2.02b Histogram: area represents frequency

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]

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]

The histogram shows information about the numbers of cars in five different price ranges, sold in one year at a car showroom. \includegraphics{figure_9} It is given that 66 cars in the price range £10000 to £20000 were sold.

1. Find the number of cars sold in the price range £50000 to £90000. [1]
2. State the units of the frequency density. [1]
3. Suggest one change that the management could make to the diagram so that it would provide more information. [1]
4. Estimate the number of cars sold in the price range £50000 to £60000. [1]

The table below shows the frequencies for a set of data from a continuous variable \(X\) A histogram is drawn to represent this data. Find the frequency density of the bar in the histogram representing the class \(24 < x \leq 42\) Circle your answer. [1 mark] 2 \qquad 18 \qquad 36 \qquad 70

The histogram below shows the heights, in cm, of male A-level students at a particular school. \includegraphics{figure_12} Which class interval contains the median height? Circle your answer. [1 mark] \([155, 160)\) \quad \([160, 170)\) \quad \([170, 180)\) \quad \([180, 190]\)

The masses of a random sample of 120 boulders in a certain area were recorded. The results are summarized in the histogram. \includegraphics{figure_5}

1. Calculate the number of boulders with masses between 60 and 65 kg. [2]
2. 1. Use midpoints to find estimates of the mean and standard deviation of the masses of the boulders in the sample. [3]
   2. Explain why your answers are only estimates. [1]
3. Use your answers to part (b)(i) to determine an estimate of the number of outliers, if any, in the distribution. [2]
4. Give one advantage of using a histogram rather than a pie chart in this context. [1]

Doug has a list of times taken by competitors in a 'fun run'. He has grouped the data and calculated the frequency densities in order to draw a histogram to represent the information. Some of the data are presented in Fig. 2.

Time in minutes	\(15-\)	\(20-\)	\(25-\)	\(35-\)	\(45-60\)
Number of runners	12	23	59	71
Frequency density	2.4		5.9	7.1	1.4

Fig. 2

1. Write down the missing values in the copy of Fig. 2 in the Printed Answer Booklet. [2]
2. Doug labels the horizontal axis on the histogram 'time in minutes' and the vertical axis 'number of minutes per runner'. State which one of these labels is incorrect and write down a correct version. [1]

Kareem bought some tomatoes. He recorded the mass of each tomato and displayed the results in a histogram, which is shown below. \includegraphics{figure_7} Determine how many tomatoes Kareem bought. [2]

A quality control department checks the lifetimes of batteries produced by a company. The lifetimes, \(x\) minutes, for a random sample of 80 'Superstrength' batteries are shown in the table below.

Lifetime	\(160 \leq x < 165\)	\(165 \leq x < 168\)	\(168 \leq x < 170\)	\(170 \leq x < 172\)	\(172 \leq x < 175\)	\(175 \leq x < 180\)
Frequency	5	14	20	21	16	4

1. Estimate the proportion of these batteries which have a lifetime of at least 174.0 minutes. [2]
2. Use the data in the table to estimate
   [3]

The data in the table on the previous page are represented in the following histogram, Fig 15. \includegraphics{figure_15} A quality control manager models the data by a Normal distribution with the mean and standard deviation you calculated in part (b).

1. Comment briefly on whether the histogram supports this choice of model. [2]
2. 1. Use this model to estimate the probability that a randomly selected battery will have a lifetime of more than 174.0 minutes.
   2. Compare your answer with your answer to part (a). [3]

The company also manufactures 'Ultrapower' batteries, which are stated to have a mean lifetime of 210 minutes.

1. A random sample of 8 Ultrapower batteries is selected. The mean lifetime of these batteries is 207.3 minutes. Carry out a hypothesis test at the 5% level to investigate whether the mean lifetime is as high as stated. You should use the following hypotheses \(\text{H}_0 : \mu = 210\), \(\text{H}_1 : \mu < 210\), where \(\mu\) represents the population mean for Ultrapower batteries. You should assume that the population is Normally distributed with standard deviation 3.4. [5]

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{figure_2} One of the 150 plants is chosen at random, and its height, \(X\) cm, is noted.

1. Show that P\((20 < X < 30) = 0.147\), correct to 3 significant figures. [2]

Sam suggests that the distribution of \(X\) can be well modelled by the distribution N\((40, 100)\).

1. 1. Give a brief justification for the use of the normal distribution in this context. [1]
   2. Give a brief justification for the choice of the parameter values 40 and 100. [2]
2. Use Sam's model to find P\((20 < X < 30)\). [1]

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 N\((m, s^2)\) as her model.

1. Use Nina's model to find P\((20 < X < 30)\). [4]
2. 1. Complete the table in the Printed Answer Booklet to show the probabilities obtained from Sam's model and Nina's model. [2]
   2. By considering the different ranges of values of \(X\) given in the table, discuss how well the two models fit the original distribution. [2]

The histogram shows information about the lengths, \(l\) centimetres, of a sample of worms of a certain species. \includegraphics{figure_2} The number of worms in the sample with lengths in the class \(3 \leq l < 4\) is 30.

1. Find the number of worms in the sample with lengths in the class \(0 \leq l < 2\). [2]
2. Find an estimate of the number of worms in the sample with lengths in the range \(4.5 \leq l < 5.5\). [3]

The histogram shows information about the lengths, \(l\) centimetres, of a sample of worms of a certain species. \includegraphics{figure_1} The number of worms in the sample with lengths in the class \(3 \leqslant l < 4\) is 30.

1. Find the number of worms in the sample with lengths in the class \(0 \leqslant l < 2\). [2]
2. Find an estimate of the number of worms in the sample with lengths in the range \(4.5 \leqslant l < 5.5\). [3]