Questions - OCR S3 - A-Level Maths

Calculate a 95\% confidence interval for the proportion of all householders who are in favour of the new method. A city councillor said he believed that as many householders were in favour of the new method as were against it.
Comment on the councillor's belief.

OCR S3 2006 January Q2

2 A particular type of engine used in rockets is designed to have a mean lifetime of at least 2000 hours. A check of four randomly chosen engines yielded the following lifetimes in hours. $$\begin{array} { l l l l } 1896.4 & 2131.5 & 1903.3 & 1901.6 \end{array}$$ A significance test of whether engines meet the design is carried out. It may be assumed that lifetimes have a normal distribution.

Give a reason why a $t$-test should be used.
Carry out the test at the $10 \%$ significance level.

OCR S3 2006 January Q3

3 For a restaurant with a home-delivery service, the delivery time in minutes can be modelled by a continuous random variable $T$ with probability density function given by $$f ( t ) = \begin{cases} \frac { \pi } { 90 } \sin \left( \frac { \pi t } { 60 } \right) & 20 \leqslant t \leqslant 60
0 & \text { otherwise. } \end{cases}$$

Given that $20 \leqslant a \leqslant 60$, show that $\mathrm { P } ( T \leqslant a ) = \frac { 1 } { 3 } \left( 1 - 2 \cos \left( \frac { \pi a } { 60 } \right) \right)$. There is a delivery charge of $\pounds 3$ but this is reduced to $\pounds 2$ if the delivery time exceeds a minutes.
Find the value of $a$ for which the expected value of the delivery charge for a home-delivery is £2.80.

OCR S3 2006 January Q4

4 A multi-storey car park has two entrances and one exit. During a morning period the numbers of cars using the two entrances are independent Poisson variables with means 2.3 and 3.2 per minute. The number leaving is an independent Poisson variable with mean 1.8 per minute. For a randomly chosen 10-minute period the total number of cars that enter and the number of cars that leave are denoted by the random variables $X$ and $Y$ respectively.

Use a suitable approximation to calculate $\mathrm { P } ( X \geqslant 40 )$.
Calculate $\mathrm { E } ( X - Y )$ and $\operatorname { Var } ( X - Y )$.
State, giving a reason, whether $X - Y$ has a Poisson distribution.

OCR S3 2006 January Q5

5 The continuous random variable $X$ has cumulative distribution function given by $$F ( x ) = \begin{cases} 0 & x < 1 ,
\frac { 1 } { 8 } ( x - 1 ) ^ { 2 } & 1 \leqslant x < 3 ,
a ( x - 2 ) & 3 \leqslant x < 4 ,
1 & x \geqslant 4 , \end{cases}$$ where $a$ is a positive constant.

Find the value of $a$.
Verify that $C _ { X } ( 8 )$, the 8th percentile of $X$, is 1.8 .
Find the cumulative distribution function of $Y$, where $Y = \sqrt { X - 1 }$.
Find $C _ { Y } ( 8 )$ and verify that $C _ { Y } ( 8 ) = \sqrt { C _ { X } ( 8 ) - 1 }$.

OCR S3 2006 January Q6

6 A company with a large fleet of cars compared two types of tyres, $A$ and $B$. They measured the stopping distances of cars when travelling at a fixed speed on a dry road. They selected 20 cars at random from the fleet and divided them randomly into two groups of 10 , one group being fitted with tyres of type $A$ and the other group with tyres of type $B$. One of the cars fitted with tyres of type $A$ broke down so these tyres were tested on only 9 cars. The stopping distances, $x$ metres, for the two samples are summarised by $$n _ { A } = 9 , \quad \bar { x } _ { A } = 17.30 , \quad s _ { A } ^ { 2 } = 0.7400 , \quad n _ { B } = 10 , \quad \bar { x } _ { B } = 14.74 , \quad s _ { B } ^ { 2 } = 0.8160 ,$$ where $s _ { A } ^ { 2 }$ and $s _ { B } ^ { 2 }$ are unbiased estimates of the two population variances.
It is given that the two populations have the same variance.

Show that an unbiased estimate of this variance is 0.780 , correct to 3 decimal places. The population mean stopping distances for cars with tyres of types $A$ and $B$ are denoted by $\mu _ { A }$ metres and $\mu _ { B }$ metres respectively.
Stating any further assumption you need to make, calculate a $98 \%$ confidence interval for $\mu _ { A } - \mu _ { B }$. The manufacturers of Type $B$ tyres assert that $\mu _ { B } < \mu _ { A } - 2$.
Carry out a significance test of this assertion at the $5 \%$ significance level. \section*{[Question 7 is printed overleaf.]}

OCR S3 2007 January Q1

1 The marks obtained by a randomly chosen student in the two papers of an examination are denoted by the random variables $X$ and $Y$, where $X \sim \mathrm {~N} ( 45,81 )$ and $Y \sim \mathrm {~N} ( 33,63 )$. The student's overall mark for the examination, $T$, is given by $T = X + \lambda Y$, where the constant $\lambda$ is chosen such that $\mathrm { E } ( T ) = 100$.

Show that $\lambda = \frac { 5 } { 3 }$.
Assuming that $X$ and $Y$ are independent, state the distribution of $T$, giving the values of its parameters.
Comment on the assumption of independence.

OCR S3 2007 January Q2

2 The continuous random variable $X$ takes values in the interval $0 \leqslant x \leqslant 3$ only with probability density function f . The graph of $y = \mathrm { f } ( x )$ consists of the two line segments shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{4a6d94a2-66e1-449a-ac0e-1fbada74bb3b-2_524_1287_950_429}

Show that $a = \frac { 2 } { 3 }$.
Find the equations of the two line segments.
Hence write down the probability density function of $X$.
Find $\mathrm { E } ( X )$.

OCR S3 2007 January Q3

3 A new treatment of cotton thread, designed to increase the breaking strength, was tested on a random sample of 6 pieces of a standard length. The breaking strengths, in grams, were as follows. $$\begin{array} { l l l l l l } 17.3 & 18.4 & 18.6 & 17.2 & 17.5 & 19.3 \end{array}$$ The breaking strengths of a random sample of 5 similar pieces of the thread which had not been treated were as follows. \section*{$\begin{array} { l l l l l } 18.6 & 17.2 & 16.3 & 17.4 & 16.8 \end{array}$} A test of whether the treatment has been successful is to be carried out.

State what distributional assumptions are needed.
Carry out the test at the $10 \%$ significance level.

OCR S3 2007 January Q4

4 A machine is set to produce metal discs with mean diameter 15.4 mm . In order to test the correctness of the setting, a random sample of 12 discs was selected and the diameters, $x \mathrm {~mm}$, were measured. The results are summarised by $\Sigma x = 177.6$ and $\Sigma x ^ { 2 } = 2640.40$. Diameters may be assumed to be normally distributed with mean $\mu \mathrm { mm }$.

Find a $95 \%$ confidence interval for $\mu$.
Test, at the $5 \%$ significance level, the null hypothesis $\mu = 15.4$ against the alternative hypothesis $\mu < 15.4$.

OCR S3 2007 January Q5

5 Each person in a random sample of 1200 people was asked whether he or she approved of certain proposals to reduce atmospheric pollution. It was found that 978 people approved. The proportion of people in the whole population who would approve is denoted by $p$.

Write down an estimate $\hat { p }$ of $p$.
Find a 90\% confidence interval for $p$.
Explain, in the context of the question, the meaning of a $90 \%$ confidence interval.
Estimate the sample size that would give a value for $\hat { p }$ that differs from the value of $p$ by less than 0.01 with probability $90 \%$.

OCR S3 2007 January Q6

6 The lifetime of a particular machine, in months, can be modelled by the random variable $T$ with probability density function given by $$\mathrm { f } ( t ) = \begin{cases} \frac { 3 } { t ^ { 4 } } & t \geqslant 1
0 & \text { otherwise. } \end{cases}$$

Obtain the (cumulative) distribution function of $T$.
Show that the probability density function of the random variable $Y$, where $Y = T ^ { 3 }$, is given by $\mathrm { g } ( y ) = \frac { 1 } { y ^ { 2 } }$, for $y \geqslant 1$.
Find $\mathrm { E } ( \sqrt { Y } )$.

OCR S3 2007 January Q7

7 It is thought that a person's eye colour is related to the reaction of the person's skin to ultra-violet light. As part of a study, a random sample of 140 people were treated with a standard dose of ultra-violet light. The degree of reaction was classified as None, Mild or Strong. The results are given in Table 1. The corresponding expected frequencies for a $\chi ^ { 2 }$ test of association between eye colour and reaction are shown in Table 2. \begin{table}[h]

\captionsetup{labelformat=empty} \caption{Table 1
Observed frequencies}

		Eye colour
		Blue	Brown	Other	Total
	None	12	17	10	39
Reaction	Mild	31	21	11	63
	Strong	22	4	12	38
	Total	65	42	33	140

\end{table} \begin{table}[h]

\captionsetup{labelformat=empty} \caption{Table 2
Expected frequencies}

		Eye colour
		Blue	Brown	Other
	None	18.11	11.70	9.19
Reaction	Mild	29.25	18.90	14.85
	Strong	17.64	11.40	8.96

\end{table}

(a) State suitable hypotheses for the test.
(b) Show how the expected frequency of 18.11 in Table 2 is obtained.
(c) Show that the three cells in the top row together contribute 4.53 to the calculated value of $\chi ^ { 2 }$, correct to 2 decimal places.
(d) You are given that the total calculated value of $\chi ^ { 2 }$ is 12.78 , correct to 2 decimal places. Give the smallest value of $\alpha$ obtained from the tables for which the null hypothesis would be rejected at the $\alpha \%$ significance level.
Test, at the $5 \%$ significance level, whether the proportions of people in the whole population with blue eyes, brown eyes and other colours are in the ratios $2 : 2 : 1$. \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. OCR is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }

OCR S3 2008 January Q1

1 A blueberry farmer increased the amount of water sprayed over his berries to see what effect this had on their weight. The farmer weighed each of a random sample of 80 berries of the previous season's crop and each of a random sample of 100 berries of the new crop. The results are summarised in the following table, in which $\bar { x }$ denotes the sample mean weight in grams, and $s ^ { 2 }$ denotes an unbiased estimate of the relevant population variance.

	Sample size	$\bar { x }$	$s ^ { 2 }$
Previous season's crop $( P )$	80	1.24	0.00356
New crop $( N )$	100	1.36	0.00340

Calculate an estimate of $\operatorname { Var } \left( \bar { X } _ { N } - \bar { X } _ { P } \right)$.
Calculate a $95 \%$ confidence interval for the difference in population mean weights.
Give a reason why it is unnecessary to use a $t$-distribution in calculating the confidence interval.

OCR S3 2008 January Q2

2 The times taken for customers' phone complaints to be handled were monitored regularly by a company. During a particular week a researcher checked a random sample of 20 complaints and the times, $x$ minutes, taken to handle the complaints are summarised by $\Sigma x = 337.5$. Handling times may be assumed to have a normal distribution with mean $\mu$ minutes and standard deviation 3.8 minutes.

Calculate a $98 \%$ confidence interval for $\mu$. During the same week two other researchers each calculated a $98 \%$ confidence interval for $\mu$ based on independent samples.
Calculate the probability that at least one of the three intervals does not contain $\mu$.
State two ways in which the calculation in part (i) would differ if the standard deviation were unknown.

OCR S3 2008 January Q3

3 A transport authority wished to compare the performance of two rail companies, Western and Northern. They noted that the number of 'on-time' arrivals for a random sample of 80 Western trains over a particular route was 71 . The corresponding number for a random sample of 90 Northern trains over a similar route was 73 .

Test, at the $5 \%$ significance level, whether the population proportion of on-time Western trains exceeds the population proportion of on-time Northern trains.
Ranjit wishes to test whether the population proportion of on-time Western trains exceeds the population proportion of on-time Northern trains by more than 0.01 . What variance estimate should she use?

OCR S3 2008 January Q4

4 Eezimix flour is sold in small bags of weight $S$ grams, where $S \sim \mathrm {~N} \left( 502.1,0.31 ^ { 2 } \right)$. It is also sold in large bags of weight $L$ grams, where $L \sim \mathrm {~N} \left( 1004.9,0.58 ^ { 2 } \right)$.

Find the probability that a randomly chosen large bag weighs at least 1 gram more than two randomly chosen small bags.
Find the probability that a randomly chosen large bag weighs less than twice the weight of a randomly chosen small bag.

OCR S3 2008 January Q5

5 Of two brands of lawnmower, $A$ and $B$, brand $A$ was claimed to take less time, on average, than brand $B$ to mow similar stretches of lawn. In order to test this claim, 9 randomly selected gardeners were each given the task of mowing two regions of lawn, one with each brand of mower. All the regions had the same size and shape and had grass of the same height. The times taken, in seconds, are given in the table.

Gardener	1	2	3	4	5	6	7	8	9
Brand $A$	412	386	389	401	396	394	397	411	391
Brand $B$	422	394	385	408	394	399	397	410	397

Test the claim using a paired-sample $t$-test at the $5 \%$ significance level. State a distributional assumption required for the test to be valid.
Give a reason why a paired-sample $t$-test should be used, rather than a 2 -sample $t$-test, in this case.

OCR S3 2008 January Q6

6 The Research and Development department of a paint manufacturer has produced paint of three different shades of grey, $G _ { 1 } , G _ { 2 }$ and $G _ { 3 }$. In order to find the reaction of the public to these shades, each of a random sample of 120 people was asked to state which shade they preferred. The results, classified by gender, are shown in Table 1. \begin{table}[h]

Shade
\cline { 2 - 5 }		$G _ { 1 }$	$G _ { 2 }$	$G _ { 3 }$
\cline { 2 - 5 } Gender	Male	11	24	23
Female	18	13	31
\cline { 2 - 5 }
\cline { 2 - 5 }

\captionsetup{labelformat=empty} \caption{Table 1}

\end{table} Table 2 shows the corresponding expected values, correct to 2 decimal places, for a test of independence. \begin{table}[h]

Shade
\cline { 2 - 5 }		$G _ { 1 }$	$G _ { 2 }$	$G _ { 3 }$
\cline { 2 - 5 } Gender	Male	14.02	17.88	26.10
	Female	14.98	19.12	27.90
\cline { 2 - 5 }
\cline { 2 - 5 }

\captionsetup{labelformat=empty} \caption{Table 2}

\end{table}

Show how the value 17.88 for Male, $G _ { 2 }$ was obtained.
Test, at the $5 \%$ significance level, whether gender and preferred shade are independent.
Determine the smallest significance level obtained from tables or calculator for which there is evidence that not all shades are equally preferred by people in general, irrespective of gender.

OCR S3 2008 January Q7

7 The continuous random variable $T$ has probability density function given by $$f ( t ) = \begin{cases} 4 t ^ { 3 } & 0 < t \leqslant 1
0 & \text { otherwise } \end{cases}$$

Obtain the cumulative distribution function of $T$.
Find the cumulative distribution function of $H$, where $H = \frac { 1 } { T ^ { 4 } }$, and hence show that the probability density function of $H$ is given by $\mathrm { g } ( h ) = \frac { 1 } { h ^ { 2 } }$ over an interval to be stated.
Find $\mathrm { E } \left( 1 + 2 H ^ { - 1 } \right)$.

OCR S3 2011 January Q1

1 A random variable has a normal distribution with unknown mean $\mu$ and known standard deviation 0.19 . In order to estimate $\mu$ a random sample of five observations of the random variable was taken. The values were as follows. $$\begin{array} { l l l l l } 5.44 & 4.93 & 5.12 & 5.36 & 5.40 \end{array}$$ Using these five values, calculate,

an estimate of $\mu$,
a 95\% confidence interval for $\mu$.

OCR S3 2011 January Q2

2 In a Year 8 internal examination in a large school the Geography marks, $G$, and Mathematics marks, $M$, had means and standard deviations as follows.

	Mean	Standard deviation
$G$	36.42	6.87
$M$	42.65	10.25

Assuming that $G$ and $M$ have independent normal distributions, find the probability that a randomly chosen Geography candidate scores at least 10 marks more than a randomly chosen Mathematics candidate. Do not use a continuity correction.

OCR S3 2011 January Q3

3 The continuous random variable $T$ has probability density function given by $$\mathrm { f } ( t ) = \begin{cases} 0 & t < 0 ,
\frac { a } { \mathrm { e } } & 0 \leqslant t < 2 ,
a \mathrm { e } ^ { - \frac { 1 } { 2 } t } & t \geqslant 2 , \end{cases}$$ where $a$ is a positive constant.

Show that $a = \frac { 1 } { 4 } \mathrm { e }$.
Find the upper quartile of $T$.

OCR S3 2011 January Q4

4 A study in 1981 investigated the effect of water fluoridation on children's dental health. In a town with fluoridation, 61 out of a random sample of 107 children showed signs of increased tooth decay after six months. In a town without fluoridation the corresponding number was 106 out of a random sample of 143 children. The population proportions of children with increased tooth decay are denoted by $p _ { 1 }$ and $p _ { 2 }$ for the towns with fluoridation and without fluoridation respectively. A test is carried out of the null hypothesis $p _ { 1 } = p _ { 2 }$ against the alternative hypothesis $p _ { 1 } < p _ { 2 }$. Find the smallest significance level at which the null hypothesis is rejected.

OCR S3 2011 January Q5

5 An experiment with hybrid corn resulted in yellow kernels and purple kernels. Of a random sample of 90 kernels, 18 were yellow and 72 were purple.

Calculate an approximate $90 \%$ confidence interval for the proportion of yellow kernels produced in all such experiments.
Deduce an approximate $90 \%$ confidence interval for the proportion of purple kernels produced in all such experiments.
Explain what is meant by a $90 \%$ confidence interval for a population proportion.
Mendel's theory of inheritance predicts that $25 \%$ of all such kernels will be yellow. State, giving a reason, whether or not your calculations support the theory.

	Sample size	\(\bar { x }\)	\(s ^ { 2 }\)
Previous season's crop \(( P )\)	80	1.24	0.00356
New crop \(( N )\)	100	1.36	0.00340

Gardener	1	2	3	4	5	6	7	8	9
Brand \(A\)	412	386	389	401	396	394	397	411	391
Brand \(B\)	422	394	385	408	394	399	397	410	397

Shade
\cline { 2 - 5 }		\(G _ { 1 }\)	\(G _ { 2 }\)	\(G _ { 3 }\)
\cline { 2 - 5 } Gender	Male	11	24	23
Female	18	13	31
\cline { 2 - 5 }
\cline { 2 - 5 }

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