Questions S1 - A-Level Maths

Without carrying out any calculation, state the value of Spearman's rank correlation coefficient for the following ranks. Give a reason for your answer. \href{http://physicsandmathstutor.com}{physicsandmathstutor.com}

OCR S1 2010 June Q5

12 marks Moderate -0.8

3 (ii) (b)





3 (ii) (c)





3 (ii) (d)

\href{http://physicsandmathstutor.com}{physicsandmathstutor.com}

OCR S1 2010 June Q7

8 marks Moderate -0.8

7 The menu below shows all the dishes available at a certain restaurant.

Rice dishes	Main dishes	Vegetable dishes
Boiled rice	Chicken	Mushrooms
Fried rice	Beef	Cauliflower
Pilau rice	Lamb	Spinach
Keema rice	Mixed grill	Lentils
	Prawn	Potatoes
	Vegetarian

A group of friends decide that they will share a total of 2 different rice dishes, 3 different main dishes and 4 different vegetable dishes from this menu. Given these restrictions,

find the number of possible combinations of dishes that they can choose to share,
assuming that all choices are equally likely, find the probability that they choose boiled rice. The friends decide to add a further restriction as follows. If they choose boiled rice, they will not choose potatoes.
Find the number of possible combinations of dishes that they can now choose.

OCR S1 2010 June Q8

12 marks Moderate -0.8

8 The proportion of people who watch West Street on television is $30 \%$. A market researcher interviews people at random in order to contact viewers of West Street. Each day she has to contact a certain number of viewers of West Street.

Near the end of one day she finds that she needs to contact just one more viewer of West Street. Find the probability that the number of further interviews required is
(a) 4 ,

OCR S1 2010 June Q10

Moderate -0.5

(a) 8

(b) \section*{PLEASE DO NOT WRITE ON THIS PAGE} RECOGNISING ACHIEVEMENT

OCR S1 2016 June Q1

8 marks Moderate -0.3

1 The table shows the probability distribution of a random variable $X$.

$x$	1	2	3	4
$\mathrm { P } ( X = x )$	0.1	0.3	0.4	0.2

Find $\mathrm { E } ( X )$ and $\operatorname { Var } ( X )$.
Three values of $X$ are chosen at random. Find the probability that $X$ takes the value 2 at least twice.

OCR S1 2016 June Q2

10 marks Moderate -0.3

The table shows the amount, $x$, in hundreds of pounds, spent on heating and the number of absences, $y$, at a factory during each month in 2014.
Amount, $x$, spent on
heating (£ hundreds)
21 23 19 15 14 5 2 10 9 20 18 23
Number of absences, $y$ 23 25 18 18 12 10 4 9 11 15 20 26
$n = 12 \quad \Sigma x = 179 \quad \Sigma x ^ { 2 } = 3215 \quad \Sigma y = 191 \quad \Sigma y ^ { 2 } = 3565 \quad \Sigma x y = 3343$
(a) Calculate $r$, the product moment correlation coefficient, showing that $r > 0.92$.
(b) A manager says, 'The value of $r$ shows that spending more money on heating causes more absences, so we should spend less on heating.' Comment on this claim.
The months in 2014 were numbered $1,2,3 , \ldots , 12$. The output, $z$, in suitable units was recorded along with the month number, $n$, for each month in 2014. The equation of the regression line of $z$ on $n$ was found to be $z = 0.6 n + 17$.
(a) Use this equation to explain whether output generally increased or decreased over these months.
(b) Find the mean of $n$ and use the equation of the regression line to calculate the mean of $z$.
(c) Hence calculate the total output in 2014.

OCR S1 2016 June Q3

13 marks Moderate -0.8

3 The masses, $m$ grams, of 52 apples of a certain variety were found and summarised as follows. $$n = 52 \quad \Sigma ( m - 150 ) = - 182 \quad \Sigma ( m - 150 ) ^ { 2 } = 1768$$

Find the mean and variance of the masses of these 52 apples.
Use your answers from part (i) to find the exact value of $\Sigma m ^ { 2 }$. The masses of the apples are illustrated in the box-and-whisker plot below.
\includegraphics[max width=\textwidth, alt={}, center]{b5ce3230-7528-439c-9e85-ef159a49cba3-3_250_1310_662_383}
How many apples have masses in the interval $130 \leqslant m < 140$ ?
An 'outlier' is a data item that lies more than 1.5 times the interquartile range above the upper quartile, or more than 1.5 times the interquartile range below the lower quartile. Explain whether any of the masses of these apples are outliers.

OCR S1 2016 June Q4

8 marks Moderate -0.3

4 In this question the product moment correlation coefficient is denoted by $r$ and Spearman's rank correlation coefficient is denoted by $r _ { s }$.

The scatter diagram in Fig. 1 shows the results of an experiment involving some bivariate data. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b5ce3230-7528-439c-9e85-ef159a49cba3-4_597_595_434_733} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} Write down the value of $r _ { s }$ for these data.
On the diagram in the Answer Booklet, draw five points such that $r _ { s } = 1$ and $r \neq 1$.
The scatter diagram in Fig. 2 shows the results of another experiment involving 5 items of bivariate data. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b5ce3230-7528-439c-9e85-ef159a49cba3-4_604_608_1484_731} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} Calculate the value of $r _ { s }$.
A random variable $X$ has the distribution $\mathrm { B } ( 25,0.6 )$. Find
(a) $\mathrm { P } ( X \leqslant 14 )$,
(b) $\mathrm { P } ( X = 14 )$,
(c) $\quad \operatorname { Var } ( X )$.
A random variable $Y$ has the distribution $\mathrm { B } ( 24,0.3 )$. Write down an expression for $\mathrm { P } ( Y = y )$ and evaluate this probability in the case where $y = 8$.
A random variable $Z$ has the distribution $\mathrm { B } ( 2,0.2 )$. Find the probability that two randomly chosen values of $Z$ are equal.
(a) Find the number of ways in which 12 people can be divided into three groups containing 5 people, 4 people and 3 people, without regard to order.
(b) The diagram shows 7 cards, each with a letter on it. $$\mathrm { A } \mathrm {~A} \mathrm {~A} \mathrm {~B} \text { } \mathrm { B } \text { } \mathrm { R } \text { } \mathrm { R }$$ The 7 cards are arranged in a random order in a straight line.
Find the number of possible arrangements of the 7 letters.
Find the probability that the 7 letters form the name BARBARA. The 7 cards are shuffled. Now 4 of the 7 cards are chosen at random and arranged in a random order in a straight line.
Find the probability that the letters form the word ABBA .

OCR S1 Specimen Q1

5 marks Easy -1.2

1 Janet and John wanted to compare their daily journey times to work, so they each kept a record of their journey times for a few weeks.

Janet's daily journey times, $x$ minutes, for a period of 25 days, were summarised by $\Sigma x = 2120$ and $\Sigma x ^ { 2 } = 180044$. Calculate the mean and standard deviation of Janet's journey times.
John's journey times had a mean of 79.7 minutes and a standard deviation of 6.22 minutes. Describe briefly, in everyday terms, how Janet and John's journey times compare.

OCR S1 Specimen Q2

7 marks Standard +0.3

2 Two independent assessors awarded marks to each of 5 projects. The results were as shown in the table.

Project	$A$	$B$	$C$	$D$	$E$
First assessor	38	91	62	83	61
Second assessor	56	84	41	85	62

Calculate Spearman's rank correlation coefficient for the data.
Show, by sketching a suitable scatter diagram, how two assessors might have assessed 5 projects in such a way that Spearman's rank correlation coefficient for their marks was + 1 while the product moment correlation coefficient for their marks was not + 1 . (Your scatter diagram need not be drawn accurately to scale.)

OCR S1 Specimen Q3

8 marks Standard +0.3

3 Five friends, Ali, Bev, Carla, Don and Ed, stand in a line for a photograph.

How many different possible arrangements are there if Ali, Bev and Carla stand next to each other?
How many different possible arrangements are there if none of Ali, Bev and Carla stand next to each other?
If all possible arrangements are equally likely, find the probability that two of Ali, Bev and Carla are next to each other, but the third is not next to either of the other two.

OCR S1 Specimen Q4

8 marks Moderate -0.8

4 Each packet of the breakfast cereal Fizz contains one plastic toy animal. There are five different animals in the set, and the cereal manufacturers use equal numbers of each. Without opening a packet it is impossible to tell which animal it contains. A family has already collected four different animals at the start of a year and they now need to collect an elephant to complete their set. The family is interested in how many packets they will need to buy before they complete their set.

Name an appropriate distribution with which to model this situation. State the value(s) of any parameter(s) of the distribution, and state also any assumption(s) needed for the distribution to be a valid model.
Find the probability that the family will complete their set with the third packet they buy after the start of the year.
Find the probability that, in order to complete their collection, the family will need to buy more than 4 packets after the start of the year.

OCR S1 Specimen Q5

10 marks Moderate -0.8

5 A sixth-form class consists of 7 girls and 5 boys. Three students from the class are chosen at random. The number of boys chosen is denoted by the random variable $X$. Show that

$\quad \mathrm { P } ( X = 0 ) = \frac { 7 } { 44 }$,
$\mathrm { P } ( X = 2 ) = \frac { 7 } { 22 }$. The complete probability distribution of $X$ is shown in the following table.
$x$ 0 1 2 3
$\mathrm { P } ( X = x )$ $\frac { 7 } { 44 }$ $\frac { 21 } { 44 }$ $\frac { 7 } { 22 }$ $\frac { 1 } { 22 }$
Calculate $\mathrm { E } ( X )$ and $\operatorname { Var } ( X )$.

OCR S1 Specimen Q6

11 marks Moderate -0.5

6
\includegraphics[max width=\textwidth, alt={}, center]{2fb25fc5-0445-44fa-a23e-647d14b1a376-3_803_1180_1018_413} The diagram shows the cumulative frequency graphs for the marks scored by the candidates in an examination. The 2000 candidates each took two papers; the upper curve shows the distribution of marks on paper 1 and the lower curve shows the distribution on paper 2. The maximum mark on each paper was 100.

Use the diagram to estimate the median mark for each of paper 1 and paper 2.
State with a reason which of the two papers you think was the easier one.
To achieve grade A on paper 1 candidates had to score 66 marks out of 100. What mark on paper 2 gives equal proportions of candidates achieving grade A on the two papers? What is this proportion?
The candidates' marks for the two papers could also be illustrated by means of a pair of box-and whisker plots. Give two brief comments comparing the usefulness of cumulative frequency graphs and box-and-whisker plots for representing the data.

OCR S1 Specimen Q7

10 marks Moderate -0.3

7 Items from a production line are examined for any defects. The probability that any item will be found to be defective is 0.15 , independently of all other items.

A batch of 16 items is inspected. Using tables of cumulative binomial probabilities, or otherwise, find the probability that
(a) at least 4 items in the batch are defective,
(b) exactly 4 items in the batch are defective.
Five batches, each containing 16 items, are taken.
(a) Find the probability that at most 2 of these 5 batches contain at least 4 defective items.
(b) Find the expected number of batches that contain at least 4 defective items.

OCR S1 Specimen Q8

13 marks Moderate -0.8

8 An experiment was conducted to see whether there was any relationship between the maximum tidal current, $y \mathrm {~cm} \mathrm {~s} ^ { - 1 }$, and the tidal range, $x$ metres, at a particular marine location. [The tidal range is the difference between the height of high tide and the height of low tide.] Readings were taken over a period of 12 days, and the results are shown in the following table.

$x$	2.0	2.4	3.0	3.1	3.4	3.7	3.8	3.9	4.0	4.5	4.6	4.9
$y$	15.2	22.0	25.2	33.0	33.1	34.2	51.0	42.3	45.0	50.7	61.0	59.2

$$\left[ \Sigma x = 43.3 , \Sigma y = 471.9 , \Sigma x ^ { 2 } = 164.69 , \Sigma y ^ { 2 } = 20915.75 , \Sigma x y = 1837.78 . \right]$$ The scatter diagram below illustrates the data.
\includegraphics[max width=\textwidth, alt={}, center]{2fb25fc5-0445-44fa-a23e-647d14b1a376-4_462_793_1464_644}

Calculate the product moment correlation coefficient for the data, and comment briefly on your answer with reference to the appearance of the scatter diagram.
Calculate the equation of the regression line of maximum tidal current on tidal range.
Estimate the maximum tidal current on a day when the tidal range is 4.2 m , and comment briefly on how reliable you consider your estimate is likely to be.
It is suggested that the equation found in part (ii) could be used to predict the maximum tidal current on a day when the tidal range is 15 m . Comment briefly on the validity of this suggestion.

OCR MEI S1 2005 January Q1

7 marks Easy -1.8

1 The number of minutes of recorded music on a sample of 100 CDs is summarised below.

Time ( $t$ minutes)	$40 \leqslant t < 45$	$45 \leqslant t < 50$	$50 \leqslant t < 60$	$60 \leqslant t < 70$	$70 \leqslant t < 90$
Number of CDs	26	18	31	16	9

Illustrate the data by means of a histogram.
Identify two features of the distribution.

OCR MEI S1 2005 January Q2

7 marks Moderate -0.8

2 A sprinter runs many 100 -metre trials, and the time, $x$ seconds, for each is recorded. A sample of eight of these times is taken, as follows. $$\begin{array} { l l l l l l l l } 10.53 & 10.61 & 10.04 & 10.49 & 10.63 & 10.55 & 10.47 & 10.63 \end{array}$$

Calculate the sample mean, $\bar { x }$, and sample standard deviation, $s$, of these times.
Show that the time of 10.04 seconds may be regarded as an outlier.
Discuss briefly whether or not the time of 10.04 seconds should be discarded.

OCR MEI S1 2005 January Q3

3 marks Moderate -0.3

3 The Venn diagram illustrates the occurrence of two events $A$ and $B$.
\includegraphics[max width=\textwidth, alt={}, center]{b35b2b3b-0d26-4a35-b4d2-110bf270d5dc-2_513_826_1713_658} You are given that $\mathrm { P } ( A \cap B ) = 0.3$ and that the probability that neither $A$ nor $B$ occurs is 0.1 . You are also given that $\mathrm { P } ( A ) = 2 \mathrm { P } ( B )$. Find $\mathrm { P } ( B )$.

OCR MEI S1 2005 January Q4

6 marks Moderate -0.8

4 The number, $X$, of children per family in a certain city is modelled by the probability distribution $\mathrm { P } ( X = r ) = k ( 6 - r ) ( 1 + r )$ for $r = 0,1,2,3,4$.

Copy and complete the following table and hence show that the value of $k$ is $\frac { 1 } { 50 }$.
$r$ 0 1 2 3 4
$\mathrm { P } ( X = r )$ $6 k$ $10 k$
Calculate $\mathrm { E } ( X )$.
Hence write down the probability that a randomly selected family in this city has more than the mean number of children.

OCR MEI S1 2005 January Q5

5 marks Easy -1.2

5 A rugby union team consists of 15 players made up of 8 forwards and 7 backs. A manager has to select his team from a squad of 12 forwards and 11 backs.

In how many ways can the manager select the forwards?
In how many ways can the manager select the team?

OCR MEI S1 2005 January Q6

8 marks Easy -1.2

6 An amateur weather forecaster describes each day as either sunny, cloudy or wet. He keeps a record each day of his forecast and of the actual weather. His results for one particular year are given in the table.

Weather Forecast

\multirow{2}{*}{Total}

\cline { 3 - 6 } \multicolumn{2}{|c|}{}

A day is selected at random from that year.

Show that the probability that the forecast is correct is $\frac { 264 } { 365 }$. Find the probability that
the forecast is correct, given that the forecast is sunny,
the forecast is correct, given that the weather is wet,
the weather is cloudy, given that the forecast is correct.

OCR MEI S1 2005 January Q7

12 marks Easy -1.2

7 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]{b35b2b3b-0d26-4a35-b4d2-110bf270d5dc-4_1073_1571_580_340}

\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 2005 January Q8

19 marks Standard +0.3

8 At a doctor's surgery, records show that $20 \%$ of patients who make an appointment fail to turn up. During afternoon surgery the doctor has time to see 16 patients. There are 16 appointments to see the doctor one afternoon.

Find the probability that all 16 patients turn up.
Find the probability that more than 3 patients do not turn up. To improve efficiency, the doctor decides to make more than 16 appointments for afternoon surgery, although there will still only be enough time to see 16 patients. There must be a probability of at least 0.9 that the doctor will have enough time to see all the patients who turn up.
The doctor makes 17 appointments for afternoon surgery. Find the probability that at least one patient does not turn up. Hence show that making 17 appointments is satisfactory.
Now find the greatest number of appointments the doctor can make for afternoon surgery and still have a probability of at least 0.9 of having time to see all patients who turn up. A computerised appointment system is introduced at the surgery. It is decided to test, at the 5\% level, whether the proportion of patients failing to turn up for their appointments has changed. There are always 20 appointments to see the doctor at morning surgery. On a randomly chosen morning, 1 patient does not turn up.
Write down suitable hypotheses and carry out the test.

\(x\)	0	1	2	3
\(\mathrm { P } ( X = x )\)	\(\frac { 7 } { 44 }\)	\(\frac { 21 } { 44 }\)	\(\frac { 7 } { 22 }\)	\(\frac { 1 } { 22 }\)

Time ( \(t\) minutes)	\(40 \leqslant t < 45\)	\(45 \leqslant t < 50\)	\(50 \leqslant t < 60\)	\(60 \leqslant t < 70\)	\(70 \leqslant t < 90\)
Number of CDs	26	18	31	16	9

\(r\)	0	1	2	3	4
\(\mathrm { P } ( X = r )\)	\(6 k\)	\(10 k\)

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

Project	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)
First assessor	38	91	62	83	61
Second assessor	56	84	41	85	62

\(x\)	2.0	2.4	3.0	3.1	3.4	3.7	3.8	3.9	4.0	4.5	4.6	4.9
\(y\)	15.2	22.0	25.2	33.0	33.1	34.2	51.0	42.3	45.0	50.7	61.0	59.2

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

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