2.02b Histogram: area represents frequency

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 The heights \(x \mathrm {~cm}\) of 100 boys in Year 7 at a school are summarised in the table below.

1. Estimate the number of boys who have heights of at least 155 cm .
2. Calculate an estimate of the median height of the 100 boys.
3. Draw a histogram to illustrate the data. The histogram below shows the heights of 100 girls in Year 7 at the same school. \includegraphics[max width=\textwidth, alt={}, center]{76283206-687f-45d6-9204-952d60843cf1-3_865_1349_1297_349}
4. How many more girls than boys had heights exceeding 160 cm ?
5. Calculate an estimate of the mean height of the 100 girls.

5 The frequency table below shows the distance travelled by 1200 visitors to a particular UK tourist destination in August 2008.

1. Draw a histogram on graph paper to illustrate these data.
2. Calculate an estimate of the median distance.

3 The lifetimes in hours of 90 components are summarised in the table.

1. Draw a histogram to illustrate these data.
2. In which class interval does the median lie? Justify your answer.

6 The engine sizes \(x \mathrm {~cm} ^ { 3 }\) of a sample of 80 cars are summarised in the table below.

1. Draw a histogram to illustrate the distribution.
2. A student claims that the midrange is \(2750 \mathrm {~cm} ^ { 3 }\). Discuss briefly whether he is likely to be correct.
3. Calculate estimates of the mean and standard deviation of the engine sizes. Explain why your answers are only estimates.
4. Hence investigate whether there are any outliers in the sample.
5. A vehicle duty of \(\pounds 1000\) is proposed for all new cars with engine size greater than \(2000 \mathrm {~cm} ^ { 3 }\). Assuming that this sample of cars is representative of all new cars in Britain and that there are 2.5 million new cars registered in Britain each year, calculate an estimate of the total amount of money that this vehicle duty would raise in one year.
6. Why in practice might your estimate in part (v) turn out to be too high?

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. Joshua is investigating the daily total rainfall in Hurn for May to October 2015

Using the information from the large data set, Joshua wishes to calculate the mean of the daily total rainfall in Hurn for May to October 2015

1. Using your knowledge of the large data set, explain why Joshua needs to clean the data before calculating the mean. Using the information from the large data set, he produces the grouped frequency table below.

   Daily total rainfall ( \(r \mathrm {~mm}\) )	Frequency	Midpoint ( \(\boldsymbol { x } \mathbf { m m }\) )
   \(0 \leqslant r < 0.5\)	121	0.25
   \(0.5 \leqslant r < 1.0\)	10	0.75
   \(1.0 \leqslant r < 5.0\)	24	3.0
   \(5.0 \leqslant r < 10.0\)	12	7.5
   \(10.0 \leqslant r < 30.0\)	17	20.0

   $$\text { You may use } \sum \mathrm { f } x = 539.75 \text { and } \sum \mathrm { f } x ^ { 2 } = 7704.1875$$
2. Use linear interpolation to calculate an estimate for the upper quartile of the daily total rainfall.
3. Calculate an estimate for the standard deviation of the daily total rainfall in Hurn for May to October 2015
4. 1. State the assumption involved with using class midpoints to calculate an estimate of a mean from a grouped frequency table.
   2. Using your knowledge of the large data set, explain why this assumption does not hold in this case.
   3. State, giving a reason, whether you would expect the actual mean daily total rainfall in Hurn for May to October 2015 to be larger than, smaller than or the same as an estimate based on the grouped frequency table.

1. \begin{figure}[h] \end{figure} The histogram in Figure 1 shows the times taken to complete a crossword by a random sample of students. The number of students who completed the crossword in more than 15 minutes is 78
Estimate the percentage of students who took less than 11 minutes to complete the crossword.

1. Customers in a shop have to queue to pay.

The partially completed table below and partially completed histogram opposite, give information about the time, \(x\) minutes, spent in the queue by each of 112 customers one day. No customer spent less than 1 minute or longer than 8 minutes in the queue.

1. Complete the table.
2. Complete the histogram. Ting decides to model the frequency density for these 112 customers by a curve with equation $$y = \frac { k } { x ^ { 2 } } \quad 1 \leqslant x \leqslant 8$$ where \(k\) is a constant.
3. Find the value of \(k\)
   \includegraphics[max width=\textwidth, alt={}]{6a0b46f8-7a6a-4ed8-8c7a-9772787f155a-07_1584_1189_285_443}
   Only use this grid if you need to redraw your histogram. \includegraphics[max width=\textwidth, alt={}, center]{6a0b46f8-7a6a-4ed8-8c7a-9772787f155a-09_1582_1192_367_440} \includegraphics[max width=\textwidth, alt={}, center]{6a0b46f8-7a6a-4ed8-8c7a-9772787f155a-09_2267_51_307_36}

1. The partially completed table and partially completed histogram give information about the ages of passengers on an airline.

There were no passengers aged 90 or over. \includegraphics[max width=\textwidth, alt={}, center]{6dfefd72-338f-40be-ac37-aef56bfaccaa-04_1173_1792_721_139}

1. Complete the histogram.
2. Use linear interpolation to estimate the median age. An outlier is defined as a value greater than \(Q _ { 3 } + 1.5 \times\) interquartile range.
   Given that \(Q _ { 1 } = 27.3\) and \(Q _ { 3 } = 58.9\)
3. determine, giving a reason, whether or not the oldest passenger could be considered as an outlier.
   (2)

1. A medical researcher is studying the number of hours, \(T\), a patient stays in hospital following a particular operation.

The histogram on the page opposite summarises the results for a random sample of 90 patients.

1. Use the histogram to estimate \(\mathrm { P } ( 10 < T < 30 )\) For these 90 patients the time spent in hospital following the operation had
   • a mean of 14.9 hours
   • a standard deviation of 9.3 hours
   Tomas suggests that \(T\) can be modelled by \(\mathrm { N } \left( 14.9,9.3 ^ { 2 } \right)\)
2. With reference to the histogram, state, giving a reason, whether or not Tomas' model could be suitable. Xiang suggests that the frequency polygon based on this histogram could be modelled by a curve with equation $$y = k x \mathrm { e } ^ { - x } \quad 0 \leqslant x \leqslant 4$$ where
   • \(x\) is measured in tens of hours
   • \(k\) is a constant
   • Use algebraic integration to show that
   $$\int _ { 0 } ^ { n } x \mathrm { e } ^ { - x } \mathrm {~d} x = 1 - ( n + 1 ) \mathrm { e } ^ { - n }$$
3. Show that, for Xiang's model, \(k = 99\) to the nearest integer.
4. Estimate \(\mathrm { P } ( 10 < T < 30 )\) using
   1. Tomas' model of \(T \sim \mathrm {~N} \left( 14.9,9.3 ^ { 2 } \right)\)
   2. Xiang's curve with equation \(y = 99 x \mathrm { e } ^ { - x }\) and the answer to part (c) The researcher decides to use Xiang's curve to model \(\mathrm { P } ( a < T < b )\)
5. State one limitation of Xiang's model. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Question 6 continued} \includegraphics[alt={},max width=\textwidth]{a067577e-e2a6-440b-9d22-d558fade15f0-17_1164_1778_294_146}
   \end{figure} Time in hours

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.

8 The histogram shows information about the lengths, \(l\) centimetres, of a sample of worms of a certain species. \includegraphics[max width=\textwidth, alt={}, center]{4c6b7c92-2fc9-4d4f-a199-8e70f34e5eed-5_904_1284_488_242} 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. Find an estimate of the number of worms in the sample with lengths in the range \(4.5 \leqslant l < 5.5\).

12 Doctors are investigating the weights of adult males registered at their surgery. One week they collect a sample by noting the weight in kilograms of all the adult males who have an appointment at their surgery.

1. State the sampling method they use.
2. Explain why this method will not generate a simple random sample of all the adult males registered at their surgery. They represent the data using a histogram. \includegraphics[max width=\textwidth, alt={}, center]{82438df0-6550-4ffd-92d8-3c67bec59a6b-09_1166_1243_726_233} An incomplete frequency table for the data is shown below.

   Weight in kg	\(50 -\)	\(65 -\)	\(75 -\)	\(80 -\)	\(90 -\)	\(100 - 120\)
   Frequency		8

3. Complete the copy of the frequency table in the Printed Answer Booklet. One of these patients is selected at random.
4. Determine an estimate of the probability that he weighs either less than 60 kg or more than 110 kg .
5. Explain why your answer to part (d) is an estimate and not exact.

11 James is investigating the amount of time retired people spend each day using social media. He collects a sample by advertising in a local newspaper for people to complete an online survey.

1. State
   James processes his data in order to draw a histogram. His table of results is shown below.

   Time spent using social media in minutes	\(0 -\)	\(15 -\)	\(30 -\)	\(60 -\)	\(120 - 240\)
   Number of people per minute	12.2	14.0	8.4	7.3	3.1

2. Show that the size of the sample is 1455 .
3. Calculate an estimate of the probability that a retired person spends more than an hour per day using social media.

3 The histogram shows the amount spent on electricity in pounds in a sample of households in March 2023. \includegraphics[max width=\textwidth, alt={}, center]{8e48bbd3-2166-49e7-8906-833261f331ca-04_542_1276_1133_244}

1. Describe the shape of the distribution. A total of 16 households each spent between \(\pounds 60\) and \(\pounds 65\) on electricity.
2. Determine how many households were in the sample altogether.

8 Rosella is carrying out an investigation into the age at which adults retire from work in the city where she lives. She collects a sample of size 50 , ensuring this comprises of 25 randomly selected retired men and 25 randomly selected retired women.

1. State the name of the sampling method she uses. Fig. 8.1 shows the data she obtains in a frequency table and Fig. 8.2 shows these data displayed in a histogram. \begin{table}[h]

   Age in years at retirement	\(45 -\)	\(50 -\)	\(55 -\)	\(60 -\)	\(65 -\)	\(70 -\)	\(75 - 80\)
   Frequency density	0.4	1.8	2.4	2.2	1.8	1.2	0.2

   \captionsetup{labelformat=empty} \caption{Fig. 8.1}
   \end{table} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{cea67565-8074-4703-8e1a-09b98e380baf-08_805_1006_1160_244} \captionsetup{labelformat=empty} \caption{Fig. 8.2}
   \end{figure}
2. How many people in the sample are aged between 50 and 55? Rosella obtains a list of the names of all 4960 people who have retired in the city during the previous month.
3. Describe how Rosella could collect a sample of size 200 from her list using
   Rosella collects two simple random samples, one of size 200 and one of size 500, from her list. The histograms in Fig. 8.3 show the data from the sample of size 200 on the left and the data from the sample of 500 on the right. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{cea67565-8074-4703-8e1a-09b98e380baf-09_659_1909_388_77} \captionsetup{labelformat=empty} \caption{Fig. 8.3}
   \end{figure}
4. With reference to the histograms shown in Fig. 8.2 and Fig. 8.3, explain why it appears reasonable to model the age of retirement in this city using the Normal distribution. Summary statistics for the sample of 500 are shown in Fig. 8.4. \begin{table}[h]

   Statistics
   n	500
   Mean	60.0515
   \(\sigma\)	6.5717
   s	6.5783
   \(\Sigma x\)	30025.7601
   \(\Sigma \mathrm { x } ^ { 2 }\)	1824686.322
   Min	36.0793
   Q1	55.2573
   Median	59.9202
   Q3	64.4239
   Max	81.742

   \captionsetup{labelformat=empty} \caption{Fig. 8.4}
   \end{table}
5. Use an appropriate Normal model based on the information in Fig. 8.4 to estimate the number of people aged over 65 who retired in the city in the previous month.
6. Identify a limitation in using this model to predict the number of people aged over 65 retiring in the following month.

2. 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.
(3)
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5. The weights, in grams, of a random sample of 48 broad beans are summarised in the table. (You may assume \(\sum \mathrm { fy } { } ^ { 2 } = 101.56\) ) A histogram was drawn to represent these data. The \(2.1 < x \leqslant 2.7\) class was represented by a bar of width 1.5 cm and height 1 cm .

1. Find the width and height of the \(0.9 < x \leqslant 1.1\) class.
2. Give a reason to justify the use of a histogram to represent these data.
3. Estimate the mean and the standard deviation of the weights of these broad beans.
4. Use linear interpolation to estimate the median of the weights of these broad beans. One of these broad beans is selected at random.
5. Estimate the probability that its weight lies between 1.1 grams and 1.6 grams. One of these broad beans having a recorded weight of 0.95 grams was incorrectly weighed. The correct weight is 1.4 grams.
6. State, giving a reason, the effect this would have on your answers to part (c). Do not carry out any further calculations.

1. A random sample of 100 carrots is taken from a farm and their lengths, \(L \mathrm {~cm}\), recorded. The data are summarised in the following table.

A histogram is drawn to represent these data.
The bar representing the class \(5 \leqslant L < 8\) is 1.5 cm wide and 1 cm high.

1. Find the width and height of the bar representing the class \(15 \leqslant L < 20\)
2. Use linear interpolation to estimate the median length of these carrots.
3. Estimate
   1. the mean length of these carrots,
   2. the standard deviation of the lengths of these carrots. A supermarket will only buy carrots with length between 9 cm and 22 cm .
4. Estimate the proportion of carrots from the farm that the supermarket will buy. Any carrots that the supermarket does not buy are sold as animal feed. The farm makes a profit of 2.2 pence on each carrot sold to the supermarket, a profit of 0.8 pence on each carrot longer than 22 cm and a loss of 1.2 pence on each carrot shorter than 9 cm .
5. Find an estimate of the mean profit per carrot made by the farm.

1. Gill buys a bag of logs to use in her stove. The lengths, \(l \mathrm {~cm}\), of the 88 logs in the bag are summarised in the table below.

A histogram is drawn to represent these data.
The bar representing logs with length \(27 < l \leqslant 30\) has a width of 1.5 cm and a height of 4 cm .

1. Calculate the width and height of the bar representing log lengths of \(20 < l \leqslant 25\)
2. Use linear interpolation to estimate the median of \(l\) The maximum length of log Gill can use in her stove is 26 cm .
   Gill estimates, using linear interpolation, that \(x\) logs from the bag will fit into her stove.
3. Show that \(x = 62\) Gill randomly selects 4 logs from the bag.
4. Using \(x = 62\), find the probability that all 4 logs will fit into her stove. The weights, \(W\) grams, of the logs in the bag are coded using \(y = 0.5 w - 255\) and summarised by $$n = 88 \quad \sum y = 924 \quad \sum y ^ { 2 } = 12862$$
5. Calculate
   1. the mean of \(W\)
   2. the variance of \(W\)

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 \(w\)

3. The parking times, \(t\) hours, for cars in a car park are summarised below. $$\text { (You may use } \sum \mathrm { fm } = 182 \text { and } \sum \mathrm { fm } ^ { 2 } = 883 \text { ) }$$ A histogram is drawn to represent these data.
The bar representing the time \(1 \leqslant t < 2\) has a width of 1.5 cm and a height of 6 cm .

1. Calculate the width and the height of the bar representing the time \(4 \leqslant t < 6\)
2. Use linear interpolation to estimate the median parking time for the cars in the car park.
3. Estimate the mean and the standard deviation of the parking time for the cars in the car park.
4. Describe, giving a reason, the skewness of the data. One of these cars is selected at random.
5. Estimate the probability that this car is parked for more than 75 minutes.