Calculate frequency density from frequency

2 The lengths of cars travelling on a car ferry are noted. The data are summarised in the following table.

1. Find the value of \(a\).
2. Draw a histogram on graph paper to represent the data.
3. Find the probability that a randomly chosen car on the ferry is less than 3.20 m in length.

3 Robert has a part-time job delivering newspapers. On a number of days he noted the time, correct to the nearest minute, that it took him to do his job. Robert used his results to draw up the following table; two of the values in the table are denoted by \(a\) and \(b\).

1. Find the values of \(a\) and \(b\).
2. On graph paper, draw a histogram to represent Robert's times.

3 Robert has a part-time job delivering newspapers. On a number of days he noted the time, correct to the nearest minute, that it took him to do his job. Robert used his results to draw up the following table; two of the values in the table are denoted by \(a\) and \(b\).

1. Find the values of \(a\) and \(b\).
2. Draw a histogram to represent Robert's times.

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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. 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.

4 The table shows information about the time, \(t\) minutes correct to the nearest minute, taken by 50 people to complete a race.

1. In a histogram illustrating the data, the height of the block for the \(31 \leqslant t \leqslant 35\) class is 5.6 cm . Find the height of the block for the \(28 \leqslant t \leqslant 30\) class. (There is no need to draw the histogram.)
2. The data in the table are used to estimate the median time. State, with a reason, whether the estimated median time is more than 33 minutes, less than 33 minutes or equal to 33 minutes.
3. Calculate estimates of the mean and standard deviation of the data.
4. It was found that the winner's time had been incorrectly recorded and that it was actually less than 27 minutes 30 seconds. State whether each of the following will increase, decrease or remain the same:
   1. the mean,
   2. the standard deviation,
   3. the median,
   4. the interquartile range.

6 An online store has a total of 930 different types of women's running shoe on sale. The prices in pounds of the types of women's running shoe are summarised in the table below.

1. Calculate estimates of the mean and standard deviation of the shoe prices.
2. Calculate an estimate of the percentage of types of shoe that cost at least \(\pounds 100\).
3. Draw a histogram to illustrate the data. The corresponding histogram below shows the prices in pounds of the 990 types of men's running shoe on sale at the same online store. \includegraphics[max width=\textwidth, alt={}, center]{aff0c5b2-011b-49a0-bf05-6d905f890eba-4_643_1192_340_440}
4. State the type of skewness shown by the histogram for men's running shoes.
5. Martin is investigating the percentage of types of shoe on sale at the store that cost more than \(\pounds 100\). He believes that this percentage is greater for men's shoes than for women's shoes. Estimate the percentage for men's shoes and comment on whether you can be certain which percentage is higher.
6. You are given that the mean and standard deviation of the prices of men's running shoes are \(\pounds 68.83\) and \(\pounds 42.93\) respectively. Compare the central tendency and variation of the prices of men's and women's running shoes at the store.

1. The lengths, \(x \mathrm {~mm}\), of 50 pebbles are summarised in the table below.

A histogram is drawn to represent these data.
The bar representing the class \(32 \leqslant x < 36\) is 2.5 cm wide and 7.5 cm tall.

1. Calculate the width and the height of the bar representing the class \(30 \leqslant x < 32\)
2. Using linear interpolation, estimate the median of \(x\) The weight, \(w\) grams, of each of the 50 pebbles is coded using \(10 y = w - 20\) These coded data are summarised by $$\sum y = 104 \quad \sum y ^ { 2 } = 233.54$$
3. Show that the mean of \(w\) is 40.8
4. Calculate the standard deviation of \(w\) The weight of a pebble recorded as 40.8 grams is added to the sample.
5. Without carrying out any further calculations, state, giving a reason, what effect this would have on the value of
   1. the mean of \(w\)
   2. the standard deviation of \(w\)

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.

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).

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.

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.

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.

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]

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