Questions - OCR S3 - A-Level Maths

Given that $Y = \frac { 1 } { X }$, find the (cumulative) distribution function of $Y$, and deduce that $Y$ and $X$ have identical distributions.
Find $\mathrm { E } ( X + 1 )$ and deduce the value of $\mathrm { E } \left( \frac { 1 } { X } \right)$.

OCR S3 2011 January Q7

When should Yates' correction be applied when carrying out a $\chi ^ { 2 }$ test? Two vaccines against typhoid fever, $A$ and $B$, were tested on a total of 700 people in Nepal during a particular year. The vaccines were allocated randomly and whether or not typhoid had developed was noted during the following year. The results are shown in the table.
\multirow{2}{*}{} Vaccines
\cline { 2 - 3 } $A$ $B$
Developed typhoid 19 4
Did not develop typhoid 310 367
Carry out a suitable $\chi ^ { 2 }$ test at the $1 \%$ significance level to determine whether the outcome depends on the vaccine used. Comment on the result.

OCR S3 2011 January Q8

State circumstances under which it would be necessary to calculate a pooled estimate of variance when carrying out a two-sample hypothesis test.
An investigation into whether passive smoking affects lung capacity considered a random sample of 20 children whose parents did not smoke and a random sample of 22 children whose parents did smoke. None of the children themselves smoked. The lung capacity, in litres, of each child was measured and the results are summarised as follows. For the children whose parents did not smoke: $n _ { 1 } = 20 , \Sigma x _ { 1 } = 42.4$ and $\Sigma x _ { 1 } ^ { 2 } = 90.43$.
For the children whose parents did smoke: $\quad n _ { 2 } = 22 , \Sigma x _ { 2 } = 42.5$ and $\Sigma x _ { 2 } ^ { 2 } = 82.93$.
The means of the two populations are denoted by $\mu _ { 1 }$ and $\mu _ { 2 }$ respectively.
(a) State conditions for which a $t$-test would be appropriate for testing whether $\mu _ { 1 }$ exceeds $\mu _ { 2 }$.
(b) Assuming the conditions are valid, carry out the test at the $1 \%$ significance level and comment on the result.
(c) Calculate a 99\% confidence interval for $\mu _ { 1 } - \mu _ { 2 }$.

OCR S3 2012 January Q1

1 In a test of association of two factors, $A$ and $B$, a $2 \times 2$ contingency table yielded 5.63 for the value of $\chi ^ { 2 }$ with Yates’ correction.

State the null hypothesis and alternative hypothesis for the test.
State how Yates' correction is applied, and whether it increases or decreases the value of $\chi ^ { 2 }$.
Carry out the test at the $2 \frac { 1 } { 2 } \%$ significance level.

OCR S3 2012 January Q2

2 An investigation in 2007 into the incidence of tuberculosis (TB) in badgers in a certain area found that 42 out of a random sample of 190 badgers tested positive for TB.
In 2010, 48 out of a random sample of 150 badgers tested positive for TB.

Assuming that the population proportions of badgers with TB are the same in 2007 and 2010, obtain the best estimate of this proportion.
Carry out a test at the $2 \frac { 1 } { 2 } \%$ significance level of whether the population proportion of badgers with TB increased from 2007 to 2010.

OCR S3 2012 January Q5

5 A statistician suggested that the weekly sales $X$ thousand litres at a petrol station could be modelled by the following probability density function. $$f ( x ) = \begin{cases} \frac { 1 } { 40 } ( 2 x + 3 ) & 0 \leqslant x < 5
0 & \text { otherwise } \end{cases}$$

Show that, using this model, $\mathrm { P } ( a \leqslant X < a + 1 ) = \frac { a + 2 } { 20 }$ for $0 \leqslant a \leqslant 4$. Sales in 100 randomly chosen weeks gave the following grouped frequency table.
$x$ $0 \leqslant x < 1$ $1 \leqslant x < 2$ $2 \leqslant x < 3$ $3 \leqslant x < 4$ $4 \leqslant x < 5$
Frequency 16 12 18 30 24
Carry out a goodness of fit test at the $10 \%$ significance level of whether $\mathrm { f } ( x )$ fits the data.

OCR S3 2012 January Q6

6 The continuous random variable $Y$ has probability density function given by $$f ( y ) = \begin{cases} - \frac { 1 } { 4 } y & - 2 \leqslant y < 0
\frac { 1 } { 4 } y & 0 \leqslant y \leqslant 2
0 & \text { otherwise. } \end{cases}$$ Find

the interquartile range of $Y$,
$\operatorname { Var } ( Y )$,
$\mathrm { E } ( | Y | )$.

OCR S3 2012 January Q7

7 The manufacturer's specification for batteries used in a certain electronic game is that the mean lifetime should be 32 hours. The manufacturer tests a random sample of 10 batteries made in Factory $A$, and the lifetimes ( $x$ hours) are summarised by $$n = 10 , \sum x = 289.0 \text { and } \sum x ^ { 2 } = 8586.19 .$$ It may be assumed that the population of lifetimes has a normal distribution.

Carry out a one-tail test at the $5 \%$ significance level of whether the specification is being met.
Justify the use of a one-tail test in this context. Batteries made with the same specification are also made in Factory $B$. The lifetimes of these batteries are also normally distributed. A random sample of 12 batteries from this factory was tested. The lifetimes are summarised by $$n = 12 , \sum x = 363.0 \text { and } \sum x ^ { 2 } = 11290.95 \text {. }$$
(a) State what further assumption must be made in order to test whether there is any difference in the mean lifetimes of batteries made at the two factories.
Use the data to comment on whether this assumption is reasonable.
(b) Carry out the test at the $10 \%$ significance level.

OCR S3 2006 June Q1

1 The numbers of $\alpha$-particles emitted per minute from two types of source, $A$ and $B$, have the distributions $\operatorname { Po } ( 1.5 )$ and $\operatorname { Po } ( 2 )$ respectively. The total number of $\alpha$-particles emitted over a period of 2 minutes from three sources of type $A$ and two sources of type $B$, all of which are independent, is denoted by $X$. Calculate $\mathrm { P } ( X = 27 )$.

OCR S3 2006 June Q2

2 The manager of a factory with a large number of employees investigated when accidents to employees occurred during 8-hour shifts. An analysis was made of 600 randomly chosen accidents that occurred over a year. The following table shows the numbers of accidents occurring in the four consecutive 2-hour periods of the 8-hour shifts.

Period	1	2	3	4
Number of accidents	138	127	165	170

Test, at the $5 \%$ significance level, whether the proportions of all accidents that occur in the four time periods differ.

OCR S3 2006 June Q3

3 Ten randomly chosen athletes were coached for a 200 m event. For each athlete, the times taken to run 200 m before and after coaching were measured. The sample mean times before and after coaching were 23.43 seconds and 22.84 seconds respectively. For each athlete the difference, $d$ seconds, in the times before and after coaching was calculated and an unbiased estimate of the population variance of $d$ was found to be 0.548 . Stating any required assumption, test at the $5 \%$ significance level whether the population mean time for the 200 m run decreased after coaching.

OCR S3 2006 June Q4

4 The continuous random variable $X$ has probability density function given by $$f ( x ) = \begin{cases} \frac { 4 } { 3 x ^ { 3 } } & 1 \leqslant x < 2
\frac { 1 } { 12 } x & 2 \leqslant x \leqslant 4
0 & \text { otherwise } \end{cases}$$

Find the upper quartile of $X$.
Find the value of $a$ for which $\mathrm { E } \left( X ^ { 2 } \right) = a \mathrm { E } ( X )$.

OCR S3 2006 June Q5

5 Gloria is a market trader who sells jeans. She trades on Mondays, Wednesdays and Fridays. Wishing to investigate whether the volume of trade depends on the day of the week, Gloria analysed a random sample of 150 days' sales and classified them by day and volume (low, medium and high). The results are given in the table below.

		Day
		Monday	Wednesday	Friday
\multirow{3}{*}{Volume}	Low	15	13	2
	Medium	23	26	23
	High	12	9	27

Gloria asked a statistician to perform a suitable test of independence and, as part of this test, expected frequencies were calculated. These are shown in the table below.

		Day
		Monday	Wednesday	Friday
	Low	10.00	9.60	10.40
Volume	Medium	24.00	23.04	24.96
	High	16.00	15.36	16.64

Show how the value 23.04 for medium volume on Wednesday has been obtained.
State, giving a reason, if it is necessary to combine any rows or columns in order to carry out the test. The value of the test statistic is found to be 21.15, correct to 2 decimal places.
Stating suitable hypotheses for the test, give its conclusion using a $1 \%$ significance level. Gloria wishes to hold a sale and asks the statistician to advise her on which day to hold it in order to sell as much as possible.
State the day that the statistician should advise and give a reason for the choice.

OCR S3 2006 June Q6

6 An anthropologist was studying the inhabitants of two islands, Raloa and Tangi. Part of the study involved the incidence of blood group type A. The blood of 80 randomly chosen inhabitants of Raloa and 85 randomly chosen inhabitants of Tangi was tested. The number of inhabitants with type A blood was 28 for the Raloa sample and 46 for the Tangi sample. The anthropologist calculated $90 \%$ confidence intervals for the population proportions of inhabitants with type A blood. They were $( 0.262,0.438 )$ for Raloa and $( 0.452,0.630 )$ for Tangi, where each figure is correct to 3 decimal places. It is known that $43 \%$ of the world's population have type A blood.

State, giving your reasons, whether there is evidence for the following assertions about the proportions of people with type A blood.
(a) The proportion in Raloa is different from the world proportion.
(b) The proportion in Tangi is different from the world proportion.
Carry out a suitable test, at the $2 \%$ significance level, of whether the proportions of people with type A blood differ on the two islands.

OCR S3 2006 June Q7

7 A queue of cars has built up at a set of traffic lights which are at red. When the lights turn green, the time for the first car to start to move has a normal distribution with mean 2.2 s and standard deviation 0.75 s . This time is the reaction time for the first car. For each subsequent car the reaction time is the time taken for it to start to move after the car in front starts to move. These reaction times have identical normal distributions with mean 1.8 s and standard deviation 0.70 s . It may be assumed that all reaction times are independent.

Calculate the probability that the reaction time for the second car in the queue is less than half of the reaction time for the first car.
Calculate the probability that the fifth car in the queue starts to move less than 10 seconds after the lights turn green.
State where, in part (i), independence is required.

OCR S3 2006 June Q8

8 Two machines, $A$ and $B$, produce metal rods. Machine $B$ is new and it is required that its accuracy should be checked against that of machine $A$. The observed variable is the length of a rod. Random samples of rods, 40 from machine $A$ and 50 from machine $B$, are taken and their lengths, $x _ { A } \mathrm {~cm}$ and $x _ { B } \mathrm {~cm}$, are measured. The results are summarised by $$\Sigma x _ { A } = 136.48 , \quad \Sigma x _ { B } = 176.35 , \quad \Sigma x _ { B } ^ { 2 } = 630.1940 .$$ The variance of the length of the rods produced by machine $A$ is known to be $0.0490 \mathrm {~cm} ^ { 2 }$. The mean lengths of the rods produced by the machines are denoted by $\mu _ { A } \mathrm {~cm}$ and $\mu _ { B } \mathrm {~cm}$ respectively.

Test, at the $5 \%$ significance level, the hypothesis $\mu _ { B } > \mu _ { A }$.
Find the set of values of $a$ for which the null hypothesis $\mu _ { B } - \mu _ { A } = 0.025$ would not be rejected in favour of the alternative hypothesis $\mu _ { B } - \mu _ { A } > 0.025$ at the $a \%$ significance level.
For the test in part (i) to be valid,
(a) state whether it is necessary to assume that the two population variances are equal,
(b) state, giving a reason, whether it is necessary to assume that the lengths of rods are normally distributed.

OCR S3 2007 June Q1

1 The continuous random variable $X$ has probability density function given by $$\mathrm { f } ( x ) = \begin{cases} a & 0 \leqslant x \leqslant 1 ,
\frac { a } { x ^ { 2 } } & x > 1 ,
0 & \text { otherwise. } \end{cases}$$ Find the value of the constant $a$.

OCR S3 2007 June Q2

2 Two brands of car battery, ‘Invincible’ and ‘Excelsior’, have lifetimes which are normally distributed. Invincible batteries have a mean lifetime of 5 years with standard deviation 0.7 years. Excelsior batteries have a mean lifetime of 4.5 years with standard deviation 0.5 years. Random samples of 20 Invincible batteries and 25 Excelsior batteries are selected and the sample mean lifetimes are $\bar { X } _ { I }$ years and $\bar { X } _ { E }$ years respectively.

State the distributions of $\bar { X } _ { I }$ and $\bar { X } _ { E }$.
Calculate $\mathrm { P } \left( \bar { X } _ { I } - \bar { X } _ { E } \geqslant 1 \right)$.

OCR S3 2007 June Q3

3 A nurse was asked to measure the blood pressure of 12 patients using an aneroid device. The nurse's readings were immediately checked using an accurate electronic device. The differences, $x$, given by $x =$ (aneroid reading - electronic reading), in appropriate units, are shown below. $$\begin{array} { c c c c c c c c c c c } - 1.3 & 4.7 & - 0.9 & 3.8 & - 1.5 & 4.0 & - 1.9 & 4.4 & - 0.8 & 5.5 & - 2.9 \end{array} 4.1$$ Stating any assumption you need to make, test, at the $10 \%$ significance level, whether readings with an aneroid device, on average, overestimate patients’ blood pressure.

OCR S3 2007 June Q4

4 The students in a large university department take a trial examination some time before the proper examination. A random sample of 60 students took both examinations during a particular course. 42 students passed the trial examination, 36 passed the proper examination and 13 failed both examinations.

Copy and complete the following contingency table.
Proper
\cline { 2 - 4 } \multicolumn{1}{l}{} Pass Fail Total
\cline { 2 - 5 } Pass 42
\cline { 2 - 5 } Trial Fail 13
\cline { 2 - 5 } Total 36 60
Carry out a test of independence at the $\frac { 1 } { 2 } \%$ level of significance.

OCR S3 2007 June Q5

5 A music store sells both upright and grand pianos. Grand pianos are sold at random times and at a constant average weekly rate $\lambda$. The probability that in one week no grand pianos are sold is 0.45 .

Show that $\lambda = 0.80$, correct to 2 decimal places. Upright pianos are sold, independently, at random times and at a constant average weekly rate $\mu$. During a period of 100 weeks the store sold 180 upright pianos.
Calculate the probability that the total number of pianos sold in a randomly chosen week will exceed 3.
Calculate the probability that over a period of 3 weeks the store sells a total of 6 pianos during the first week and a total of 4 pianos during the next fortnight.

OCR S3 2007 June Q6

6 Random samples of 200 'Alpha' and 150 'Beta' vacuum cleaners were monitored for reliability. It was found that 62 Alpha and 35 Beta cleaners required repair during the guarantee period of one year. The proportions of all Alpha and Beta cleaners that require repair during the guarantee period are $p _ { \alpha }$ and $p _ { \beta }$ respectively.

Find a $95 \%$ confidence interval for $p _ { \alpha }$.
Give a reason why, apart from rounding, the interval is approximate.
Test, at the $5 \%$ significance level, whether $p _ { \alpha }$ differs from $p _ { \beta }$.

OCR S3 2007 June Q7

7 The continuous random variable $X$ has (cumulative) distribution function given by $$\mathrm { F } ( x ) = \begin{cases} 0 & x < 1
1 - \frac { 1 } { x ^ { 4 } } & x \geqslant 1 \end{cases}$$

Find the (cumulative) distribution function, $\mathrm { G } ( y )$, of the random variable $Y$, where $Y = \frac { 1 } { X ^ { 2 } }$.
Hence show that the probability density function of $Y$ is given by $$g ( y ) = \begin{cases} 2 y & 0 < y \leqslant 1
0 & \text { otherwise } \end{cases}$$
Find $\mathrm { E } ( \sqrt [ 3 ] { Y } )$.

OCR S3 2007 June Q8

8 The continuous random variable $Y$ has a distribution with mean $\mu$ and variance 20. A random sample of 50 observations of $Y$ is selected and these observations are summarised in the following grouped frequency table.

Values	$y < 20$	$20 \leqslant y < 25$	$25 \leqslant y < 30$	$y \geqslant 30$
Frequency	3	27	12	8

Assuming that $Y \sim \mathrm {~N} ( 25,20 )$, show that the expected frequency for the interval $20 \leqslant y < 25$ is 18.41, correct to 2 decimal places, and obtain the remaining expected frequencies.
Test, at the $5 \%$ significance level, whether the distribution $\mathrm { N } ( 25,20 )$ fits the data.
Given that the sample mean is 24.91 , find a $98 \%$ confidence interval for $\mu$.
Does the outcome of the test in part (ii) affect the validity of the confidence interval found in part (iii)? Justify your answer.

OCR S3 2011 June Q1

1 The random variables $X$ and $Y$ are independent with $X \sim \operatorname { Po } ( 5 )$ and $Y \sim \operatorname { Po } ( 4 )$. $S$ denotes the sum of 2 observations of $X$ and 3 observations of $Y$.

Find $\mathrm { E } ( S )$ and $\operatorname { Var } ( S )$.
The random variable $T$ is defined by $\frac { 1 } { 2 } X - \frac { 1 } { 4 } Y$. Show that $\mathrm { E } ( T ) = \operatorname { Var } ( T )$.
State which of $S$ and $T$ (if either) does not have a Poisson distribution, giving a reason for your answer.

\multirow{2}{*}{}	Vaccines
\cline { 2 - 3 }	\(A\)	\(B\)
Developed typhoid	19	4
Did not develop typhoid	310	367

\(x\)	\(0 \leqslant x < 1\)	\(1 \leqslant x < 2\)	\(2 \leqslant x < 3\)	\(3 \leqslant x < 4\)	\(4 \leqslant x < 5\)
Frequency	16	12	18	30	24

				Proper
\cline { 2 - 4 } \multicolumn{1}{l}{}	Pass	Fail	Total
\cline { 2 - 5 }	Pass			42
\cline { 2 - 5 } Trial	Fail		13
\cline { 2 - 5 }	Total	36		60

Values	\(y < 20\)	\(20 \leqslant y < 25\)	\(25 \leqslant y < 30\)	\(y \geqslant 30\)
Frequency	3	27	12	8

Questions — OCR S3 (139 questions)

Browse by module

Browse by board

OCR S3 2011 January Q6

OCR S3 2011 January Q7

OCR S3 2011 January Q8

OCR S3 2012 January Q1

OCR S3 2012 January Q2

OCR S3 2012 January Q5

OCR S3 2012 January Q6

OCR S3 2012 January Q7

OCR S3 2006 June Q1

OCR S3 2006 June Q2

OCR S3 2006 June Q3

OCR S3 2006 June Q4

OCR S3 2006 June Q5

OCR S3 2006 June Q6

OCR S3 2006 June Q7

OCR S3 2006 June Q8

OCR S3 2007 June Q1

OCR S3 2007 June Q2

OCR S3 2007 June Q3

OCR S3 2007 June Q4

OCR S3 2007 June Q5

OCR S3 2007 June Q6

OCR S3 2007 June Q7

OCR S3 2007 June Q8

OCR S3 2011 June Q1