Questions - OCR S2 - A-Level Maths

OCR S2 2011 June Q2

2 The random variable $Y$ has the distribution $\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)$. It is given that $$\mathrm { P } ( Y < 48.0 ) = \mathrm { P } ( Y > 57.0 ) = 0.0668 .$$ Find the value $y _ { 0 }$ such that $\mathrm { P } \left( Y > y _ { 0 } \right) = 0.05$.

OCR S2 2011 June Q3

3 The random variable $X$ has the distribution $\mathrm { N } \left( \mu , 5 ^ { 2 } \right)$. A hypothesis test is carried out of $\mathrm { H } _ { 0 } : \mu = 20.0$ against $\mathrm { H } _ { 1 } : \mu < 20.0$, at the $1 \%$ level of significance, based on the mean of a sample of size 16. Given that in fact $\mu = 15.0$, find the probability that the test results in a Type II error.

OCR S2 2011 June Q4

4 A continuous random variable $X$ has probability density function $$f ( x ) = \begin{cases} \frac { 3 } { 16 } ( x - 2 ) ^ { 2 } & 0 \leqslant x \leqslant 4
0 & \text { otherwise } \end{cases}$$

Sketch the graph of $y = \mathrm { f } ( x )$.
Calculate the variance of $X$.
A student writes " $X$ is more likely to occur when $x$ takes values further away from 2 ". Explain whether you agree with this statement.

OCR S2 2011 June Q5

5 A travel company finds from its records that $40 \%$ of its customers book with travel agents. The company redesigns its website, and then carries out a survey of 10 randomly chosen customers. The result of the survey is that 1 of these customers booked with a travel agent.

Test at the $5 \%$ significance level whether the percentage of customers who book with travel agents has decreased.
The managing director says that "Our redesigned website has resulted in a decrease in the percentage of our customers who book with travel agents." Comment on this statement.

OCR S2 2011 June Q6

6 Records show that before the year 1990 the maximum daily temperature $T ^ { \circ } \mathrm { C }$ at a seaside resort in August can be modelled by a distribution with mean 24.3. The maximum temperatures of a random sample of 50 August days since 1990 can be summarised by $$n = 50 , \quad \Sigma t = 1314.0 , \quad \Sigma t ^ { 2 } = 36602.17 .$$

Test, at the $1 \%$ significance level, whether there is evidence of a change in the mean maximum daily temperature in August since 1990.
Give a reason why it is possible to use the Central Limit Theorem in your test.

OCR S2 2011 June Q7

7 The number of customer complaints received by a company per day is denoted by $X$. Assume that $X$ has the distribution $\operatorname { Po } ( 2.2 )$.

In a week of 5 working days, the probability there are at least $n$ customer complaints is 0.146 correct to 3 significant figures. Use tables to find the value of $n$.
Use a suitable approximation to find the probability that in a period of 20 working days there are fewer than 38 customer complaints. A week of 5 working days in which at least $n$ customer complaints are received, where $n$ is the value found in part (i), is called a 'bad' week.
Use a suitable approximation to find the probability that, in 40 randomly chosen weeks, more than 7 are bad.

OCR S2 2011 June Q8

8

A group of students is discussing the conditions that are needed if a Poisson distribution is to be a good model for the number of telephone calls received by a fire brigade on a working day.
1. Alice says "Events must be independent". Explain why this condition may not hold in this context.
2. State a different condition that is needed. Explain whether it is likely to hold in this context.
The random variables $R , S$ and $T$ have independent Poisson distributions with means $\lambda , \mu$ and $\lambda + \mu$ respectively.
1. In the case $\lambda = 2.74$, find $\mathrm { P } ( R > 2 )$.
2. In the case $\lambda = 2$ and $\mu = 3$, find $\mathrm { P } ( R = 0$ and $S = 1 ) + \mathrm { P } ( R = 1$ and $S = 0 )$. Give your answer correct to 4 decimal places.
3. In the general case, show algebraically that $$\mathrm { P } ( R = 0 \text { and } S = 1 ) + \mathrm { P } ( R = 1 \text { and } S = 0 ) = \mathrm { P } ( T = 1 ) .$$

OCR S2 2012 June Q1

1 In one day's production, a machine produces 1000 CDs . Explain how to take a random sample of 15 CDs chosen from one day's production.

OCR S2 2012 June Q2

2

For the continuous random variable $V$, it is known that $\mathrm { E } ( V ) = 72.0$. The mean of a random sample of 40 observations of $V$ is denoted by $\bar { V }$. Given that $\mathrm { P } ( \bar { V } < 71.2 ) = 0.35$, estimate the value of $\operatorname { Var } ( V )$.
Explain why you need to use the Central Limit Theorem in part (i), and why its use is justified.

OCR S2 2012 June Q3

3 It is known that on average one person in three prefers the colour of a certain object to be blue. In a psychological test, 12 randomly chosen people were seated in a room with blue walls, and asked to state independently which colour they preferred for the object. Seven of the 12 people said that they preferred blue. Carry out a significance test, at the $5 \%$ level, of whether the statement "on average one person in three prefers the colour of the object to be blue" is true for people who are seated in a room with blue walls.

OCR S2 2012 June Q4

4 In a rock, small crystal formations occur at a constant average rate of 3.2 per cubic metre.

State a further assumption needed to model the number of crystal formations in a fixed volume of rock by a Poisson distribution. In the remainder of the question, you should assume that a Poisson model is appropriate.
Calculate the probability that in one cubic metre of rock there are exactly 5 crystal formations.
Calculate the probability that in 0.74 cubic metres of rock there are at least 3 crystal formations.
Use a suitable approximation to calculate the probability that in 10 cubic metres of rock there are at least 36 crystal formations.

OCR S2 2012 June Q5

5 The acidity $A$ (measured in pH ) of soil of a particular type has a normal distribution. The pH values of a random sample of 80 soil samples from a certain region can be summarised as $$\Sigma a = 496 , \quad \Sigma a ^ { 2 } = 3126 .$$ Test, at the $10 \%$ significance level, whether in this region the mean pH of soil is 6.1 .

OCR S2 2012 June Q6

6 At a tourist car park, a survey is made of the regions from which cars come.

It is given that $40 \%$ of cars come from the London region. Use a suitable approximation to find the probability that, in a random sample of 32 cars, more than 17 come from the London region. Justify your approximation.
It is given that $1 \%$ of cars come from France. Use a suitable approximation to find the probability that, in a random sample of 90 cars, exactly 3 come from France.

OCR S2 2012 June Q7

7 The continuous random variable $X$ has probability density function $$f ( x ) = \begin{cases} k x ^ { 2 } & 0 \leqslant x \leqslant a
0 & \text { otherwise } \end{cases}$$ where $a$ and $k$ are constants.

Sketch the graph of $y = \mathrm { f } ( x )$ and explain in non-technical language what this tells you about $X$.
Given that $\mathrm { E } ( X ) = 4.5$, find
(a) the value of $a$,
(b) $\operatorname { Var } ( X )$.

OCR S2 2012 June Q8

8 The random variable $X$ has the distribution $\mathrm { N } \left( \mu , 8 ^ { 2 } \right)$. A test is carried out, at the $5 \%$ significance level, of $\mathrm { H } _ { 0 } : \mu = 30$ against $\mathrm { H } _ { 1 } : \mu > 30$, based on a random sample of size 18 .

Find the critical region for the test.
If $\mu = 30$ and the outcome of the test is that $\mathrm { H } _ { 0 }$ is rejected, state the type of error that is made. On a particular day this test is carried out independently a total of 20 times, and for 4 of these tests the outcome is that $\mathrm { H } _ { 0 }$ is rejected. It is known that the value of $\mu$ remains the same throughout these 20 tests.
Find the probability that $\mathrm { H } _ { 0 }$ is rejected at least 4 times if $\mu = 30$. Hence state whether you think that $\mu = 30$, giving a reason.
Given that the probability of making an error of the type different from that stated in part (ii) is 0.4 , calculate the actual value of $\mu$, giving your answer correct to 4 significant figures. \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE}

OCR S2 2013 June Q1

1 It is required to select a random sample of 30 pupils from a school with 853 pupils. A student suggests the following method.
"Give each pupil sequentially a three-digit number from 001 to 853 . Use a calculator to generate random three-digit numbers from 0.000 to 0.999 inclusive, multiply the answer by 853 , add 1 and round off to the nearest whole number. Select the corresponding pupil, and repeat as necessary".

Determine which pupil would be picked for each of the following calculator outputs: $$0.103 , \quad 0.104 , \quad 0.105 , \quad 0.106 , \quad 0.107$$
Use your answers to part (i) to show that this method is biased, and suggest an improvement.

OCR S2 2013 June Q2

2 The number of neutrinos that pass through a certain region in one second is a random variable with the distribution $\operatorname { Po } \left( 5 \times 10 ^ { 4 } \right)$. Use a suitable approximation to calculate the probability that the number of neutrinos passing through the region in 40 seconds is less than $1.999 \times 10 ^ { 6 }$.

OCR S2 2013 June Q3

3 The mean of a sample of 80 independent observations of a continuous random variable $Y$ is denoted by $\bar { Y }$. It is given that $\mathrm { P } ( \bar { Y } \leqslant 157.18 ) = 0.1$ and $\mathrm { P } ( \bar { Y } \geqslant 164.76 ) = 0.7$.

Calculate $\mathrm { E } ( Y )$ and the standard deviation of $Y$.
State
(a) where in your calculations you have used the Central Limit Theorem,
(b) why it was necessary to use the Central Limit Theorem,
(c) why it was possible to use the Central Limit Theorem.

OCR S2 2013 June Q4

4 The number of floods in a certain river plain is known to have a Poisson distribution. It is known that up until 10 years ago the mean number of floods per year was 0.32 . During the last 10 years there were 6 floods. Test at the $1 \%$ significance level whether there is evidence of an increase in the mean number of floods per year.

OCR S2 2013 June Q5

5 Two random variables $S$ and $T$ have probability density functions given by $$\begin{aligned} & f _ { S } ( x ) = \begin{cases} \frac { 3 } { a ^ { 3 } } ( x - a ) ^ { 2 } & 0 \leqslant x \leqslant a
0 & \text { otherwise } \end{cases}
& f _ { T } ( x ) = \begin{cases} c & 0 \leqslant x \leqslant a
0 & \text { otherwise } \end{cases} \end{aligned}$$ where $a$ and $c$ are constants.

On a single diagram sketch both probability density functions.
Calculate the mean of $S$, in terms of $a$.
Use your diagram to explain which of $S$ or $T$ has the bigger variance. (Answers obtained by calculation will score no marks.)

OCR S2 2013 June Q6

6 The random variable $X$ denotes the yield, in kilograms per acre, of a certain crop. Under the standard treatment it is known that $\mathrm { E } ( X ) = 38.4$. Under a new treatment, the yields of 50 randomly chosen regions can be summarised as $$n = 50 , \quad \sum x = 1834.0 , \quad \sum x ^ { 2 } = 70027.37 .$$ Test at the $1 \%$ level whether there has been a change in the mean crop yield.

OCR S2 2013 June Q7

7 Past experience shows that $35 \%$ of the senior pupils in a large school know the regulations about bringing cars to school. The head teacher addresses this subject in an assembly, and afterwards a random sample of 120 senior pupils is selected. In this sample it is found that 50 of these pupils know the regulations. Use a suitable approximation to test, at the $10 \%$ significance level, whether there is evidence that the proportion of senior pupils who know the regulations has increased. Justify your approximation.

OCR S2 2013 June Q8

8 The random variable $R$ has the distribution $\mathrm { B } ( 14 , p )$. A test is carried out at the $\alpha \%$ significance level of the null hypothesis $\mathrm { H } _ { 0 } : p = 0.25$, against $\mathrm { H } _ { 1 } : p > 0.25$.

Given that $\alpha$ is as close to 5 as possible, find the probability of a Type II error when the true value of $p$ is 0.4 .
State what happens to the probability of a Type II error as
(a) $p$ increases from 0.4,
(b) $\alpha$ increases, giving a reason.

OCR S2 2013 June Q9

9 The managers of a car breakdown recovery service are discussing whether the number of breakdowns per day can be modelled by a Poisson distribution. They agree that breakdowns occur randomly. Manager $A$ says, "it must be assumed that breakdowns occur at a constant rate throughout the day".

Give an improved version of Manager $A$ 's statement, and explain why the improvement is necessary.
Explain whether you think your improved statement is likely to hold in this context. Assume now that the number $B$ of breakdowns per day can be modelled by the distribution $\operatorname { Po } ( \lambda )$.
Given that $\lambda = 9.0$ and $\mathrm { P } \left( B > B _ { 0 } \right) < 0.1$, use tables to find the smallest possible value of $B _ { 0 }$, and state the corresponding value of $\mathrm { P } \left( B > B _ { 0 } \right)$.
Given that $\mathrm { P } ( B = 2 ) = 0.0072$, show that $\lambda$ satisfies an equation of the form $\lambda = 0.12 \mathrm { e } ^ { k \lambda }$, for a value of $k$ to be stated. Evaluate the expression $0.12 \mathrm { e } ^ { k \lambda }$ for $\lambda = 8.5$ and $\lambda = 8.6$, giving your answers correct to 4 decimal places. What can be deduced about a possible value of $\lambda$ ?

OCR S2 2007 June Q4

State two conditions needed for $X$ to be well modelled by a normal distribution.
It is given that $X \sim \mathrm {~N} \left( 50.0,8 ^ { 2 } \right)$. The mean of 20 random observations of $X$ is denoted by $\bar { X }$. Find $\mathrm { P } ( \bar { X } > 47.0 )$. 5 The number of system failures per month in a large network is a random variable with the distribution $\operatorname { Po } ( \lambda )$. A significance test of the null hypothesis $\mathrm { H } _ { 0 } : \lambda = 2.5$ is carried out by counting $R$, the number of system failures in a period of 6 months. The result of the test is that $\mathrm { H } _ { 0 }$ is rejected if $R > 23$ but is not rejected if $R \leqslant 23$.
State the alternative hypothesis.
Find the significance level of the test.
Given that $\mathrm { P } ( R > 23 ) < 0.1$, use tables to find the largest possible actual value of $\lambda$. You should show the values of any relevant probabilities. 6 In a rearrangement code, the letters of a message are rearranged so that the frequency with which any particular letter appears is the same as in the original message. In ordinary German the letter $e$ appears $19 \%$ of the time. A certain encoded message of 20 letters contains one letter $e$.
Using an exact binomial distribution, test at the $10 \%$ significance level whether there is evidence that the proportion of the letter $e$ in the language from which this message is a sample is less than in German, i.e., less than $19 \%$.
Give a reason why a binomial distribution might not be an appropriate model in this context. 7 Two continuous random variables $S$ and $T$ have probability density functions as follows. $$\begin{array} { l l } S : & f ( x ) = \begin{cases} \frac { 1 } { 2 } & - 1 \leqslant x \leqslant 1
0 & \text { otherwise } \end{cases}
T : & g ( x ) = \begin{cases} \frac { 3 } { 2 } x ^ { 2 } & - 1 \leqslant x \leqslant 1
0 & \text { otherwise } \end{cases} \end{array}$$
Sketch on the same axes the graphs of $y = \mathrm { f } ( x )$ and $y = \mathrm { g } ( x )$. [You should not use graph paper or attempt to plot points exactly.]
Explain in everyday terms the difference between the two random variables.
Find the value of $t$ such that $\mathrm { P } ( T > t ) = 0.2$. 8 A random variable $Y$ is normally distributed with mean $\mu$ and variance 12.25. Two statisticians carry out significance tests of the hypotheses $\mathrm { H } _ { 0 } : \mu = 63.0 , \mathrm { H } _ { 1 } : \mu > 63.0$.
Statistician $A$ uses the mean $\bar { Y }$ of a sample of size 23, and the critical region for his test is $\bar { Y } > 64.20$. Find the significance level for $A$ 's test.
Statistician $B$ uses the mean of a sample of size 50 and a significance level of $5 \%$.
(a) Find the critical region for $B$ 's test.
(b) Given that $\mu = 65.0$, find the probability that $B$ 's test results in a Type II error.
Given that, when $\mu = 65.0$, the probability that $A$ 's test results in a Type II error is 0.1365 , state with a reason which test is better. 9 (a) The random variable $G$ has the distribution $\mathrm { B } ( n , 0.75 )$. Find the set of values of $n$ for which the distribution of $G$ can be well approximated by a normal distribution.
(b) The random variable $H$ has the distribution $\mathrm { B } ( n , p )$. It is given that, using a normal approximation, $\mathrm { P } ( H \geqslant 71 ) = 0.0401$ and $\mathrm { P } ( H \leqslant 46 ) = 0.0122$.
Find the mean and standard deviation of the approximating normal distribution.
Hence find the values of $n$ and $p$.

Questions — OCR S2 (167 questions)

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