Draw histogram then perform calculations

Edexcel S1 2014 January Q8

8. A manager records the number of hours of overtime claimed by 40 staff in a month. The histogram in Figure 1 represents the results. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{a839a89a-17f0-473b-ac10-bcec3dbe97f7-26_1107_1513_406_210} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure}

Calculate the number of staff who have claimed less than 10 hours of overtime in the month.
Estimate the median number of hours of overtime claimed by these 40 staff in the month.
Estimate the mean number of hours of overtime claimed by these 40 staff in the month. The manager wants to compare these data with overtime data he collected earlier to find out if the overtime claimed by staff has decreased.
State, giving a reason, whether the manager should use the median or the mean to compare the overtime claimed by staff.
(2)

Edexcel S1 2004 January Q5

5. The values of daily sales, to the nearest $\pounds$, taken at a newsagents last year are summarised in the table below.

Sales	Number of days
$1 - 200$	166
$201 - 400$	100
$401 - 700$	59
$701 - 1000$	30
$1001 - 1500$	5

Draw a histogram to represent these data.
Use interpolation to estimate the median and inter-quartile range of daily sales.
Estimate the mean and the standard deviation of these data. The newsagent wants to compare last year's sales with other years.
State whether the newsagent should use the median and the inter-quartile range or the mean and the standard deviation to compare daily sales. Give a reason for your answer.
(2)

OCR MEI S1 2011 January Q7

7 The incomes of a sample of 918 households on an island are given in the table below.

Income

$( x$ thousand pounds $)$

$0 \leqslant x \leqslant 20$

$20 < x \leqslant 40$

$40 < x \leqslant 60$

$60 < x \leqslant 100$

$100 < x \leqslant 200$

Frequency

238

365

142

128

45

Draw a histogram to illustrate the data.
Calculate an estimate of the mean income.
Calculate an estimate of the standard deviation of the incomes.
Use your answers to parts (ii) and (iii) to show there are almost certainly some outliers in the sample. Explain whether or not it would be appropriate to exclude the outliers from the calculation of the mean and the standard deviation.
The incomes were converted into another currency using the formula $y = 1.15 x$. Calculate estimates of the mean and variance of the incomes in the new currency.

OCR MEI S1 2009 June Q5

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

Distance $( d$ miles $)$	$0 \leqslant d < 50$	$50 \leqslant d < 100$	$100 \leqslant d < 200$	$200 \leqslant d < 400$
Frequency	360	400	307	133

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

OCR MEI S1 2010 June Q3

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

Lifetime $( x$ hours $)$	$0 < x \leqslant 20$	$20 < x \leqslant 30$	$30 < x \leqslant 50$	$50 < x \leqslant 65$	$65 < x \leqslant 100$
Frequency	24	13	14	21	18

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

OCR MEI S1 2012 June Q6

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

Engine size $x$	$500 \leqslant x \leqslant 1000$	$1000 < x \leqslant 1500$	$1500 < x \leqslant 2000$	$2000 < x \leqslant 3000$	$3000 < x \leqslant 5000$
Frequency	7	22	26	18	7

Draw a histogram to illustrate the distribution.
A student claims that the midrange is $2750 \mathrm {~cm} ^ { 3 }$. Discuss briefly whether he is likely to be correct.
Calculate estimates of the mean and standard deviation of the engine sizes. Explain why your answers are only estimates.
Hence investigate whether there are any outliers in the sample.
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.
Why in practice might your estimate in part (v) turn out to be too high?

Edexcel S1 2022 June Q3

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.

Length $( \boldsymbol { l } )$	Frequency $( \boldsymbol { f } )$
$15 < l \leqslant 20$	19
$20 < l \leqslant 25$	35
$25 < l \leqslant 27$	16
$27 < l \leqslant 30$	15
$30 < l \leqslant 40$	3

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 .

Calculate the width and height of the bar representing log lengths of $20 < l \leqslant 25$
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.
Show that $x = 62$ Gill randomly selects 4 logs from the bag.
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 $$n = 88 \quad \sum y = 924 \quad \sum y ^ { 2 } = 12862$$
Calculate
1. the mean of $W$
2. the variance of $W$

Edexcel S1 2002 June Q6

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.

Weight in kg	Frequency
14.6-14.8	1
14.8-18.0	0
18.0-18.5	5
18.5-20.0	6
20.0-20.2	22
20.2-20.4	15
20.4-21.0	1

On graph paper, draw a histogram of these data.
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]
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.
Write down which of these averages you would recommend the company to use. Give a reason for your answer.

Edexcel S1 2010 June Q5

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.

Hours	$1 - 10$	$11 - 20$	$21 - 25$	$26 - 30$	$31 - 40$	$41 - 59$
Frequency	6	15	11	13	8	3
Mid-point	5.5	15.5		28		50

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 .
Find the width and height of the 26-30 group.
Estimate the mean and standard deviation of the time spent watching television by these students.
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.
State, giving a reason, the skewness of these data.

Edexcel S1 Q6

6. The number of people visiting a new art gallery each day is recorded over a three-month period and the results are summarised in the table below.

Number of visitors	Number of days
400-459	3
460-479	8
480-499	13
500-519	12
520-539	18
540-559	11
560-599	9
600-699	5

Draw a histogram on graph paper to illustrate these data. In order to calculate summary statistics for the data it is coded using $y = \frac { x - 509.5 } { 10 }$, where $x$ is the mid-point of each class.
Find $\sum$ fy. You may assume that $\sum f y ^ { 2 } = 2041$.
Using these values for $\sum f y$ and $\sum f y ^ { 2 }$, calculate estimates of the mean and standard deviation of the number of visitors per day.
(6 marks)

Edexcel S1 Q6

6. A cinema recorded the number of people at each showing of each film during a one-week period. The results are summarised in the table below.

Number of people	Number of showings
1-40	36
41-60	20
61-80	33
81-100	24
101-150	36
151-200	39
201-300	52

Draw a histogram on graph paper to illustrate these data.
Calculate estimates of the median and quartiles of these data.
Use your answers to part (b) to show that the data is positively skewed.

OCR MEI S1 Q3

3 Answer part (i) of this question on the insert provided. A taxi driver operates from a taxi rank at a main railway station in London. During one particular week he makes 120 journeys, the lengths of which are summarised in the table.

Length

$( x$ miles $)$

$0 < x \leqslant 1$

$1 < x \leqslant 2$

$2 < x \leqslant 3$

$3 < x \leqslant 4$

$4 < x \leqslant 6$

$6 < x \leqslant 10$

Number of

journeys

38

30

21

14

9

8

On the insert, draw a cumulative frequency diagram to illustrate the data.
Use your graph to estimate the median length of journey and the quartiles. Hence find the interquartile range.
State the type of skewness of the distribution of the data.

OCR MEI S1 Q4

4 Answer part (i) of this question on the insert provided. A taxi driver operates from a taxi rank at a main railway station in London. During one particular week he makes 120 journeys, the lengths of which are summarised in the table.