Questions - OCR - A-Level Maths

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.

OCR S3 2011 January Q6

6 The continuous random variable $X$ has (cumulative) distribution function given by $$\mathrm { F } ( x ) = \begin{cases} 0 & x < \frac { 1 } { 2 }
\frac { 2 x - 1 } { x + 1 } & \frac { 1 } { 2 } \leqslant x \leqslant 2
1 & x > 2 . \end{cases}$$

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.

	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 }

\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

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