's average GCSE score.
4. What AS grade would the department predict for a student with an average GCSE score of 7.4 ?
5. What do you think the prediction should be for a student with an average GCSE score of 5.5? Give a reason for your answer.
6. Using your answers to part (iii), estimate the sample mean and sample standard deviation of the predicted AS scores of the 60 students in the department.

3 The times taken for 480 university students to travel from their accommodation to lectures are summarised below.

1. Illustrate these data by means of a histogram.
2. Identify the type of skewness of the distribution.

6 The maximum temperatures \(x\) degrees Celsius recorded during each month of 2005 in Cambridge are given in the table below. These data are summarised by \(n = 12 , \Sigma x = 180.6 , \Sigma x ^ { 2 } = 3107.56\).

1. Calculate the mean and standard deviation of the data.
2. Determine whether there are any outliers.
3. The formula \(y = 1.8 x + 32\) is used to convert degrees Celsius to degrees Fahrenheit. Find the mean and standard deviation of the 2005 maximum temperatures in degrees Fahrenheit.
4. In New York, the monthly maximum temperatures are recorded in degrees Fahrenheit. In 2005 the mean was 63.7 and the standard deviation was 16.0 . Briefly compare the maximum monthly temperatures in Cambridge and New York in 2005. The total numbers of hours of sunshine recorded in Cambridge during the month of January for each of the last 48 years are summarised below.

   Hours \(h\)	\(70 \leqslant h < 100\)	\(100 \leqslant h < 110\)	\(110 \leqslant h < 120\)	\(120 \leqslant h < 150\)	\(150 \leqslant h < 170\)	\(170 \leqslant h < 190\)
   Number of years	6	8	10	11	10	3

5. Draw a cumulative frequency graph for these data.
6. Use your graph to estimate the 90th percentile.

2 Answer part (i) of this question on the insert provided.
A taxi driver operates from a taxi rank at a main railway station in London. During one particular week he makes 120 journeys, the lengths of which are summarised in the table.

1. On the insert, draw a cumulative frequency diagram to illustrate the data.
2. Use your graph to estimate the median length of journey and the quartiles. Hence find the interquartile range.
3. State the type of skewness of the distribution of the data.

2 The histogram shows the amount of money, in pounds, spent by the customers at a supermarket on a particular day. \includegraphics[max width=\textwidth, alt={}, center]{5e4f3310-b96e-43db-9b6d-61da3270db06-2_977_1132_808_340}
□ represents 20 customers

1. Express the data in the form of a grouped frequency table.
2. Use your table to estimate the total amount of money spent by customers on that day.

7 The histogram shows the age distribution of people living in Inner London in 2001. \includegraphics[max width=\textwidth, alt={}, center]{be764df3-ff20-415d-9c5c-10edabf350de-5_814_1383_349_379} Data sourced from the 2001 Census, \href{http://www.statistics.gov.uk}{www.statistics.gov.uk}

1. State the type of skewness shown by the distribution.
2. Use the histogram to estimate the number of people aged under 25.
3. The table below shows the cumulative frequency distribution.

   Age	20	30	40	50	65	100
   Cumulative frequency (thousands)	660	1240	1810	\(a\)	2490	2770

   (A) Use the histogram to find the value of \(a\).
   (B) Use the table to calculate an estimate of the median age of these people. The ages of people living in Outer London in 2001 are summarised below.

   Age ( \(x\) years)	\(0 \leqslant x < 20\)	\(20 \leqslant x < 30\)	\(30 \leqslant x < 40\)	\(40 \leqslant x < 50\)	\(50 \leqslant x < 65\)	\(65 \leqslant x < 100\)
   Frequency (thousands)	1120	650	770	590	680	610

4. Illustrate these data by means of a histogram.
5. Make two brief comments on the differences between the age distributions of the populations of Inner London and Outer London.
6. The data given in the table for Outer London are used to calculate the following estimates. Mean 38.5, median 35.7, midrange 50, standard deviation 23.7, interquartile range 34.4.
   The final group in the table assumes that the maximum age of any resident is 100 years. These estimates are to be recalculated, based on a maximum age of 105, rather than 100. For each of the five estimates, state whether it would increase, decrease or be unchanged.

4 The weights, \(w\) grams, of a random sample of 60 carrots of variety A are summarised in the table below.

1. Draw a histogram to illustrate these data.
2. Calculate estimates of the mean and standard deviation of \(w\).
3. Use your answers to part (ii) to investigate whether there are any outliers. The weights, \(x\) grams, of a random sample of 50 carrots of variety B are summarised as follows. $$n = 50 \quad \sum x = 3624.5 \quad \sum x ^ { 2 } = 265416$$
4. Calculate the mean and standard deviation of \(x\).
5. Compare the central tendency and variation of the weights of varieties A and B .