2.02b Histogram: area represents frequency

1. The stem lengths of a sample of 120 tulips are recorded in the grouped frequency table below.

A histogram is drawn to represent these data.
The area of the bar representing the \(40 \leqslant x < 42\) class is \(16.5 \mathrm {~cm} ^ { 2 }\)

1. Calculate the exact area of the bar representing the \(42 \leqslant x < 45\) class. The height of the tallest bar in the histogram is 10 cm .
2. Find the exact height of the second tallest bar. \(Q _ { 1 }\) for these data is 45 cm .
3. Use linear interpolation to find an estimate for
   1. \(Q _ { 2 }\)
   2. the interquartile range. One measure of skewness is given by $$\frac { Q _ { 3 } - 2 Q _ { 2 } + Q _ { 1 } } { Q _ { 3 } - Q _ { 1 } }$$
4. By calculating this measure, describe the skewness of these data.

1. The time taken to complete a puzzle, in minutes, is recorded for each person in a club. The times are summarised in a grouped frequency distribution and represented by a histogram.

One of the class intervals has a frequency of 20 and is shown by a bar of width 1.5 cm and height 12 cm on the histogram. The total area under the histogram is \(94.5 \mathrm {~cm} ^ { 2 }\) Find the number of people in the club.

1. A teacher selects a random sample of 56 students and records, to the nearest hour, the time spent watching television in a particular week.

1. Find the mid-points of the 21-25 hour and 31-40 hour groups. A histogram was drawn to represent these data. The 11-20 group was represented by a bar of width 4 cm and height 6 cm .
2. Find the width and height of the 26-30 group.
3. Estimate the mean and standard deviation of the time spent watching television by these students.
4. Use linear interpolation to estimate the median length of time spent watching television by these students. The teacher estimated the lower quartile and the upper quartile of the time spent watching television to be 15.8 and 29.3 respectively.
5. State, giving a reason, the skewness of these data.

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 $$\frac { 3 ( \text { mean - median } ) } { \text { standard deviation } } .$$
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. The total amount of time a secretary spent on the telephone in a working day was recorded to the nearest minute. The data collected over 40 days are summarised in the table below.

Draw a histogram to illustrate these data

1. A teacher recorded, to the nearest hour, the time spent watching television during a particular week by each child in a random sample. The times were summarised in a grouped frequency table and represented by a histogram.

One of the classes in the grouped frequency distribution was 20-29 and its associated frequency was 9. On the histogram the height of the rectangle representing that class was 3.6 cm and the width was 2 cm .

1. Give a reason to support the use of a histogram to represent these data.
2. Write down the underlying feature associated with each of the bars in a histogram.
3. Show that on this histogram each child was represented by \(0.8 \mathrm {~cm} ^ { 2 }\). The total area under the histogram was \(24 \mathrm {~cm} ^ { 2 }\).
4. Find the total number of children in the group.

3. The histogram in Figure 1 shows the time taken, to the nearest minute, for 140 runners to complete a fun run. \begin{figure}[h] \end{figure} Use the histogram to calculate the number of runners who took between 78.5 and 90.5 minutes to complete the fun run.

5. In a shopping survey a random sample of 104 teenagers were asked how many hours, to the nearest hour, they spent shopping in the last month. The results are summarised in the table below. A histogram was drawn and the group ( \(8 - 10\) ) hours was represented by a rectangle that was 1.5 cm wide and 3 cm high.

1. Calculate the width and height of the rectangle representing the group (16-25) hours.
2. Use linear interpolation to estimate the median and interquartile range.
3. Estimate the mean and standard deviation of the number of hours spent shopping.
4. State, giving a reason, the skewness of these data.
5. State, giving a reason, which average and measure of dispersion you would recommend to use to summarise these data.

1. The histogram in Figure 1 shows the time, to the nearest minute, that a random sample of 100 motorists were delayed by roadworks on a stretch of motorway.

\begin{figure}[h] \end{figure}

1. Complete the table.

   Delay (minutes)	Number of motorists
   4-6	6
   7-8
   9	21
   10-12	45
   13-15	9
   16-20

2. Estimate the number of motorists who were delayed between 8.5 and 13.5 minutes by the roadworks.

1. A survey of 100 households gave the following results for weekly income \(\pounds y\).

(You may use \(\sum f y ^ { 2 } = 12452\) 800)
A histogram was drawn and the class \(200 \leqslant y < 240\) was represented by a rectangle of width 2 cm and height 7 cm .

1. Calculate the width and the height of the rectangle representing the class $$320 \leqslant y < 400$$
2. Use linear interpolation to estimate the median weekly income to the nearest pound.
3. Estimate the mean and the standard deviation of the weekly income for these data. One measure of skewness is \(\frac { 3 ( \text { mean } - \text { median } ) } { \text { standard deviation } }\).
4. Use this measure to calculate the skewness for these data and describe its value. Katie suggests using the random variable \(X\) which has a normal distribution with mean 320 and standard deviation 150 to model the weekly income for these data.
5. Find \(\mathrm { P } ( 240 < X < 400 )\).
6. With reference to your calculations in parts (d) and (e) and the data in the table, comment on Katie's suggestion.

6. The labelling on bags of garden compost indicates that the bags weigh 20 kg . The weights of a random sample of 50 bags are summarised in the table below.

1. On graph paper, draw a histogram of these data.
2. Using the coding \(y = 10\) (weight in \(\mathrm { kg } - 14\) ), find an estimate for the mean and standard deviation of the weight of a bag of compost.
   [0pt] [Use \(\Sigma f y ^ { 2 } = 171\) 503.75]
3. Using linear interpolation, estimate the median. The company that produces the bags of compost wants to improve the accuracy of the labelling. The company decides to put the average weight in kg on each bag.
4. Write down which of these averages you would recommend the company to use. Give a reason for your answer.

2. The following table summarises the distances, to the nearest km , that 134 examiners travelled to attend a meeting in London.

1. Give a reason to justify the use of a histogram to represent these data.
2. Calculate the frequency densities needed to draw a histogram for these data.
   (DO NOT DRAW THE HISTOGRAM)
3. Use interpolation to estimate the median \(Q _ { 2 }\), the lower quartile \(Q _ { 1 }\), and the upper quartile \(Q _ { 3 }\) of these data. The mid-point of each class is represented by \(x\) and the corresponding frequency by \(f\). Calculations then give the following values $$\Sigma f _ { x } = 8379.5 \quad \text { and } \quad \Sigma f _ { x ^ { 2 } } = 557489.75$$
4. Calculate an estimate of the mean and an estimate of the standard deviation for these data. One coefficient of skewness is given by $$\frac { Q _ { 3 } - 2 Q _ { 2 } + Q _ { 1 } } { Q _ { 3 } - Q _ { 1 } }$$
5. Evaluate this coefficient and comment on the skewness of these data.
6. Give another justification of your comment in part (e).

5. \begin{figure}[h] \end{figure} Figure 2 shows a histogram for the variable \(t\) which represents the time taken, in minutes, by a group of people to swim 500 m .

1. Complete the frequency table for \(t\).

   \(t\)	\(5 - 10\)	\(10 - 14\)	\(14 - 18\)	\(18 - 25\)	\(25 - 40\)
   Frequency	10	16	24

2. Estimate the number of people who took longer than 20 minutes to swim 500 m .
3. Find an estimate of the mean time taken.
4. Find an estimate for the standard deviation of \(t\).
5. Find the median and quartiles for \(t\). One measure of skewness is found using \(\frac { 3 ( \text { mean } - \text { median } ) } { \text { standard deviation } }\).
6. Evaluate this measure and describe the skewness of these data.

3. The variable \(x\) was measured to the nearest whole number. Forty observations are given in the table below. A histogram was drawn and the bar representing the \(10 - 15\) class has a width of 2 cm and a height of 5 cm . For the \(16 - 18\) class find

1. the width,
2. the height
   of the bar representing this class.

5. A teacher selects a random sample of 56 students and records, to the nearest hour, the time spent watching television in a particular week.

1. Find the mid-points of the 21-25 hour and 31-40 hour groups. A histogram was drawn to represent these data. The \(11 - 20\) group was represented by a bar of width 4 cm and height 6 cm .
2. Find the width and height of the 26-30 group.
3. Estimate the mean and standard deviation of the time spent watching television by these students.
4. Use linear interpolation to estimate the median length of time spent watching television by these students. The teacher estimated the lower quartile and the upper quartile of the time spent watching television to be 15.8 and 29.3 respectively.
5. State, giving a reason, the skewness of these data.

5. \begin{figure}[h] \end{figure} A policeman records the speed of the traffic on a busy road with a 30 mph speed limit. He records the speeds of a sample of 450 cars. The histogram in Figure 2 represents the results.

1. Calculate the number of cars that were exceeding the speed limit by at least 5 mph in the sample.
2. Estimate the value of the mean speed of the cars in the sample.
3. Estimate, to 1 decimal place, the value of the median speed of the cars in the sample.
4. Comment on the shape of the distribution. Give a reason for your answer.
5. State, with a reason, whether the estimate of the mean or the median is a better representation of the average speed of the traffic on the road.

3. An agriculturalist is studying the yields, \(y \mathrm {~kg}\), from tomato plants. The data from a random sample of 70 tomato plants are summarised below. $$\text { (You may use } \sum \mathrm { f } x = 755 \text { and } \sum \mathrm { f } x ^ { 2 } = 12037.5 \text { ) }$$ A histogram has been drawn to represent these data. The bar representing the yield \(5 \leqslant y < 10\) has a width of 1.5 cm and a height of 8 cm .

1. Calculate the width and the height of the bar representing the yield \(15 \leqslant y < 25\)
2. Use linear interpolation to estimate the median yield of the tomato plants.
3. Estimate the mean and the standard deviation of the yields of the tomato plants.
4. Describe, giving a reason, the skewness of the data.
5. Estimate the number of tomato plants in the sample that have a yield of more than 1 standard deviation above the mean.

1. The table shows the time, to the nearest minute, spent waiting for a taxi by each of 80 people one Sunday afternoon.

1. Write down the upper class boundary for the \(2 - 4\) minute interval. A histogram is drawn to represent these data. The height of the tallest bar is 6 cm .
2. Calculate the height of the second tallest bar.
3. Estimate the number of people with a waiting time between 3.5 minutes and 7 minutes.
4. Use linear interpolation to estimate the median, the lower quartile and the upper quartile of the waiting times.
5. Describe the skewness of these data, giving a reason for your answer.

6. The times, in seconds, spent in a queue at a supermarket by 85 randomly selected customers, are summarised in the table below. A histogram was drawn to represent these data. The \(30 - 60\) group was represented by a bar of width 1.5 cm and height 1 cm .

1. Find the width and the height of the \(70 - 80\) group.
2. Use linear interpolation to estimate the median of this distribution. Given that \(x\) denotes the midpoint of each group in the table and $$\sum f x = 6460 \quad \sum f x ^ { 2 } = 529400$$
3. calculate an estimate for
   1. the mean,
   2. the standard deviation,
      for the above data. One measure of skewness is given by $$\text { coefficient of skewness } = \frac { 3 ( \text { mean } - \text { median } ) } { \text { standard deviation } }$$
4. Evaluate this coefficient and comment on the skewness of these data.

5. A midwife records the weights, in kg , of a sample of 50 babies born at a hospital. Her results are given in the table below. [You may use \(\sum \mathrm { f } x ^ { 2 } = 611.375\) ] A histogram has been drawn to represent these data. The bar representing the weight \(2 \leqslant w < 3\) has a width of 1 cm and a height of 4 cm .

1. Calculate the width and height of the bar representing a weight of \(3 \leqslant w < 3.5\)
2. Use linear interpolation to estimate the median weight of these babies.
3. 1. Show that an estimate of the mean weight of these babies is 3.43 kg .
   2. Find an estimate of the standard deviation of the weights of these babies. Shyam decides to model the weights of babies born at the hospital, by the random variable \(W\), where \(W \sim \mathrm {~N} \left( 3.43,0.65 ^ { 2 } \right)\)
4. Find \(\mathrm { P } ( W < 3 )\)
5. With reference to your answers to (b), (c)(i) and (d) comment on Shyam's decision. A newborn baby weighing 3.43 kg is born at the hospital.
6. Without carrying out any further calculations, state, giving a reason, what effect the addition of this newborn baby to the sample would have on your estimate of the
   1. mean,
   2. standard deviation.

2. An estate agent is studying the cost of office space in London. He takes a random sample of 90 offices and calculates the cost, \(\pounds x\) per square foot. His results are given in the table below. A histogram is drawn for these data and the bar representing \(50 \leqslant x < 60\) is 2 cm wide and 8 cm high.

1. Calculate the width and height of the bar representing \(20 \leqslant x < 40\)
2. Use linear interpolation to estimate the median cost.
3. Estimate the mean cost of office space for these data.
4. Estimate the standard deviation for these data.
5. Describe, giving a reason, the skewness. Rika suggests that the cost of office space in London can be modelled by a normal distribution with mean \(\pounds 50\) and standard deviation \(\pounds 10\)
6. With reference to your answer to part (e), comment on Rika's suggestion.
7. Use Rika's model to estimate the 80th percentile of the cost of office space in London.

2. The following grouped frequency distribution summarises the number of minutes, to the nearest minute, that a random sample of 100 motorists were delayed by roadworks on a stretch of motorway one Monday. (You may use \(\sum \mathrm { f } x ^ { 2 } = 8096.25\) ) A histogram has been drawn to represent these data. The bar representing a delay of (3-6) minutes has a width of 2 cm and a height of 9.5 cm .

1. Calculate the width and the height of the bar representing a delay of (11-15) minutes.
2. Use linear interpolation to estimate the median delay.
3. Calculate an estimate of the mean delay.
4. Calculate an estimate of the standard deviation of the delays. One coefficient of skewness is given by \(\frac { 3 ( \text { mean } - \text { median } ) } { \text { standard deviation } }\)
5. Evaluate this coefficient for the above data, giving your answer to 2 significant figures. On the following Friday, the coefficient of skewness for the delays on this stretch of motorway was - 0.22
6. State, giving a reason, how the delays on this stretch of motorway on Friday are different from the delays on Monday.

7. A college organised a 'fun run'. The times, to the nearest minute, of a random sample of 100 students who took part are summarised in the table below.

1. Give a reason to support the use of a histogram to represent these data.
2. Write down the upper class boundary and the lower class boundary of the class 40-44.
3. On graph paper, draw a histogram to represent these data. END

1. A histogram is to be drawn to represent the following grouped continuous data:

The ' \(10 - 20\) ' bar has height 6 cm and width 4 cm . Calculate

1. the height of the ' \(20 - 25\) ' bar,
2. the total area under the histogram.

3. The frequency distribution for the lengths of 108 fish in an aquarium is given by the following table. The lengths of the fish ranged from 5 cm to 90 cm .

1. Calculate estimates of the three quartiles of the distribution.
2. On graph paper, draw a box and whisker plot of the data.
3. Hence describe the skewness of the distribution.
4. If the data were represented by a histogram, what would be the ratio of the heights of the shortest and highest bars?