2.02b Histogram: area represents frequency

163 questions

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OCR MEI S1 Q1
18 marks Moderate -0.3
1 The heights \(x \mathrm {~cm}\) of 100 boys in Year 7 at a school are summarised in the table below.
Height\(125 \leqslant x \leqslant 140\)\(140 < x \leqslant 145\)\(145 < x \leqslant 150\)\(150 < x \leqslant 160\)\(160 < x \leqslant 170\)
Frequency252924184
  1. Estimate the number of boys who have heights of at least 155 cm .
  2. Calculate an estimate of the median height of the 100 boys.
  3. Draw a histogram to illustrate the data. The histogram below shows the heights of 100 girls in Year 7 at the same school. \includegraphics[max width=\textwidth, alt={}, center]{ab4d5ab1-e3b7-495f-9142-d37df7e712de-1_868_1361_1015_381}
  4. How many more girls than boys had heights exceeding 160 cm ?
  5. Calculate an estimate of the mean height of the 100 girls.
OCR MEI S1 Q2
18 marks Moderate -0.8
2 The engine sizes \(x \mathrm {~cm} ^ { 3 }\) of a sample of 80 cars are summarised in the table below.
Engine size \(x\)\(500 \leqslant x \leqslant 1000\)\(1000 < x \leqslant 1500\)\(1500 < x \leqslant 2000\)\(2000 < x \leqslant 3000\)\(3000 < x \leqslant 5000\)
Frequency72226187
  1. Draw a histogram to illustrate the distribution.
  2. A student claims that the midrange is \(2750 \mathrm {~cm} ^ { 3 }\). Discuss briefly whether he is likely to be correct.
  3. Calculate estimates of the mean and standard deviation of the engine sizes. Explain why your answers are only estimates.
  4. Hence investigate whether there are any outliers in the sample.
  5. A vehicle duty of \(\pounds 1000\) is proposed for all new cars with engine size greater than \(2000 \mathrm {~cm} ^ { 3 }\). Assuming that this sample of cars is representative of all new cars in Britain and that there are 2.5 million new cars registered in Britain each year, calculate an estimate of the total amount of money that this vehicle duty would raise in one year.
  6. Why in practice might your estimate in part (v) turn out to be too high?
OCR MEI S1 Q1
5 marks Easy -1.3
1 In the Paris-Roubaix cycling race, there are a number of sections of cobbled road. The lengths of these sections, measured in metres, are illustrated in the histogram. \includegraphics[max width=\textwidth, alt={}, center]{3aabac69-ead8-40e4-b06f-5e812bb02906-1_897_1398_494_410}
  1. Find the number of sections which are between 1000 and 2000 metres in length.
  2. Name the type of skewness suggested by the histogram.
  3. State the minimum and maximum possible values of the midrange.
OCR MEI S1 Q4
19 marks Moderate -0.3
4 The incomes of a sample of 918 households on an island are given in the table below.
Income
\(( x\) thousand pounds \()\)
\(0 \leqslant x \leqslant 20\)\(20 < x \leqslant 40\)\(40 < x \leqslant 60\)\(60 < x \leqslant 100\)\(100 < x \leqslant 200\)
Frequency23836514212845
  1. Draw a histogram to illustrate the data.
  2. Calculate an estimate of the mean income.
  3. Calculate an estimate of the standard deviation of the incomes.
  4. Use your answers to parts (ii) and (iii) to show there are almost certainly some outliers in the sample. Explain whether or not it would be appropriate to exclude the outliers from the calculation of the mean and the standard deviation.
  5. The incomes were converted into another currency using the formula \(y = 1.15 x\). Calculate estimates of the mean and variance of the incomes in the new currency.
OCR MEI S1 Q1
8 marks Easy -1.2
1 A business analyst collects data about the distribution of hourly wages, in \(\pounds\), of shop-floor workers at a factory. These data are illustrated in the box and whisker plot. \includegraphics[max width=\textwidth, alt={}, center]{56f1bd5c-4b45-4e36-a324-e7e0edbb5bdd-1_206_1420_505_397}
  1. Name the type of skewness of the distribution.
  2. Find the interquartile range and hence show that there are no outliers at the lower end of the distribution, but there is at least one outlier at the upper end.
  3. Suggest possible reasons why this may be the case.
OCR MEI S1 Q2
7 marks Moderate -0.8
2 The lifetimes in hours of 90 components are summarised in the table.
Lifetime \(( x\) hours \()\)\(0 < x \leqslant 20\)\(20 < x \leqslant 30\)\(30 < x \leqslant 50\)\(50 < x \leqslant 65\)\(65 < x \leqslant 100\)
Frequency2413142118
  1. Draw a histogram to illustrate these data.
  2. In which class interval does the median lie? Justify your answer.
OCR MEI S1 Q3
19 marks Moderate -0.3
3 A pear grower collects a random sample of 120 pears from his orchard. The histogram below shows the lengths, in mm, of these pears. \includegraphics[max width=\textwidth, alt={}, center]{56f1bd5c-4b45-4e36-a324-e7e0edbb5bdd-2_825_1634_467_295}
  1. Calculate the number of pears which are between 90 and 100 mm long.
  2. Calculate an estimate of the mean length of the pears. Explain why your answer is only an estimate.
  3. Calculate an estimate of the standard deviation.
  4. Use your answers to parts (ii) and (iii) to investigate whether there are any outliers.
  5. Name the type of skewness of the distribution.
  6. Illustrate the data using a cumulative frequency diagram.
OCR MEI S1 Q4
8 marks Moderate -0.8
4 The frequency table below shows the distance travelled by 1200 visitors to a particular UK tourist destination in August 2008.
Distance \(( d\) miles \()\)\(0 \leqslant d < 50\)\(50 \leqslant d < 100\)\(100 \leqslant d < 200\)\(200 \leqslant d < 400\)
Frequency360400307133
  1. Draw a histogram on graph paper to illustrate these data.
  2. Calculate an estimate of the median distance.
OCR MEI S1 Q2
4 marks Moderate -0.8
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]{c7cb0f6b-7b6b-4c52-8287-7efc6bd70247-2_985_1473_470_379}
  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.
OCR MEI S1 Q5
6 marks Easy -1.2
5 The times taken for 480 university students to travel from their accommodation to lectures are summarised below.
Time \(( t\) minutes \()\)\(0 \leqslant t < 5\)\(5 \leqslant t < 10\)\(10 \leqslant t < 20\)\(20 \leqslant t < 30\)\(30 \leqslant t < 40\)\(40 \leqslant t < 60\)
Frequency3415318873275
  1. Illustrate these data by means of a histogram.
  2. Identify the type of skewness of the distribution.
OCR MEI S1 Q3
18 marks Moderate -0.8
3 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.
GradeA*ABCDEFGU
Points876543210
  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 scoreNumber 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 GradeABCDEU
    Score60504030200
    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.
OCR MEI S1 Q3
20 marks Moderate -0.8
3 The histogram shows the age distribution of people living in Inner London in 2001. \includegraphics[max width=\textwidth, alt={}, center]{b6d84f99-ee39-49c7-a5e8-05838efeef5a-2_804_1372_483_436} Data sourced from the 2001 Census, www.sta is \href{http://ics.gov.uk}{ics.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.
    Age2030405065100
    Cumulative frequency (thousands)66012401810\(a\)24902770
    (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)1120650770590680610
  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.
OCR MEI S1 Q7
20 marks Moderate -0.8
7 The histogram shows the age distribution of people living in Inner London in 2001. \includegraphics[max width=\textwidth, alt={}, center]{aabf9d8b-5f91-4a3b-bcf8-e46cb45127c4-4_805_1372_392_401} Data sourced from he 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.
    Age2030405065100
    Cumulative frequency (thousands)66012401810\(a\)24902770
    (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)1120650770590680610
  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.
    [0pt] [4]
OCR MEI S1 Q7
4 marks Moderate -0.8
7 The histogram shows the age distribution of people living in Inner London in 2001. \includegraphics[max width=\textwidth, alt={}, center]{93bbc0cf-d3ad-4bc2-a6c6-36a3b8e394a9-4_805_1372_392_401} Data sourced from he 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.
    Age2030405065100
    Cumulative frequency (thousands)66012401810\(a\)24902770
    (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)1120650770590680610
  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.
    [0pt] [4]
OCR MEI S1 Q5
20 marks Moderate -0.3
5 A pear grower collects a random sample of 120 pears from his orchard. The histogram below shows the lengths, in mm , of these pears. \includegraphics[max width=\textwidth, alt={}, center]{056d3e9a-088d-4c97-9546-7cecb59b8727-3_815_1628_505_304}
  1. Calculate the number of pears which are between 90 and 100 mm long.
  2. Calculate an estimate of the mean length of the pears. Explain why your answer is only an estimate.
  3. Calculate an estimate of the standard deviation.
  4. Use your answers to parts (ii) and (iii) to investigate whether there are any outliers.
  5. Name the type of skewness of the distribution.
  6. Illustrate the data using a cumulative frequency diagram.
OCR MEI S1 Q6
6 marks Easy -1.2
6 The times taken for 480 university students to travel from their accommodation to lectures are summarised below.
Time \(( t\) minutes \()\)\(0 \leqslant t < 5\)\(5 \leqslant t < 10\)\(10 \leqslant t < 20\)\(20 \leqslant t < 30\)\(30 \leqslant t < 40\)\(40 \leqslant t < 60\)
Frequency3415318873275
  1. Illustrate these data by means of a histogram.
  2. Identify the type of skewness of the distribution.
OCR MEI S1 Q5
22 marks Moderate -0.3
5 A pear grower collects a random sample of 120 pears from his orchard. The histogram below shows the lengths, in mm , of these pears. \includegraphics[max width=\textwidth, alt={}, center]{99c502aa-2c9f-461d-9dc0-ed55e3df32a2-3_815_1628_505_304}
  1. Calculate the number of pears which are between 90 and 100 mm long.
  2. Calculate an estimate of the mean length of the pears. Explain why your answer is only an estimate.
  3. Calculate an estimate of the standard deviation.
  4. Use your answers to parts (ii) and (iii) to investigate whether there are any outliers.
  5. Name the type of skewness of the distribution.
  6. Illustrate the data using a cumulative frequency diagram.
Edexcel S1 2014 January Q8
10 marks Moderate -0.8
8. A manager records the number of hours of overtime claimed by 40 staff in a month. The histogram in Figure 1 represents the results. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a839a89a-17f0-473b-ac10-bcec3dbe97f7-26_1107_1513_406_210} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure}
  1. Calculate the number of staff who have claimed less than 10 hours of overtime in the month.
  2. Estimate the median number of hours of overtime claimed by these 40 staff in the month.
  3. Estimate the mean number of hours of overtime claimed by these 40 staff in the month. The manager wants to compare these data with overtime data he collected earlier to find out if the overtime claimed by staff has decreased.
  4. State, giving a reason, whether the manager should use the median or the mean to compare the overtime claimed by staff.
    (2)
Edexcel S1 2017 January Q1
12 marks Easy -1.3
  1. Ralph records the weights, in grams, of 100 tomatoes. This information is displayed in the histogram below. \includegraphics[max width=\textwidth, alt={}, center]{1130517e-33d0-41b1-9303-2d981379954d-02_981_1268_338_274}
Given that 5 of the tomatoes have a weight between 2 and 3 grams,
  1. find the number of tomatoes with a weight between 0 and 2 grams. One of the tomatoes is selected at random.
  2. Find the probability that it weighs more than 3 grams.
  3. Estimate the proportion of the tomatoes with a weight greater than 6.25 grams.
  4. Using your answer to part (c), explain whether or not the median is greater than 6.25 grams. Given that the mean weight of these tomatoes is 6.25 grams and using your answer to part (d),
  5. describe the skewness of the distribution of the weights of these tomatoes. Give a reason for your answer. Two of these 100 tomatoes are selected at random.
  6. Estimate the probability that both tomatoes weigh within 0.75 grams of the mean.
Edexcel S1 2019 January Q4
13 marks Moderate -0.8
4. A group of 100 adults recorded the amount of time, \(t\) minutes, they spent exercising each day. Their results are summarised in the table below.
Time (t minutes)Frequency (f)Time midpoint (x)
\(0 \leqslant t < 15\)257.5
\(15 \leqslant t < 30\)1722.5
\(30 \leqslant t < 60\)2845
\(60 \leqslant t < 120\)2490
\(120 \leqslant t \leqslant 240\)6180
[You may use \(\sum \mathrm { f } x ^ { 2 } = 455\) 512.5]
A histogram is drawn to represent these data.
The bar representing the time \(0 \leqslant t < 15\) has width 0.5 cm and height 6 cm .
  1. Calculate the width and height of the bar representing a time of \(60 \leqslant t < 120\)
  2. Use linear interpolation to estimate the median time spent exercising by these adults each day.
  3. Find an estimate of the mean time spent exercising by these adults each day.
  4. Calculate an estimate for the standard deviation of these times.
  5. Describe, giving a reason, the skewness of these data. Further analysis of the above data revealed that 18 of the 25 adults in the \(0 \leqslant t < 15\) group took no exercise each day.
  6. State, giving a reason, what effect, if any, this new information would have on your answers to
    1. the estimate of the median in part (b),
    2. the estimate of the mean in part (c),
    3. the estimate of the standard deviation in part (d).
Edexcel S1 2023 January Q1
10 marks Moderate -0.3
  1. The histogram shows the times taken, \(t\) minutes, by each of 100 people to swim 500 metres. \includegraphics[max width=\textwidth, alt={}, center]{c316fa29-dedc-4890-bd82-31eb0bb819f9-02_986_1070_342_424}
    1. Use the histogram to complete the frequency table for the times taken by the 100 people to swim 500 metres.
    Time taken ( \(\boldsymbol { t }\) minutes)\(5 - 10\)\(10 - 14\)\(14 - 18\)\(18 - 25\)\(25 - 40\)
    Frequency ( \(\boldsymbol { f }\) )101624
  2. Estimate the number of people who took less than 16 minutes to swim 500 metres.
  3. Find an estimate for the mean time taken to swim 500 metres. Given that \(\sum f t ^ { 2 } = 41033\)
  4. find an estimate for the standard deviation of the times taken to swim 500 metres. Given that \(Q _ { 3 } = 23\)
  5. use linear interpolation to estimate the interquartile range of the times taken to swim 500 metres.
Edexcel S1 2024 January Q1
8 marks Moderate -0.8
  1. The histogram below shows the distribution of the heights, to the nearest cm , of 408 plants. \includegraphics[max width=\textwidth, alt={}, center]{86446ce3-496a-4f02-9566-9b207bac9efa-02_1001_1473_340_296}
    1. Use the histogram to complete the following table.
    Height \(( h\) cm)\(5 \leqslant h < 9\)\(9 \leqslant h < 13\)\(13 \leqslant h < 15\)\(15 \leqslant h < 17\)\(17 \leqslant h < 25\)
    Frequency32152120
  2. Use interpolation to estimate the median. The mean height of these plants is 13.2 cm correct to one decimal place.
  3. Describe the skew of these data. Give a reason for your answer. Two of these plants are chosen at random.
  4. Estimate the probability that both of their heights are between 8 cm and 14 cm
Edexcel S1 2014 June Q2
14 marks Moderate -0.8
  1. The table below shows the distances (to the nearest km ) travelled to work by the 50 employees in an office.
Distance (km)Frequency (f)Distance midpoint (x)
0-2161.25
3-5124
6-10108
11-20815.5
21-40430.5
$$\text { [You may use } \left. \sum \mathrm { f } x = 394 , \quad \sum \mathrm { f } x ^ { 2 } = 6500 \right]$$ A histogram has been drawn to represent these data.
The bar representing the distance of \(3 - 5\) has a width of 1.5 cm and a height of 6 cm .
  1. Calculate the width and height of the bar representing the distance of 6-10
  2. Use linear interpolation to estimate the median distance travelled to work.
    1. Show that an estimate of the mean distance travelled to work is 7.88 km .
    2. Estimate the standard deviation of the distances travelled to work.
  4. Describe, giving a reason, the skewness of these data. Peng starts to work in this office as the \(51 ^ { \text {st } }\) employee.
    She travels a distance of 7.88 km to work.
  5. Without carrying out any further calculations, state, giving a reason, what effect Peng's addition to the workforce would have on your estimates of the
    1. mean,
    2. median,
    3. standard deviation
      of the distances travelled to work.
Edexcel S1 2004 January Q5
18 marks Moderate -0.3
5. The values of daily sales, to the nearest \(\pounds\), taken at a newsagents last year are summarised in the table below.
SalesNumber of days
\(1 - 200\)166
\(201 - 400\)100
\(401 - 700\)59
\(701 - 1000\)30
\(1001 - 1500\)5
  1. Draw a histogram to represent these data.
  2. Use interpolation to estimate the median and inter-quartile range of daily sales.
  3. Estimate the mean and the standard deviation of these data. The newsagent wants to compare last year's sales with other years.
  4. State whether the newsagent should use the median and the inter-quartile range or the mean and the standard deviation to compare daily sales. Give a reason for your answer.
    (2)
OCR S1 2012 January Q5
11 marks Moderate -0.8
5 At a certain resort the number of hours of sunshine, measured to the nearest hour, was recorded on each of 21 days. The results are summarised in the table.
Hours of sunshine0\(1 - 3\)\(4 - 6\)\(7 - 9\)\(10 - 15\)
Number of days06942
The diagram shows part of a histogram to illustrate the data. The scale on the frequency density axis is 2 cm to 1 unit. \includegraphics[max width=\textwidth, alt={}, center]{56ca7462-d061-48d3-bc5f-274d925e4e34-3_944_1778_699_148}
  1. (a) Calculate the frequency density of the \(1 - 3\) class.
    (b) Fred wishes to draw the block for the 10 - 15 class on the same diagram. Calculate the height, in centimetres, of this block.
  2. A cumulative frequency graph is to be drawn. Write down the coordinates of the first two points that should be plotted. You are not asked to draw the graph.
  3. (a) Calculate estimates of the mean and standard deviation of the number of hours of sunshine.
    (b) Explain why your answers are only estimates.