Questions - OCR MEI - A-Level Maths

Name the type of skewness of the distribution.
Find the interquartile range and hence show that there are no outliers at the lower end of the distribution, but there is at least one outlier at the upper end.
Suggest possible reasons why this may be the case.

OCR MEI S1 Q2

2 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 Q3

3 A pear grower collects a random sample of 120 pears from his orchard. The histogram below shows the lengths, in mm, of these pears.
\includegraphics[max width=\textwidth, alt={}, center]{56f1bd5c-4b45-4e36-a324-e7e0edbb5bdd-2_825_1634_467_295}

Calculate the number of pears which are between 90 and 100 mm long.
Calculate an estimate of the mean length of the pears. Explain why your answer is only an estimate.
Calculate an estimate of the standard deviation.
Use your answers to parts (ii) and (iii) to investigate whether there are any outliers.
Name the type of skewness of the distribution.
Illustrate the data using a cumulative frequency diagram.

OCR MEI S1 Q4

4 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 Q1

1 The temperature of a supermarket fridge is regularly checked to ensure that it is working correctly. Over a period of three months the temperature (measured in degrees Celsius) is checked 600 times. These temperatures are displayed in the cumulative frequency diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{c7cb0f6b-7b6b-4c52-8287-7efc6bd70247-1_1052_1647_549_289}

Use the diagram to estimate the median and interquartile range of the data.
Use your answers to part (i) to show that there are very few, if any, outliers in the sample.
Suppose that an outlier is identified in these data. Discuss whether it should be excluded from any further analysis.
Copy and complete the frequency table below for these data.
Temperature
$( t$ degrees Celsius $)$
$3.0 \leqslant t \leqslant 3.4$ $3.4 < t \leqslant 3.8$ $3.8 < t \leqslant 4.2$ $4.2 < t \leqslant 4.6$ $4.6 < t \leqslant 5.0$
Frequency 243 157
Use your table to calculate an estimate of the mean.
The standard deviation of the temperatures in degrees Celsius is 0.379 . The temperatures are converted from degrees Celsius into degrees Fahrenheit using the formula $F = 1.8 C + 32$. Hence estimate the mean and find the standard deviation of the temperatures in degrees Fahrenheit.

OCR MEI S1 Q2

2 The histogram shows the amount of money, in pounds, spent by the customers at a supermarket on a particular day.
\includegraphics[max width=\textwidth, alt={}, center]{c7cb0f6b-7b6b-4c52-8287-7efc6bd70247-2_985_1473_470_379}

Express the data in the form of a grouped frequency table.
Use your table to estimate the total amount of money spent by customers on that day.

OCR MEI S1 Q3

3 A GCSE geography student is investigating a claim that global warming is causing summers in Britain to have more rainfall. He collects rainfall data from a local weather station for 2001 and 2006. The vertical line chart shows the number of days per week on which some rainfall was recorded during the 22 weeks of summer 2001.
\includegraphics[max width=\textwidth, alt={}, center]{c7cb0f6b-7b6b-4c52-8287-7efc6bd70247-3_804_1557_547_337}

Show that the median of the data is 4 , and find the interquartile range.
For summer 2006 the median is 3 and the interquartile range is also 3. The student concludes that the data demonstrate that global warming is causing summer rainfall to decrease rather than increase. Is this a valid conclusion from the data? Give two brief reasons to justify your answer.

OCR MEI S1 Q4

4 The numbers of absentees per day from Mrs Smith's reception class over a period of 50 days are summarised below.

Number of absentees	0	1	2	3	4	5	6	$> 6$
Frequency	8	15	11	8	3	4	1	0

Illustrate these data by means of a vertical line chart.
Calculate the mean and root mean square deviation of these data.
There are 30 children in Mrs Smith's class altogether. Find the mean and root mean square deviation of the number of children who are present during the 50 days.

OCR MEI S1 Q5

5 The times taken for 480 university students to travel from their accommodation to lectures are summarised below.

Time $( t$ minutes $)$	$0 \leqslant t < 5$	$5 \leqslant t < 10$	$10 \leqslant t < 20$	$20 \leqslant t < 30$	$30 \leqslant t < 40$	$40 \leqslant t < 60$
Frequency	34	153	188	73	27	5

Illustrate these data by means of a histogram.
Identify the type of skewness of the distribution.

OCR MEI S1 Q1

1 The birth weights in grams of a random sample of 1000 babies are displayed in the cumulative frequency diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{088972e9-bfcd-429c-9145-af274a4c0a58-1_1268_1548_472_335}

Use the diagram to estimate the median and interquartile range of the data.
Use your answers to part (i) to estimate the number of outliers in the sample.
Should these outliers be excluded from any further analysis? Briefly explain your answer.
Any baby whose weight is below the 10th percentile is selected for careful monitoring. Use the diagram to determine the range of weights of the babies who are selected.
$12 \%$ of new-born babies require some form of special care. A maternity unit has 17 new-born babies. You may assume that these 17 babies form an independent random sample.
Find the probability that
(A) exactly 2 of these 17 babies require special care,
(B) more than 2 of the 17 babies require special care.
On 100 independent occasions the unit has 17 babies. Find the expected number of occasions on which there would be more than 2 babies who require special care.

OCR MEI S1 Q2

2 The times taken, in minutes, by 80 people to complete a crossword puzzle are summarised by the box and whisker plot below.
\includegraphics[max width=\textwidth, alt={}, center]{088972e9-bfcd-429c-9145-af274a4c0a58-2_163_857_436_642}

Write down the range and the interquartile range of the times.
Determine whether any of the times can be regarded as outliers.
Describe the shape of the distribution of the times.

OCR MEI S1 Q3

3 At East Cornwall College, the mean GCSE score of each student is calculated. This is done by allocating a number of points to each GCSE grade in the following way.

Grade	A*	A	B	C	D	E	F	G	U
Points	8	7	6	5	4	3	2	1	0

Calculate the mean GCSE score, $X$, of a student who has the following GCSE grades: $$\mathrm { A } ^ { * } , \mathrm {~A} ^ { * } , \mathrm {~A} , \mathrm {~A} , \mathrm {~A} , \mathrm {~B} , \mathrm {~B} , \mathrm {~B} , \mathrm {~B} , \mathrm { C } , \mathrm { D } .$$ 60 students study AS Mathematics at the college. The mean GCSE scores of these students are summarised in the table below.
Mean GCSE score Number of students
$4.5 \leqslant X < 5.5$ 8
$5.5 \leqslant X < 6.0$ 14
$6.0 \leqslant X < 6.5$ 19
$6.5 \leqslant X < 7.0$ 13
$7.0 \leqslant X \leqslant 8.0$ 6
Draw a histogram to illustrate this information.
Calculate estimates of the sample mean and the sample standard deviation. The scoring system for AS grades is shown in the table below.
AS Grade A B C D E U
Score 60 50 40 30 20 0
The Mathematics department at the college predicts each student's AS score, $Y$, using the formula $Y = 13 X - 46$, where $X$ is the student's average GCSE score.
What AS grade would the department predict for a student with an average GCSE score of 7.4 ?
What do you think the prediction should be for a student with an average GCSE score of 5.5? Give a reason for your answer.
Using your answers to part (iii), estimate the sample mean and sample standard deviation of the predicted AS scores of the 60 students in the department.

OCR MEI S1 Q4

4 At a certain stage of a football league season, the numbers of goals scored by a sample of 20 teams in the league were as follows.
$\begin{array} { l l l l l l l l l l l l l l l l l l l l } 22 & 23 & 23 & 23 & 26 & 28 & 28 & 30 & 31 & 33 & 33 & 34 & 35 & 35 & 36 & 36 & 37 & 46 & 49 & 49 \end{array}$

Calculate the sample mean and sample variance, $s ^ { 2 }$, of these data.
The three teams with the most goals appear to be well ahead of the other teams. Determine whether or not any of these three pieces of data may be considered outliers.

OCR MEI S1 Q2

2 The cumulative frequency graph below illustrates the distances that 176 children live from their primary school. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Distance from school} \includegraphics[alt={},max width=\textwidth]{b4bf1bd0-f85d-42b7-ad15-6672387bb208-2_998_1466_566_367}

\end{figure}

Use the graph to estimate, to the nearest 10 metres,
(A) the median distance from school,
(B) the lower quartile, upper quartile and interquartile range.
Draw a box and whisker plot to illustrate the data. The graph on page 4 used the following grouped data.
Distance (metres) 200 400 600 800 1000 1200
Cumulative frequency 20 64 118 150 169 176
Copy and complete the grouped frequency table below describing the same data.
Distance $( d$ metres $)$ Frequency
$0 < d \leqslant 200$ 20
$200 < d \leqslant 400$
Hence estimate the mean distance these children live from school. It is subsequently found that none of the 176 children lives within 100 metres of the school.
Calculate the revised estimate of the mean distance.
Describe what change needs to be made to the cumulative frequency graph.

OCR MEI S1 Q3

3 The stem and leaf diagram illustrates the heights in metres of 25 young oak trees.

3	4	6	7	8	9	9
4	0	2	2	3	4	6	8	9
5	0	1	3	5	8
6	2	4	5
7	4	6
8	1

Key: 4 |2 represents 4.2

State the type of skewness of the distribution.
Use your calculator to find the mean and standard deviation of these data.
Determine whether there are any outliers.

OCR MEI S1 Q4

4 At a call centre, $85 \%$ of callers are put on hold before being connected to an operator. A random sample of 30 callers is selected.

Find the probability that exactly 29 of these callers are put on hold.
Find the probability that at least 29 of these callers are put on hold.
If 10 random samples, each of 30 callers, are selected, find the expected number of samples in which at least 29 callers are put on hold.

OCR MEI S1 Q1

1 Yasmin has 5 coins. One of these coins is biased with P (heads) $= 0.6$. The other 4 coins are fair. She tosses all 5 coins once and records the number of heads, $X$.

Show that $\mathrm { P } ( X = 0 ) = 0.025$.
Show that $\mathrm { P } ( X = 1 ) = 0.1375$. The table shows the probability distribution of $X$.
$r$ 0 1
$\mathrm { P } ( X = r )$ 0.025 0.1375 0.3 0.325 0.175 0.0375
Draw a vertical line chart to illustrate the probability distribution.
Comment on the skewness of the distribution.
Find $\mathrm { E } ( X )$ and $\operatorname { Var } ( X )$.
Yasmin tosses the 5 coins three times. Find the probability that the total number of heads is 3 .

OCR MEI S1 Q2

2 In a traffic survey, the number of people in each car passing the survey point is recorded. The results are given in the following frequency table.

Number of people	1	2	3	4
Frequency	50	31	16	5

Write down the median and mode of these data.
Draw a vertical line diagram for these data.
State the type of skewness of the distribution.

OCR MEI S1 Q3

3 The histogram shows the age distribution of people living in Inner London in 2001.
\includegraphics[max width=\textwidth, alt={}, center]{b6d84f99-ee39-49c7-a5e8-05838efeef5a-2_804_1372_483_436} Data sourced from the 2001 Census, www.sta is \href{http://ics.gov.uk}{ics.gov.uk}

State the type of skewness shown by the distribution.
Use the histogram to estimate the number of people aged under 25.
The table below shows the cumulative frequency distribution.
Age 20 30 40 50 65 100
Cumulative frequency (thousands) 660 1240 1810 $a$ 2490 2770
(A) Use the histogram to find the value of $a$.
(B) Use the table to calculate an estimate of the median age of these people. The ages of people living in Outer London in 2001 are summarised below.
Age ( $x$ years) $0 \leqslant x < 20$ $20 \leqslant x < 30$ $30 \leqslant x < 40$ $40 \leqslant x < 50$ $50 \leqslant x < 65$ $65 \leqslant x < 100$
Frequency (thousands) 1120 650 770 590 680 610
Illustrate these data by means of a histogram.
Make two brief comments on the differences between the age distributions of the populations of Inner London and Outer London.
The data given in the table for Outer London are used to calculate the following estimates. Mean 38.5, median 35.7, midrange 50, standard deviation 23.7, interquartile range 34.4.
The final group in the table assumes that the maximum age of any resident is 100 years. These estimates are to be recalculated, based on a maximum age of 105, rather than 100. For each of the five estimates, state whether it would increase, decrease or be unchanged.

OCR MEI S1 Q1

1 The maximum temperatures $x$ degrees Celsius recorded during each month of 2005 in Cambridge are given in the table below.

Jan	Feb	Mar	Apr	May	Jun	Jul	Aug	Sep	Oct	Nov	Dec
9.2	7.1	10.7	14.2	16.6	21.8	22.0	22.6	21.1	17.4	10.1	7.8

These data are summarised by $n = 12 , \Sigma x = 180.6 , \Sigma x ^ { 2 } = 3107.56$.

Calculate the mean and standard deviation of the data.
Determine whether there are any outliers.
The formula $y = 1.8 x + 32$ is used to convert degrees Celsius to degrees Fahrenheit. Find the mean and standard deviation of the 2005 maximum temperatures in degrees Fahrenheit.
In New York, the monthly maximum temperatures are recorded in degrees Fahrenheit. In 2005 the mean was 63.7 and the standard deviation was 16.0 . Briefly compare the maximum monthly temperatures in Cambridge and New York in 2005. The total numbers of hours of sunshine recorded in Cambridge during the month of January for each of the last 48 years are summarised below.
Hours $h$ $70 \leqslant h < 100$ $100 \leqslant h < 110$ $110 \leqslant h < 120$ $120 \leqslant h < 150$ $150 \leqslant h < 170$ $170 \leqslant h < 190$
Number of years 6 8 10 11 10 3
Draw a cumulative frequency graph for these data.
Use your graph to estimate the 90th percentile.

OCR MEI S1 Q2

2 Every day, George attempts the quiz in a national newspaper. The quiz always consists of 7 questions. In the first 25 days of January, the numbers of questions George answers correctly each day are summarised in the table below.

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 Q1

1 The amounts of electricity, $x \mathrm { kWh }$ (kilowatt hours), used by 40 households in a three-month period are summarised as follows. $$n = 40 \quad \sum x = 59972 \quad \sum x ^ { 2 } = 96767028$$

Calculate the mean and standard deviation of $x$.
The formula $y = 0.163 x + 14.5$ gives the cost in pounds of the electricity used by each household. Use your answers to part (i) to deduce the mean and standard deviation of the costs of the electricity used by these 40 households.

OCR MEI S1 Q2

2 Three fair six-sided dice are thrown. The random variable $X$ represents the highest of the three scores on the dice.

Show that $\mathrm { P } ( X = 6 ) = \frac { 91 } { 216 }$. The table shows the probability distribution of $X$.
$r$ 1 2 3 4 5 6
$\mathrm { P } ( X = r )$ $\frac { 1 } { 216 }$ $\frac { 7 } { 216 }$ $\frac { 19 } { 216 }$ $\frac { 37 } { 216 }$ $\frac { 61 } { 216 }$ $\frac { 91 } { 216 }$
Find $\mathrm { E } ( X )$ and $\operatorname { Var } ( X )$.

OCR MEI S1 Q3

3 The probability distribution of the random variable $X$ is given by the formula $$\mathrm { P } ( X = r ) = k + 0.01 r ^ { 2 } \text { for } r = 1,2,3,4,5 .$$

Show that $k = 0.09$. Using this value of $k$, display the probability distribution of $X$ in a table.
Find $\mathrm { E } ( X )$ and $\operatorname { Var } ( X )$.

OCR MEI S1 Q4

4 The probability distribution of the random variable $X$ is given by the formula $$\mathrm { P } ( X = r ) = k \left( r ^ { 2 } - 1 \right) \text { for } r = 2,3,4,5 .$$

Show the probability distribution in a table, and find the value of $k$.
Find $\mathrm { E } ( X )$ and $\operatorname { Var } ( X )$.

Time \(( t\) minutes \()\)	\(0 \leqslant t < 5\)	\(5 \leqslant t < 10\)	\(10 \leqslant t < 20\)	\(20 \leqslant t < 30\)	\(30 \leqslant t < 40\)	\(40 \leqslant t < 60\)
Frequency	34	153	188	73	27	5

Distance \(( d\) metres \()\)	Frequency
\(0 < d \leqslant 200\)	20
\(200 < d \leqslant 400\)

Hours \(h\)	\(70 \leqslant h < 100\)	\(100 \leqslant h < 110\)	\(110 \leqslant h < 120\)	\(120 \leqslant h < 150\)	\(150 \leqslant h < 170\)	\(170 \leqslant h < 190\)
Number of years	6	8	10	11	10	3

\(r\)	1	2	3	4	5	6
\(\mathrm { P } ( X = r )\)	\(\frac { 1 } { 216 }\)	\(\frac { 7 } { 216 }\)	\(\frac { 19 } { 216 }\)	\(\frac { 37 } { 216 }\)	\(\frac { 61 } { 216 }\)	\(\frac { 91 } { 216 }\)

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

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

Mean GCSE score	Number of students
\(4.5 \leqslant X < 5.5\)	8
\(5.5 \leqslant X < 6.0\)	14
\(6.0 \leqslant X < 6.5\)	19
\(6.5 \leqslant X < 7.0\)	13
\(7.0 \leqslant X \leqslant 8.0\)	6

Distance (metres)	200	400	600	800	1000	1200
Cumulative frequency	20	64	118	150	169	176

\(r\)	0	1
\(\mathrm { P } ( X = r )\)	0.025	0.1375	0.3	0.325	0.175	0.0375

Age	20	30	40	50	65	100
Cumulative frequency (thousands)	660	1240	1810	\(a\)	2490	2770

Age ( \(x\) years)	\(0 \leqslant x < 20\)	\(20 \leqslant x < 30\)	\(30 \leqslant x < 40\)	\(40 \leqslant x < 50\)	\(50 \leqslant x < 65\)	\(65 \leqslant x < 100\)
Frequency (thousands)	1120	650	770	590	680	610

AS Grade	A	B	C	D	E	U
Score	60	50	40	30	20	0

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