Questions - OCR S1 - A-Level Maths

$x$	$10 \leqslant x \leqslant 19$	$20 \leqslant x \leqslant 24$	$25 \leqslant x \leqslant 29$	$30 \leqslant x \leqslant 49$
Number of stones	25	22	29	24

These data are to be presented on a statistical diagram.

For a histogram, find the frequency density of the $10 \leqslant x \leqslant 19$ class.
For a cumulative frequency graph, state the coordinates of the first two points that should be plotted.
Why is it not possible to draw an exact box-and-whisker plot to illustrate the data?

OCR S1 2009 June Q6

6 Last year Eleanor played 11 rounds of golf. Her scores were as follows: $79 , \quad 71 , \quad 80 , \quad 67 , \quad 67 , \quad 74 , \quad 66 , \quad 65 , \quad 71 , \quad 66 , \quad 64$.

Calculate the mean of these scores and show that the standard deviation is 5.31 , correct to 3 significant figures.
Find the median and interquartile range of the scores. This year, Eleanor also played 11 rounds of golf. The standard deviation of her scores was 4.23, correct to 3 significant figures, and the interquartile range was the same as last year.
Give a possible reason why the standard deviation of her scores was lower than last year although her interquartile range was unchanged. In golf, smaller scores mean a better standard of play than larger scores. Ken suggests that since the standard deviation was smaller this year, Eleanor's overall standard has improved.
Explain why Ken is wrong.
State what the smaller standard deviation does show about Eleanor's play.

OCR S1 2009 June Q7

7 Three letters are selected at random from the 8 letters of the word COMPUTER, without regard to order.

Find the number of possible selections of 3 letters.
Find the probability that the letter P is included in the selection. Three letters are now selected at random, one at a time, from the 8 letters of the word COMPUTER, and are placed in order in a line.
Find the probability that the 3 letters form the word TOP.

OCR S1 2009 June Q8

8 A game at a charity event uses a bag containing 19 white counters and 1 red counter. To play the game once a player takes counters at random from the bag, one at a time, without replacement. If the red counter is taken, the player wins a prize and the game ends. If not, the game ends when 3 white counters have been taken. Niko plays the game once.

(a) Copy and complete the tree diagram showing the probabilities for Niko. \section*{First counter} \includegraphics[max width=\textwidth, alt={}, center]{c985b9cc-a202-4d5d-a6b3-591b0560f570-4_293_426_1231_532}
(b) Find the probability that Niko will win a prize.
The number of counters that Niko takes is denoted by $X$.
(a) Find $\mathrm { P } ( X = 3 )$.
(b) Find $\mathrm { E } ( X )$.

OCR S1 2009 June Q9

9 Repeated independent trials of a certain experiment are carried out. On each trial the probability of success is 0.12 .

Find the smallest value of $n$ such that the probability of at least one success in $n$ trials is more than 0.95.
Find the probability that the 3rd success occurs on the 7th trial.

OCR S1 2010 June Q1

1 The marks of some students in a French examination were summarised in a grouped frequency distribution and a cumulative frequency diagram was drawn, as shown below.
\includegraphics[max width=\textwidth, alt={}, center]{18d9bdc5-2758-47ec-8698-d90c1c6ac224-02_846_1404_356_367}

Estimate how many students took the examination.
How can you tell that no student scored more than 55 marks?
Find the greatest possible range of the marks.
The minimum mark for Grade C was 27 . The number of students who gained exactly Grade C was the same as the number of students who gained a grade lower than C. Estimate the maximum mark for Grade C.
In a German examination the marks of the same students had an interquartile range of 16 marks. What does this result indicate about the performance of the students in the German examination as compared with the French examination?

OCR S1 2010 June Q2

2 Three skaters, $A , B$ and $C$, are placed in rank order by four judges. Judge $P$ ranks skater $A$ in 1st place, skater $B$ in 2nd place and skater $C$ in 3rd place.

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

3 (ii) (b)





3 (ii) (c)





3 (ii) (d)

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

OCR S1 2010 June Q7

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

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

(a) 8

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

OCR S1 2016 June Q1

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

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

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

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

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

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

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

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

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

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

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

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 S1 2009 January Q1

1 Each time a certain triangular spinner is spun, it lands on one of the numbers 0,1 and 2 with probabilities as shown in the table.

Number	Probability
0	0.7
1	0.2
2	0.1

The spinner is spun twice. The total of the two numbers on which it lands is denoted by $X$.

Show that $\mathrm { P } ( X = 2 ) = 0.18$. The probability distribution of $X$ is given in the table.
$x$ 0 1 2 3 4
$\mathrm { P } ( X = x )$ 0.49 0.28 0.18 0.04 0.01
Calculate $\mathrm { E } ( X )$ and $\operatorname { Var } ( X )$.

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

\(x\)	\(10 \leqslant x \leqslant 19\)	\(20 \leqslant x \leqslant 24\)	\(25 \leqslant x \leqslant 29\)	\(30 \leqslant x \leqslant 49\)
Number of stones	25	22	29	24

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

\(x\)	0	1	2	3	4
\(\mathrm { P } ( X = x )\)	0.49	0.28	0.18	0.04	0.01

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