Questions - OCR S2 - A-Level Maths

OCR S2 2013 June Q2

4 marks Standard +0.3

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

9 marks Challenging +1.2

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
1. where in your calculations you have used the Central Limit Theorem,
2. why it was necessary to use the Central Limit Theorem,
3. why it was possible to use the Central Limit Theorem.

OCR S2 2013 June Q4

7 marks Standard +0.3

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

10 marks Moderate -0.3

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

11 marks Standard +0.3

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

11 marks Standard +0.3

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

6 marks Challenging +1.2

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
1. $p$ increases from 0.4,
2. $\alpha$ increases, giving a reason.

OCR S2 2013 June Q9

10 marks Standard +0.3

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

6 marks Moderate -0.3

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 $5 \%$.
1. Find the critical region for $B$ 's test.
2. Given that $\mu = 65.0$, find the probability that $B$ 's test results in a Type II error.
3. 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 $\mathrm { P } ( H \leqslant 46 ) = 0.0122$.
  1. Find the mean and standard deviation of the approximating normal distribution.
  2. Hence find the values of $n$ and $p$.

OCR S2 Q1

7 marks Moderate -0.3

1 In a study of urban foxes it is found that on average there are 2 foxes in every 3 acres.

Use a Poisson distribution to find the probability that, at a given moment,
1. in a randomly chosen area of 3 acres there are at least 4 foxes,
2. in a randomly chosen area of 1 acre there are exactly 2 foxes.
3. Explain briefly why a Poisson distribution might not be a suitable model.

OCR S2 Q2

7 marks Moderate -0.8

2 The random variable $W$ has the distribution $B \left( 40 , \frac { 2 } { 7 } \right)$. Use an appropriate approximation to find $\mathrm { P } ( W > 13 )$.

OCR S2 Q3

7 marks Moderate -0.3

3 The manufacturers of a brand of chocolates claim that, on average, $30 \%$ of their chocolates have hard centres. In a random sample of 8 chocolates from this manufacturer, 5 had hard centres. Test, at the $5 \%$ significance level, whether there is evidence that the population proportion of chocolates with hard centres is not $30 \%$, stating your hypotheses clearly. Show the values of any relevant probabilities.

OCR S2 Q4

7 marks Moderate -0.8

4 DVD players are tested after manufacture. The probability that a randomly chosen DVD player is defective is 0.02 . The number of defective players in a random sample of size 80 is denoted by $R$.

Use an appropriate approximation to find $\mathrm { P } ( R \geqslant 2 )$.
Find the smallest value of $r$ for which $\mathrm { P } ( R \geqslant r ) < 0.01$.

OCR S2 Q5

9 marks Challenging +1.2

5 In an investment model the increase, $Y \%$, in the value of an investment in one year is modelled as a continuous random variable with the distribution $\mathrm { N } \left( \mu , \frac { 1 } { 4 } \mu ^ { 2 } \right)$. The value of $\mu$ depends on the type of investment chosen.

Find $\mathrm { P } ( Y < 0 )$, showing that it is independent of the value of $\mu$.
Given that $\mu = 6$, find the probability that $Y < 9$ in each of three randomly chosen years.
Explain why the calculation in part (ii) might not be valid if applied to three consecutive years.

OCR S2 Q6

10 marks Standard +0.3

6 Alex obtained the actual waist measurements, $w$ inches, of a random sample of 50 pairs of jeans, each of which was labelled as having a 32 -inch waist. The results are summarised by $$n = 50 , \quad \Sigma w = 1615.0 , \quad \Sigma w ^ { 2 } = 52214.50$$ Test, at the $0.1 \%$ significance level, whether this sample provides evidence that the mean waist measurement of jeans labelled as having 32 -inch waists is in fact greater than 32 inches. State your hypotheses clearly. \section*{Jan 2006}

OCR S2 Q7

10 marks Standard +0.3

7 The random variable $X$ has the distribution $\mathrm { N } \left( \mu , 8 ^ { 2 } \right)$. The mean of a random sample of 12 observations of $X$ is denoted by $\bar { X }$. A test is carried out at the $1 \%$ significance level of the null hypothesis $\mathrm { H } _ { 0 } : \mu = 80$ against the alternative hypothesis $\mathrm { H } _ { 1 } : \mu < 80$. The test is summarised as follows: 'Reject $\mathrm { H } _ { 0 }$ if $\bar { X } < c$; otherwise do not reject $\mathrm { H } _ { 0 } { } ^ { \prime }$.

Calculate the value of $c$.
Assuming that $\mu = 80$, state whether the conclusion of the test is correct, results in a Type I error, or results in a Type II error if:
1. $\bar { X } = 74.0$,
2. $\bar { X } = 75.0$.
3. Independent repetitions of the above test, using the value of $c$ found in part (i), suggest that in fact the probability of rejecting the null hypothesis is 0.06 . Use this information to calculate the value of $\mu$.

OCR S2 Q8

15 marks Moderate -0.3

8 A continuous random variable $X$ has probability density function given by $$f ( x ) = \begin{cases} k x ^ { n } & 0 \leqslant x \leqslant 1 \\ 0 & \text { otherwise } \end{cases}$$ where $n$ and $k$ are positive constants.

Find $k$ in terms of $n$.
Show that $\mathrm { E } ( X ) = \frac { n + 1 } { n + 2 }$. It is given that $n = 3$.
Find the variance of $X$.
One hundred observations of $X$ are taken, and the mean of the observations is denoted by $\bar { X }$. Write down the approximate distribution of $\bar { X }$, giving the values of any parameters.
Write down the mean and the variance of the random variable $Y$ with probability density function given by $$g ( y ) = \begin{cases} 4 \left( y + \frac { 4 } { 5 } \right) ^ { 3 } & - \frac { 4 } { 5 } \leqslant y \leqslant \frac { 1 } { 5 } \\ 0 & \text { otherwise } \end{cases}$$

OCR S2 2010 January Q1

4 marks Easy -1.2

The values of 5 independent observations from a population can be summarised by $$\sum x = 75.8, \quad \sum x^2 = 1154.58.$$ Find unbiased estimates of the population mean and variance. [4]

OCR S2 2010 January Q2

3 marks Easy -1.2

A college has 400 students. A journalist wants to carry out a survey about food preferences and she obtains a sample of 30 pupils from the college by the following method. • Obtain a list of all the students. • Number the students, with numbers running sequentially from 0 to 399. • Select 30 random integers in the range 000 to 999 inclusive. If a random integer is in the range 0 to 399, then the student with that number is selected. If the number is greater than 399, then 400 is subtracted from the number (if necessary more than once) until an answer in the range 0 to 399 is selected, and the student with that number is selected.

Explain why this method is unsatisfactory. [2]
Explain how it could be improved. [1]

OCR S2 2010 January Q3

6 marks Moderate -0.8

In a large town, 35% of the inhabitants have access to television channel $C$. A random sample of 60 inhabitants is obtained. Use a suitable approximation to find the probability that 18 or fewer inhabitants in the sample have access to channel $C$. [6]

OCR S2 2010 January Q4

7 marks Moderate -0.3

80 randomly chosen people are asked to estimate a time interval of 60 seconds without using a watch or clock. The mean of the 80 estimates is 58.9 seconds. Previous evidence shows that the population standard deviation of such estimates is 5.0 seconds. Test, at the 5% significance level, whether there is evidence that people tend to underestimate the time interval. [7]

OCR S2 2010 January Q5

8 marks Standard +0.3

The number of customers arriving at a store between 8.50 am and 9 am on Saturday mornings is a random variable which can be modelled by the distribution Po(11.0). Following a series of price cuts, on one particular Saturday morning 19 customers arrive between 8.50 am and 9 am. The store's management claims, first, that the mean number of customers has increased, and second, that this is due to the price cuts.

Test the first part of the claim, at the 5% significance level. [7]
Comment on the second part of the claim. [1]

OCR S2 2010 January Q6

7 marks Moderate -0.8

The continuous random variable $X$ has the distribution N($\mu$, $\sigma^2$).

Each of the three following sets of probabilities is impossible. Give a reason in each case why the probabilities cannot both be correct. (You should not attempt to find $\mu$ or $\sigma$.)
1. P($X > 50$) = 0.7 and P($X < 50$) = 0.2 [1]
2. P($X > 50$) = 0.7 and P($X > 70$) = 0.8 [1]
3. P($X > 50$) = 0.3 and P($X < 70$) = 0.3 [1]
Given that P($X > 50$) = 0.7 and P($X < 70$) = 0.7, find the values of $\mu$ and $\sigma$. [4]

OCR S2 2010 January Q7

13 marks Moderate -0.3

The continuous random variable $T$ is equally likely to take any value from 5.0 to 11.0 inclusive.

Sketch the graph of the probability density function of $T$. [2]
Write down the value of E($T$) and find by integration the value of Var($T$). [5]
A random sample of 48 observations of $T$ is obtained. Find the approximate probability that the mean of the sample is greater than 8.3, and explain why the answer is an approximation. [6]

OCR S2 2010 January Q8

8 marks Standard +0.3

The random variable $R$ has the distribution B(10, $p$). The null hypothesis H$_0$: $p = 0.7$ is to be tested against the alternative hypothesis H$_1$: $p < 0.7$, at a significance level of 5%.

Find the critical region for the test and the probability of making a Type I error. [3]
Given that $p = 0.4$, find the probability that the test results in a Type II error. [3]
Given that $p$ is equally likely to take the values 0.4 and 0.7, find the probability that the test results in a Type II error. [2]

Questions — OCR S2 (169 questions)

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