Calculate using histogram bar dimensions

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.

8

1. Joseph drew a histogram to show information about one Local Authority. He used data from the "Age structure by LA 2011" tab in the large data set. The table shows an extract from the data that he used.

   Age group	0 to 4
   Frequency	2143

   Joseph used a scale of \(1 \mathrm {~cm} = 1000\) units on the frequency density axis. Calculate the height of the histogram block for the 0 to 4 class.
2. Magdalene wishes to draw a statistical diagram to illustrate some of the data from the "Method of travel by LA 2011" tab in the large data set. State why she cannot draw a histogram.

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.

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.

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]

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]

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 number of patients attending a hospital trauma clinic each day was recorded over several months, giving the data in the table below.

Number of patients	10 - 19	20 - 29	30 - 34	35 - 39	40 - 44	45 - 49	50 - 69
Frequency	2	18	24	30	27	14	5

These data are represented by a histogram. Given that the bar representing the 20 - 29 group is 2 cm wide and 7.2 cm high,

1. calculate the dimensions of the bars representing the groups
   1. 30 - 34
   2. 50 - 69
   [6]
2. Use linear interpolation to estimate the median and quartiles of these data. [6]

The lowest and highest numbers of patients recorded were 14 and 67 respectively.

1. Represent these data with a boxplot drawn on graph paper and describe the skewness of the distribution. [5]

A histogram was drawn to show the distribution of age in completed years of the participants on an outward-bound course. There were 32 people aged 30-34 years on the course. The height of the rectangle representing this group was 19.2 cm and it was 1 cm in width. Given that there were 28 people aged 35-39 years,

1. find the height of the rectangle representing this group. [3 marks]

Given that the height of the rectangle representing people aged 40-59 years was 2.7 cm,

1. find the number of people on the course in this age group. [3 marks]