2.02b Histogram: area represents frequency

6. The number of people visiting a new art gallery each day is recorded over a three-month period and the results are summarised in the table below.

1. Draw a histogram on graph paper to illustrate these data. In order to calculate summary statistics for the data it is coded using \(y = \frac { x - 509.5 } { 10 }\), where \(x\) is the mid-point of each class.
2. Find \(\sum\) fy. You may assume that \(\sum f y ^ { 2 } = 2041\).
3. Using these values for \(\sum f y\) and \(\sum f y ^ { 2 }\), calculate estimates of the mean and standard deviation of the number of visitors per day.
   (6 marks)

6. A cinema recorded the number of people at each showing of each film during a one-week period. The results are summarised in the table below.

1. Draw a histogram on graph paper to illustrate these data.
2. Calculate estimates of the median and quartiles of these data.
3. Use your answers to part (b) to show that the data is positively skewed.

1. A net was used to catch swallows so that they could be ringed and examined. The weights of 55 adult birds were recorded and the results are summarised in the table below.

1. For these data calculate estimates of
   1. the median,
   2. the \(33 ^ { \text {rd } }\) percentile. These data are represented by a histogram and the bar representing the 24-25 group is 1 cm wide and 20 cm high.
2. Calculate the dimensions of the bars representing the groups
   1. 20-21
   2. 26-29

9 The finance department of a retail firm recorded the daily income each day for 300 days. The results are summarised in the histogram. \includegraphics[max width=\textwidth, alt={}, center]{85de9a39-f8be-40ee-b0c8-e2e632be93d8-6_689_1575_488_246}

1. Find the number of days on which the daily income was between \(\pounds 4000\) and \(\pounds 6000\).
2. Calculate an estimate of the number of days on which the daily income was between \(\pounds 2700\) and \(\pounds 3600\).
3. Use the midpoints of the classes to show that an estimate of the mean daily income is \(\pounds 3275\). An estimate of the standard deviation of the daily income is \(\pounds 1060\). The finance department uses the distribution \(\mathrm { N } \left( 3275,1060 ^ { 2 } \right)\) to model the daily income, in pounds.
4. Calculate the number of days on which, according to this model, the daily income would be between \(\pounds 4000\) and \(\pounds 6000\).
5. It is given that approximately \(95 \%\) of values of the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\) lie within the range \(\mu \pm 2 \sigma\). Without further calculation, use this fact to comment briefly on whether the proposed model is a good fit to the data illustrated in the histogram.

5. The following grouped frequency distribution summarises the number of minutes, to the nearest minute, that a random sample of 200 motorists were delayed by roadworks on a stretch of motorway.

1. Using graph paper represent these data by a histogram.
2. Give a reason to justify the use of a histogram to represent these data.
3. Use interpolation to estimate the median of this distribution.
4. Calculate an estimate of the mean and an estimate of the standard deviation of these data. One coefficient of skewness is given by $$\frac { 3 ( \text { mean } - \text { median } ) } { \text { standard deviation } } .$$
5. Evaluate this coefficient for the above data.
6. Explain why the normal distribution may not be suitable to model the number of minutes that motorists are delayed by these roadworks.

1. In a particular week, a dentist treats 100 patients. The length of time, to the nearest minute, for each patient's treatment is summarised in the table below.

Draw a histogram to illustrate these data.

2 The table summarises the diameters, \(d\) millimetres, of a random sample of 60 new cricket balls to be used in junior cricket.

9 The heights, in centimetres, of a random sample of 150 plants of a certain variety were measured. The results are summarised in the histogram. \includegraphics[max width=\textwidth, alt={}, center]{cb83836f-753f-4b3a-99e8-a18aff0f49ff-08_842_1651_495_207} One of the 150 plants is chosen at random, and its height, \(X \mathrm {~cm}\), is noted.

1. Show that \(\mathrm { P } ( 20 < X < 30 ) = 0.147\), correct to 3 significant figures. Sam suggests that the distribution of \(X\) can be well modelled by the distribution \(\mathrm { N } ( 40,100 )\).
2. 1. Give a brief justification for the use of the normal distribution in this context.
   2. Give a brief justification for the choice of the parameter values 40 and 100 .
3. Use Sam's model to find \(\mathrm { P } ( 20 < X < 30 )\). Nina suggests a different model. She uses the midpoints of the classes to calculate estimates, \(m\) and \(s\), for the mean and standard deviation respectively, in centimetres, of the 150 heights. She then uses the distribution \(\mathrm { N } \left( m , s ^ { 2 } \right)\) as her model.
4. Use Nina's model to find \(\mathrm { P } ( 20 < X < 30 )\).
5. 1. Complete the table in the Printed Answer Booklet to show the probabilities obtained from Sam's model and Nina's model.
   2. By considering the different ranges of values of \(X\) given in the table, discuss how well the two models fit the original distribution.

1. The partially completed histogram and the partially completed table show the time, to the nearest minute, that a random sample of motorists was delayed by roadworks on a stretch of motorway. \includegraphics[max width=\textwidth, alt={}, center]{8f3dbcb4-3260-4493-a230-12577b4ed691-04_1227_1465_354_301}

Estimate the percentage of these motorists who were delayed by the roadworks for between 8.5 and 13.5 minutes.

1. The number of hours of sunshine each day, \(y\), for the month of July at Heathrow are summarised in the table below.

A histogram was drawn to represent these data. The \(8 \leqslant y < 11\) group was represented by a bar of width 1.5 cm and height 8 cm .

1. Find the width and the height of the \(0 \leqslant y < 5\) group.
2. Use your calculator to estimate the mean and the standard deviation of the number of hours of sunshine each day, for the month of July at Heathrow.
   Give your answers to 3 significant figures. The mean and standard deviation for the number of hours of daily sunshine for the same month in Hurn are 5.98 hours and 4.12 hours respectably.
   Thomas believes that the further south you are the more consistent should be the number of hours of daily sunshine.
3. State, giving a reason, whether or not the calculations in part (b) support Thomas' belief.
4. Estimate the number of days in July at Heathrow where the number of hours of sunshine is more than 1 standard deviation above the mean. Helen models the number of hours of sunshine each day, for the month of July at Heathrow by \(\mathrm { N } \left( 6.6,3.7 ^ { 2 } \right)\).
5. Use Helen's model to predict the number of days in July at Heathrow when the number of hours of sunshine is more than 1 standard deviation above the mean.
6. Use your answers to part (d) and part (e) to comment on the suitability of Helen's model.

1. Kaff coffee is sold in packets. A seller measures the masses of the contents of a random sample of 90 packets of Kaff coffee from her stock. The results are shown in the table below.

$$\text { (You may use } \sum \mathrm { fy } ^ { 2 } = 5644 \text { 171.75) }$$ A histogram is drawn and the class \(245 \leq w < 248\) is represented by a rectangle of width 1.2 cm and height 10 cm .

1. Calculate the width and the height of the rectangle representing the class \(255 \leq w < 260\).
2. Use linear interpolation to estimate the median mass of the contents of a packet of Kaff coffee to 1 decimal place.
3. Estimate the mean and the standard deviation of the mass of the contents of a packet of Kaff coffee to 1 decimal place. The seller claims that the mean mass of the contents of the packets is more than the stated mass. Given that the stated mass of the contents of a packet of Kaff coffee is 250 g and the actual standard deviation of the contents of a packet of Kaff coffee is 4 g ,
4. test, using a 5\% level of significance, whether or not the seller's claim is justified. State your hypotheses clearly.
   (You may assume that the mass of the contents of a packet is normally distributed.)
5. Using your answers to parts (b) and (c), comment on the assumption that the mass of the contents of a packet is normally distributed.
   (Total 14 marks)

5. The histogram shows the number of hours worked in a given week by a group of 64 freelance photographers.

1. Give a reason to justify the use of a histogram to represent these data. Given that 16 of these freelance photographers spent between 10 and 20 hours working in this week,
2. estimate the number that spent between 12 and 24 hours working in this week.
3. Find an estimate for the median time spent working in this week by these 64 freelance photographers. Charlie decides to model these data using a normal distribution. Charlie calculates an estimate of the mean to be 23.9 hours to one decimal place.
4. Comment on Charlie's decision to use a normal distribution. Give a justification for your answer.

The probability distribution for \(X\), the lifetime of a light bulb, in hours, is given below. a) Suppose that a random sample of 40 light bulbs is tested, and a histogram is drawn of their lifetimes. Calculate the expected height of the bar for the interval \(262 \leqslant x < 265\).
b) Now suppose that the last two intervals are changed to \(265 \leqslant x < 268\) and \(268 \leqslant x < 300\). Explain why it is not possible to tell what will happen to the expected heights of the last two bars.
c) Celyn collects a different random sample of 40 light bulbs to test. She draws a histogram of their lifetimes and finds that it is different to the histogram referred to in part (a). Should Celyn be concerned that the two histograms are different?

The histogram shows the distances, in km, that 274 people travel to work. \includegraphics{figure_1} Given that 60 of these people travel between 10km and 20km to work, estimate

1. the number of people who travel between 22km and 45km to work, [3]
2. the median distance travelled to work by these 274 people, [2]
3. the mean distance travelled to work by these 274 people. [3]

A meteorologist measured the number of hours of sunshine, to the nearest hour, each day for 100 days. The results are summarised in the table below.

1. On graph paper, draw a histogram to represent these data. [5]
2. Calculate an estimate of the number of days that had between 6 and 9 hours of sunshine. [2]

1. Explain briefly why, for data grouped in unequal classes, the class with the highest frequency may not be the modal class. [2 marks]

In a histogram drawn to represent the annual incomes (in thousands of pounds) of 1000 families, the modal class was 15 - 20 (i.e. \(£x\), where \(15000 \leq x < 20000\)), with frequency 300. The highest frequency in a class was 400, for the class 30 - 40, and the bar representing this class was 8 cm high. The total area under the histogram was 50 cm\(^2\).

1. Find the height and the width of the bar representing the modal class. [7 marks]

The length of time, in minutes, that visitors queued for a tourist attraction is given by the following table, where, for example, '\(20 -\)' means from 20 up to but not including 30 minutes.

Queuing time (mins)	\(0 -\)	\(10 -\)	\(15 -\)	\(20 -\)	\(30 -\)	\(40 - 60\)
Number of visitors	\(15\)	\(24\)	\(x\)	\(13\)	\(10\)	\(y\)

1. State the upper class boundary of the first class. [1 mark]

A histogram is drawn to represent this data. The total area under the histogram is \(36\) cm\(^2\). The '\(10 -\)' bar has width \(1\) cm and height \(9.6\) cm. The '\(15 -\)' bar is ten times as high as the '\(40 - 60\)' bar.

1. Find the values of \(x\) and \(y\). [7 marks]
2. On graph paper, construct the histogram accurately. [5 marks]

40 people were asked to guess the length of a certain road. Each person gave their guess, \(l\) km, correct to the nearest kilometre. The results are summarised below.

\(l\)	10-12	13-15	16-20	21-30
Frequency	1	13	20	6

1. 1. Use appropriate formulae to calculate estimates of the mean and standard deviation of \(l\). [6]
   2. Explain why your answers are only estimates. [1]
2. A histogram is to be drawn to illustrate the data. Calculate the frequency density of the block for the 16-20 class. [2]
3. Explain which class contains the median value of \(l\). [2]
4. Later, the person whose guess was between 10 km and 12 km changed his guess to between 13 km and 15 km. Without calculation state whether the following will increase, decrease or remain the same:
   1. the mean of \(l\), [1]
   2. the standard deviation of \(l\). [1]

The masses, \(x\) grams, of 800 apples are summarised in the histogram. \includegraphics{figure_6}

1. On the frequency density axis, 1 cm represents \(a\) units. Find the value of \(a\). [3]
2. Find an estimate of the median mass of the apples. [4]

The diameters of 100 pebbles were measured. The measurements rounded to the nearest millimetre, \(x\), are summarised in the table. These data are to be presented on a statistical diagram.

1. For a histogram, find the frequency density of the \(10 \leqslant x \leqslant 19\) class. [2]
2. For a cumulative frequency graph, state the coordinates of the first two points that should be plotted. [2]
3. Why is it not possible to draw an exact box-and-whisker plot to illustrate the data? [1]

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{figure_7}

1. Calculate the number of pears which are between 90 and 100 mm long. [2]
2. Calculate an estimate of the mean length of the pears. Explain why your answer is only an estimate. [4]
3. Calculate an estimate of the standard deviation. [3]
4. Use your answers to parts (ii) and (iii) to investigate whether there are any outliers. [4]
5. Name the type of skewness of the distribution. [1]
6. Illustrate the data using a cumulative frequency diagram. [5]

The incomes of a sample of 918 households on an island are given in the table below.

1. Draw a histogram to illustrate the data. [5]
2. Calculate an estimate of the mean income. [3]
3. Calculate an estimate of the standard deviation of the incomes. [4]
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. [4]
5. The incomes were converted into another currency using the formula \(y = 1.15x\). Calculate estimates of the mean and variance of the incomes in the new currency. [3]

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{figure_1}

1. Find the number of sections which are between 1000 and 2000 metres in length. [2]
2. Name the type of skewness suggested by the histogram. [1]
3. State the minimum and maximum possible values of the midrange. [2]

Two fair six-sided dice are thrown. The random variable \(X\) denotes the difference between the scores on the two dice. The table shows the probability distribution of \(X\).

\(r\)	0	1	2	3	4	5
P(X = r)	\(\frac{1}{6}\)	\(\frac{5}{18}\)	\(\frac{2}{9}\)	\(\frac{1}{6}\)	\(\frac{1}{9}\)	\(\frac{1}{18}\)

1. Draw a vertical line chart to illustrate the probability distribution. [2]
2. Use a probability argument to show that
   1. P(X = 1) = \(\frac{5}{18}\). [2]
   2. P(X = 0) = \(\frac{1}{6}\). [1]
3. Find the mean value of \(X\). [2]

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

Weight	\(30 \leqslant w < 50\)	\(50 \leqslant w < 60\)	\(60 \leqslant w < 70\)	\(70 \leqslant w < 80\)	\(80 \leqslant w < 90\)
Frequency	11	10	18	14	7

1. Draw a histogram to illustrate these data. [5]
2. Calculate estimates of the mean and standard deviation of \(w\). [4]
3. Use your answers to part (ii) to investigate whether there are any outliers. [3]

The weights, \(x\) grams, of a random sample of 50 carrots of variety B are summarised as follows. $$n = 50 \quad \sum x = 3624.5 \quad \sum x^2 = 265416$$

1. Calculate the mean and standard deviation of \(x\). [3]
2. Compare the central tendency and variation of the weights of varieties A and B. [2]