Questions - OCR MEI S2 - A-Level Maths

Find $\mathrm { P } ( X > 25000 )$.
10 vans in the fleet are selected at random. Find the probability that the tyres on exactly 7 of them last for more than 25000 miles.
The tyres of $99 \%$ of vans last for more than $k$ miles. Find the value of $k$. A tyre supplier claims that a different type of tyre will have a greater mean lifetime. A random sample of 15 vans is fitted with these tyres. For each van, the number of miles driven before the tyres are replaced is recorded. A hypothesis test is carried out to investigate the claim. You may assume that these lifetimes are also Normally distributed with standard deviation 4000.
Write down suitable null and alternative hypotheses for the test.
For the 15 vans, it is found that the mean lifetime of the tyres is 28630 miles. Carry out the test at the $5 \%$ level.

OCR MEI S2 2008 June Q4

4 A student is investigating whether there is any association between the species of shellfish that occur on a rocky shore and where they are located. A random sample of 160 shellfish is selected and the numbers of shellfish in each category are summarised in the table below.

		Location
\cline { 3 - 5 } \multicolumn{2}{\|c\|}{}	Exposed	Sheltered	Pool
\multirow{3}{*}{Species}	Limpet	24	32	16
\cline { 2 - 5 }	Mussel	24	11	3
\cline { 2 - 5 }	Other	5	22	23

Write down null and alternative hypotheses for a test to examine whether there is any association between species and location. The contributions to the test statistic for the usual $\chi ^ { 2 }$ test are shown in the table below.
Contribution Location
\cline { 3 - 5 } Exposed Sheltered Pool
\multirow{3}{*}{Species} Limpet 0.0009 0.2585 0.4450
\cline { 2 - 5 } Mussel 10.3472 1.2756 4.8773
\cline { 2 - 5 } Other 8.0719 0.1402 7.4298
The sum of these contributions is 32.85 .
Calculate the expected frequency for mussels in pools. Verify the corresponding contribution 4.8773 to the test statistic.
Carry out the test at the $5 \%$ level of significance, stating your conclusion clearly.
For each species, comment briefly on how its distribution compares with what would be expected if there were no association.
If 3 of the 160 shellfish are selected at random, one from each of the 3 types of location, find the probability that all 3 of them are limpets.

OCR MEI S2 2009 January Q1

1 A researcher is investigating whether there is a relationship between the population size of cities and the average walking speed of pedestrians in the city centres. Data for the population size, $x$ thousands, and the average walking speed of pedestrians, $y \mathrm {~m} \mathrm {~s} ^ { - 1 }$, of eight randomly selected cities are given in the table below.

$x$	18	43	52	94	98	206	784	1530
$y$	1.15	0.97	1.26	1.35	1.28	1.42	1.32	1.64

Calculate the value of Spearman's rank correlation coefficient.
Carry out a hypothesis test at the $5 \%$ significance level to determine whether there is any association between population size and average walking speed. In another investigation, the researcher selects a random sample of six adult males of particular ages and measures their maximum walking speeds. The data are shown in the table below, where $t$ years is the age of the adult and $w \mathrm {~m} \mathrm {~s} ^ { - 1 }$ is the maximum walking speed. Also shown are summary statistics and a scatter diagram on which the regression line of $w$ on $t$ is drawn.
$t$ 20 30 40 50 60 70
$w$ 2.49 2.41 2.38 2.14 1.97 2.03
$$n = 6 \quad \Sigma t = 270 \quad \Sigma w = 13.42 \quad \Sigma t ^ { 2 } = 13900 \quad \Sigma w ^ { 2 } = 30.254 \quad \Sigma t w = 584.6$$ \includegraphics[max width=\textwidth, alt={}, center]{77b97142-afb6-41d6-8fec-e982b7a7501b-2_728_1091_1379_529}
Calculate the equation of the regression line of $w$ on $t$.
(A) Use this equation to calculate an estimate of maximum walking speed of an 80 -year-old male.
(B) Explain why it might not be appropriate to use the equation to calculate an estimate of maximum walking speed of a 10 -year-old male.

OCR MEI S2 2009 January Q2

2 Clover stems usually have three leaves. Occasionally a clover stem has four leaves. This is considered by some to be lucky and is known as a four-leaf clover. On average 1 in 10000 clover stems is a four-leaf clover. You may assume that four-leaf clovers occur randomly and independently. A random sample of 5000 clover stems is selected.

State the exact distribution of $X$, the number of four-leaf clovers in the sample.
Explain why $X$ may be approximated by a Poisson distribution. Write down the mean of this Poisson distribution.
Use this Poisson distribution to find the probability that the sample contains at least one four-leaf clover.
Find the probability that in 20 samples, each of 5000 clover stems, there are exactly 9 samples which contain at least one four-leaf clover.
Find the expected number of these 20 samples which contain at least one four-leaf clover. The table shows the numbers of four-leaf clovers in these 20 samples.
Number of four-leaf clovers 0 1 2 $> 2$
Number of samples 11 7 2 0
Calculate the mean and variance of the data in the table.
Briefly comment on whether your answers to parts (v) and (vi) support the use of the Poisson approximating distribution in part (iii).

OCR MEI S2 2009 January Q3

3 The number of minutes, $X$, for which a particular model of laptop computer will run on battery power is Normally distributed with mean 115.3 and standard deviation 21.9.

(A) Find $\mathrm { P } ( X < 120 )$.
(B) Find $\mathrm { P } ( 100 < X < 110 )$.
(C) Find the value of $k$ for which $\mathrm { P } ( X > k ) = 0.9$. The number of minutes, $Y$, for which a different model of laptop computer will run on battery power is known to be Normally distributed with mean $\mu$ and standard deviation $\sigma$.
Given that $\mathrm { P } ( Y < 180 ) = 0.7$ and $\mathrm { P } ( Y < 140 ) = 0.15$, find the values of $\mu$ and $\sigma$.
Find values of $a$ and $b$ for which $\mathrm { P } ( a < Y < b ) = 0.95$.

OCR MEI S2 2009 January Q4

4 A gardening research organisation is running a trial to examine the growth and the size of flowers of various plants.

In the trial, seeds of three types of plant are sown. The growth of each plant is classified as good, average or poor. The results are shown in the table.
\multirow{2}{*}{} Growth \multirow[t]{2}{*}{Row totals}
Good Average Poor
\multirow{3}{*}{Type of plant} Coriander 12 28 15 55
Aster 7 18 23 48
Fennel 14 22 11 47
Column totals 33 68 49 150
Carry out a test at the $5 \%$ significance level to examine whether there is any association between growth and type of plant. State carefully your null and alternative hypotheses. Include a table of the contributions of each cell to the test statistic.
It is known that the diameter of marigold flowers is Normally distributed with mean 47 mm and standard deviation 8.5 mm . A certain fertiliser is expected to cause flowers to have a larger mean diameter, but without affecting the standard deviation. A large number of marigolds are grown using this fertiliser. The diameters of a random sample of 50 of the flowers are measured and the mean diameter is found to be 49.2 mm . Carry out a hypothesis test at the $1 \%$ significance level to check whether flowers grown with this fertiliser appear to be larger on average. Use hypotheses $\mathrm { H } _ { 0 } : \mu = 47 , \mathrm { H } _ { 1 } : \mu > 47$, where $\mu \mathrm { mm }$ represents the mean diameter of all marigold flowers grown with this fertiliser.

OCR MEI S2 2010 January Q1

1 A pilot records the take-off distance for his light aircraft on runways at various altitudes. The data are shown in the table below, where $a$ metres is the altitude and $t$ metres is the take-off distance. Also shown are summary statistics for these data.

$a$	0	300	600	900	1200	1500	1800
$t$	635	704	776	836	923	1008	1105

$$n = 7 \quad \Sigma a = 6300 \quad \Sigma t = 5987 \quad \Sigma a ^ { 2 } = 8190000 \quad \Sigma t ^ { 2 } = 5288931 \quad \Sigma a t = 6037800$$

Draw a scatter diagram to illustrate these data.
State which of the two variables $a$ and $t$ is the independent variable and which is the dependent variable. Briefly explain your answer.
Calculate the equation of the regression line of $t$ on $a$.
Use the equation of the regression line to calculate estimates of the take-off distance for altitudes
(A) 800 metres,
(B) 2500 metres. Comment on the reliability of each of these estimates.
Calculate the value of the residual for the data point where $a = 1200$ and $t = 923$, and comment on its sign.

OCR MEI S2 2010 January Q2

2 On average 2\% of a particular model of laptop computer are faulty. Faults occur independently and randomly.

Find the probability that exactly 1 of a batch of 10 laptops is faulty.
State the conditions under which the use of a Poisson distribution is appropriate as an approximation to a binomial distribution.
A school buys a batch of 150 of these laptops. Use a Poisson approximating distribution to find the probability that
(A) there are no faulty laptops in the batch,
(B) there are more than the expected number of faulty laptops in the batch.
A large company buys a batch of 2000 of these laptops for its staff.
(A) State the exact distribution of the number of faulty laptops in this batch.
(B) Use a suitable approximating distribution to find the probability that there are at most 50 faulty laptops in this batch.

OCR MEI S2 2010 January Q3

3 In an English language test for 12-year-old children, the raw scores, $X$, are Normally distributed with mean 45.3 and standard deviation 11.5.

Find
(A) $\mathrm { P } ( X < 50 )$,
(B) $\mathrm { P } ( 45.3 < X < 50 )$.
Find the least raw score which would be obtained by the highest scoring $10 \%$ of children.
The raw score is then scaled so that the scaled score is Normally distributed with mean 100 and standard deviation 15. This scaled score is then rounded to the nearest integer. Find the probability that a randomly selected child gets a rounded score of exactly 111 .
In a Mathematics test for 12-year-old children, the raw scores, $Y$, are Normally distributed with mean $\mu$ and standard deviation $\sigma$. Given that $\mathrm { P } ( Y < 15 ) = 0.3$ and $\mathrm { P } ( Y < 22 ) = 0.8$, find the values of $\mu$ and $\sigma$.

OCR MEI S2 2010 January Q4

4 A council provides waste paper recycling services for local businesses. Some businesses use the standard service for recycling paper, others use a special service for dealing with confidential documents, and others use both. Businesses are classified as small or large. A survey of a random sample of 285 businesses gives the following data for size of business and recycling service.

Recycling Service

\cline { 3 - 5 } \multicolumn{2}{|c|}{}

Write down null and alternative hypotheses for a test to examine whether there is any association between size of business and recycling service used. The contributions to the test statistic for the usual $\chi ^ { 2 }$ test are shown in the table below.
Recycling Service
\cline { 3 - 5 } \multicolumn{2}{|c|}{} Standard Special Both
\multirow{2}{*}{
Size of
business
} Small 0.1023 0.2607 0.0186
Large 0.0597 0.1520 0.0108
The sum of these contributions is 0.6041 .
Calculate the expected frequency for large businesses using the special service. Verify the corresponding contribution 0.1520 to the test statistic.
Carry out the test at the $5 \%$ level of significance, stating your conclusion clearly. The council is also investigating the weight of rubbish in domestic dustbins. In 2008 the average weight of rubbish in bins was 32.8 kg . The council has now started a recycling initiative and wishes to determine whether there has been a reduction in the weight of rubbish in bins. A random sample of 50 domestic dustbins is selected and it is found that the mean weight of rubbish per bin is now 30.9 kg , and the standard deviation is 3.4 kg .
Carry out a test at the $5 \%$ level to investigate whether the mean weight of rubbish has been reduced in comparison with 2008 . State carefully your null and alternative hypotheses. {www.ocr.org.uk}) after the live examination series.
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OCR MEI S2 2011 January Q1

1 The scatter diagram below shows the birth rates $x$, and death rates $y$, measured in standard units, in a random sample of 14 countries in a particular year. Summary statistics for the data are as follows. $$\Sigma x = 139.8 \quad \Sigma y = 140.4 \quad \Sigma x ^ { 2 } = 1411.66 \quad \Sigma y ^ { 2 } = 1417.88 \quad \Sigma x y = 1398.56 \quad n = 14$$ \includegraphics[max width=\textwidth, alt={}, center]{cd1a8f39-dd3c-44c9-90b0-6a919361d593-2_643_1047_488_550}

Calculate the sample product moment correlation coefficient.
Carry out a hypothesis test at the $5 \%$ significance level to determine whether there is any correlation between birth rates and death rates.
State the distributional assumption which is necessary for this test to be valid. Explain briefly in the light of the scatter diagram why it appears that the assumption may be valid.
The values of $x$ and $y$ for another country in that year are 14.4 and 7.8 respectively. If these values are included, the value of the sample product moment correlation coefficient is - 0.5694 . Explain why this one observation causes such a large change to the value of the sample product moment correlation coefficient. Discuss whether this brings the validity of the test into question.

OCR MEI S2 2011 January Q2

2 A student is investigating the numbers of sultanas in a particular brand of biscuit. The data in the table show the numbers of sultanas in a random sample of 50 of these biscuits.

Number of sultanas	0	1	2	3	4	5	$> 5$
Frequency	8	15	12	9	4	2	0

Show that the sample mean is 1.84 and calculate the sample variance.
Explain why these results support a suggestion that a Poisson distribution may be a suitable model for the distribution of the numbers of sultanas in this brand of biscuit. For the remainder of the question you should assume that a Poisson distribution with mean 1.84 is a suitable model for the distribution of the numbers of sultanas in these biscuits.
Find the probability of
(A) no sultanas in a biscuit,
(B) at least two sultanas in a biscuit.
Show that the probability that there are at least 10 sultanas in total in a packet containing 5 biscuits is 0.4389 .
Six packets each containing 5 biscuits are selected at random. Find the probability that exactly 2 of the six packets contain at least 10 sultanas.
Sixty packets each containing 5 biscuits are selected at random. Use a suitable approximating distribution to find the probability that more than half of the sixty packets contain at least 10 sultanas.

OCR MEI S2 2011 January Q3

3 The random variable $X$ represents the reaction times, in milliseconds, of men in a driving simulator. $X$ is Normally distributed with mean 355 and standard deviation 52.

Find
(A) $\mathrm { P } ( X < 325 )$,
(B) $\mathrm { P } ( 300 < X < 400 )$.
Find the value of $k$ for which $\mathrm { P } ( X < k ) = 0.2$. It is thought that women may have a different mean reaction time from men. In order to test this, a random sample of 25 women is selected. The mean reaction time of these women in the driving simulator is 344 milliseconds. You may assume that women's reaction times are also Normally distributed with standard deviation 52 milliseconds. A hypothesis test is carried out to investigate whether women have a different mean reaction time from men.
Carry out the test at the $5 \%$ significance level.

OCR MEI S2 2011 January Q4

4 A researcher is investigating the sizes of pebbles at various locations in a river. Three sites in the river are chosen and each pebble sampled at each site is classified as large, medium or small. The results are as follows.

\cline { 3 - 6 } \multicolumn{2}{|c|}{}

Carry out a test at the $5 \%$ significance level to examine whether there is any association between pebble size and site. Your working should include a table of the contributions of each cell to the test statistic.
By referring to each site, comment briefly on how the size of the pebbles compares with what would be expected if there were no association. You should support your answers by referring to your table of contributions.

OCR MEI S2 2012 January Q1

1 Nine long-distance runners are starting an exercise programme to improve their strength. During the first session, each of them has to do a 100 metre run and to do as many push-ups as possible in one minute. The times taken for the run, together with the number of push-ups each runner achieves, are shown in the table.

Runner	A	B	C	D	E	F	G	H	I
100 metre time (seconds)	13.2	11.6	10.9	12.3	14.7	13.1	11.7	13.6	12.4
Push-ups achieved	32	42	22	36	41	27	37	38	33

Draw a scatter diagram to illustrate the data.
Calculate the value of Spearman's rank correlation coefficient.
Carry out a hypothesis test at the $5 \%$ significance level to examine whether there is any association between time taken for the run and number of push-ups achieved.
Under what circumstances is it appropriate to carry out a hypothesis test based on the product moment correlation coefficient? State, with a reason, which test is more appropriate for these data.

OCR MEI S2 2012 January Q2

2 The number of printing errors per page in a book is modelled by a Poisson distribution with a mean of 0.85 .

State conditions for a Poisson distribution to be a suitable model for the number of printing errors per page.
A page is chosen at random. Find the probability of
(A) exactly 1 error on this page,
(B) at least 2 errors on this page. 10 pages are chosen at random.
Find the probability of exactly 10 errors in these 10 pages.
Find the least integer $k$ such that the probability of there being $k$ or more errors in these 10 pages is less than $1 \%$. 30 pages are chosen at random.
Use a suitable approximating distribution to find the probability of no more than 30 errors in these 30 pages.

OCR MEI S2 2012 January Q3

3 The lifetime of a particular type of light bulb is $X$ hours, where $X$ is Normally distributed with mean 1100 and variance 2000.

Find $\mathrm { P } ( 1100 < X < 1200 )$.
Use a suitable approximating distribution to find the probability that, in a random sample of 100 of these light bulbs, no more than 40 have a lifetime between 1100 and 1200 hours.
A factory has a large number of these light bulbs installed. As soon as $1 \%$ of the bulbs have come to the end of their lifetimes, it is company policy to replace all of the bulbs. After how many hours should the bulbs need to be replaced?
The bulbs are to be replaced by low-energy bulbs. The lifetime of these bulbs is Normally distributed and the mean is claimed by the manufacturer to be 7000 hours. The standard deviation is known to be 100 hours. A random sample of 25 low-energy bulbs is selected. Their mean lifetime is found to be 6972 hours. Carry out a 2 -tail test at the $10 \%$ level to investigate the claim.
[0pt] [Question 4 is printed overleaf.]

OCR MEI S2 2012 January Q4

4 Birds are observed at feeding stations in three different places - woodland, farm and garden. The numbers of finches, thrushes and tits observed at each site are summarised in the table. The birds observed are regarded as a random sample from the population of birds of these species that use these feeding stations.

\multirow{2}{*}{Observed Frequency}		Place
		Farm	Garden	Woodland	Totals
\multirow{4}{*}{Species}	Thrushes	11	74	7	92
	Tits	70	26	88	184
	Finches	17	2	10	29
	Totals	98	102	105	305

The expected frequencies under the null hypothesis for the usual $\chi ^ { 2 }$ test are shown in the table below.

\multirow{2}{*}{Expected Frequency}		Place
		Farm	Garden	Woodland
\multirow{3}{*}{Species}	Thrushes	29.5607	30.7672	31.6721
	Tits	59.1213	61.5344	63.3443
	Finches	9.3180	9.6984	9.9836

Verify that the entry 9.3180 is correct. The corresponding contributions to the test statistic are shown in the table below.
\multirow{2}{*}{Contribution} Place
Farm Garden Woodland
\multirow{3}{*}{Species} Thrushes 11.6539 60.7489 19.2192
Tits 2.0017 20.5201 9.5969
Finches 6.3332 6.1108 0.0000
Verify that the entry 6.3332 is correct.
Carry out the test at the $1 \%$ level of significance.
For each place, use the table of contributions to comment briefly on the differences between the observed and expected distributions of species.

OCR MEI S2 2013 January Q1

1 A manufacturer of playground safety tiles is testing a new type of tile. Tiles of various thicknesses are tested to estimate the maximum height at which people would be unlikely to sustain injury if they fell onto a tile. The results of the test are as follows.

Thickness $( t \mathrm {~mm} )$	20	40	60	80	100
Maximum height $( h \mathrm {~m} )$	0.72	1.09	1.62	1.97	2.34

Draw a scatter diagram to illustrate these data.
State which of the two variables is the independent variable, giving a reason for your answer.
Calculate the equation of the regression line of maximum height on thickness.
Use the equation of the regression line to calculate estimates of the maximum height for thicknesses of
(A) 70 mm ,
(B) 120 mm . Comment on the reliability of each of these estimates.
Calculate the value of the residual for the data point at which $t = 40$.
In a further experiment, the manufacturer tests a tile with a thickness of 200 mm and finds that the corresponding maximum height is 2.96 m . What can be said about the relationship between tile thickness and maximum height?

OCR MEI S2 2013 January Q2

2 John is observing butterflies being blown across a fence in a strong wind. He uses the Poisson distribution with mean 2.1 to model the number of butterflies he observes in one minute.

Find the probability that John observes
(A) no butterflies in a minute,
(B) at least 2 butterflies in a minute,
(C) between 5 and 10 butterflies inclusive in a period of 5 minutes.
Use a suitable approximating distribution to find the probability that John observes at least 130 butterflies in a period of 1 hour. In fact some of the butterflies John observes being blown across the fence are being blown in pairs.
Explain why this invalidates one of the assumptions required for a Poisson distribution to be a suitable model. John decides to revise his model for the number of butterflies he observes in one minute. In this new model, the number of pairs of butterflies is modelled by the Poisson distribution with mean 0.2 , and the number of single butterflies is modelled by an independent Poisson distribution with mean 1.7.
Find the probability that John observes no more than 3 butterflies altogether in a period of one minute.

OCR MEI S2 2013 January Q3

3 The amount of data, $X$ megabytes, arriving at an internet server per second during the afternoon is modelled by the Normal distribution with mean 435 and standard deviation 30.

Find
(A) $\mathrm { P } ( X < 450 )$,
(B) $\mathrm { P } ( 400 < X < 450 )$.
Find the probability that, during 5 randomly selected seconds, the amounts of data arriving are all between 400 and 450 megabytes. The amount of data, $Y$ megabytes, arriving at the server during the evening is modelled by the Normal distribution with mean $\mu$ and standard deviation $\sigma$.
Given that $\mathrm { P } ( Y < 350 ) = 0.2$ and $\mathrm { P } ( Y > 390 ) = 0.1$, find the values of $\mu$ and $\sigma$.
Find values of $a$ and $b$ for which $\mathrm { P } ( a < Y < b ) = 0.95$.

OCR MEI S2 2013 January Q4

A random sample of 60 students studying mathematics was selected. Their grades in the Core 1 module are summarised in the table below, classified according to whether they worked less than 5 hours per week or at least 5 hours per week. Test, at the $5 \%$ significance level, whether there is any association between grade and hours worked.
Hours worked
\cline { 3 - 4 } \multicolumn{2}{|c|}{} Less than 5 At least 5
\multirow{2}{*}{Grade} A or B 20 11
\cline { 2 - 4 } C or lower 13 16
At a canning factory, cans are filled with tomato purée. The machine which fills the cans is set so that the volume of tomato purée in a can, measured in millilitres, is Normally distributed with mean 420 and standard deviation 3.5. After the machine is recalibrated, a quality control officer wishes to check whether the mean is still 420 millilitres. A random sample of 10 cans of tomato purée is selected and the volumes, measured in millilitres, are as follows. $$\begin{array} { l l l l l l l l l l } 417.2 & 422.6 & 414.3 & 419.6 & 420.4 & 410.0 & 418.3 & 416.9 & 418.9 & 419.7 \end{array}$$ Carry out a test at the $1 \%$ significance level to investigate whether the mean is still 420 millilitres. You should assume that the volumes are Normally distributed with unchanged standard deviation.

OCR MEI S2 2009 June Q1

1 An investment analyst thinks that there may be correlation between the cost of oil, $x$ dollars per barrel, and the price of a particular share, $y$ pence. The analyst selects 50 days at random and records the values of $x$ and $y$. Summary statistics for these data are shown below, together with a scatter diagram. $$\Sigma x = 2331.3 \quad \Sigma y = 6724.3 \quad \Sigma x ^ { 2 } = 111984 \quad \Sigma y ^ { 2 } = 921361 \quad \Sigma x y = 316345 \quad n = 50$$ \includegraphics[max width=\textwidth, alt={}, center]{ae79cdd9-a57c-490e-a9f3-f47c7c8a1aa6-2_857_905_516_621}

Calculate the sample product moment correlation coefficient.
Carry out a hypothesis test at the $5 \%$ significance level to investigate the analyst's belief. State your hypotheses clearly, defining any symbols which you use.
An assumption that there is a bivariate Normal distribution is required for this test to be valid. State whether it is the sample or the population which is required to have such a distribution. State, with a reason, whether in this case the assumption appears to be justified.
Explain why a 2-tail test is appropriate even though it is clear from the scatter diagram that the sample has a positive correlation coefficient.

OCR MEI S2 2009 June Q2

2 Jess is watching a shower of meteors (shooting stars). During the shower, she sees meteors at an average rate of 1.3 per minute.

State conditions required for a Poisson distribution to be a suitable model for the number of meteors which Jess sees during a randomly selected minute. You may assume that these conditions are satisfied.
Find the probability that, during one minute, Jess sees
(A) exactly one meteor,
(B) at least 4 meteors.
Find the probability that, in a period of 10 minutes, Jess sees exactly 10 meteors.
Use a suitable approximating distribution to find the probability that Jess sees a total of at least 100 meteors during a period of one hour.
Jess watches the shower for $t$ minutes. She wishes to be at least $99 \%$ certain that she will see one or more meteors. Find the smallest possible integer value of $t$.

OCR MEI S2 2009 June Q3

3 Intensity of light is measured in lumens. The random variable $X$ represents the intensity of the light from a standard 100 watt light bulb. $X$ is Normally distributed with mean 1720 and standard deviation 90. You may assume that the intensities for different bulbs are independent.

Show that $\mathrm { P } ( X < 1700 ) = 0.4121$.
These bulbs are sold in packs of 4 . Find the probability that the intensities of exactly 2 of the 4 bulbs in a randomly chosen pack are below 1700 lumens.
Use a suitable approximating distribution to find the probability that the intensities of at least 20 out of 40 randomly selected bulbs are below 1700 lumens. A manufacturer claims that the average intensity of its 25 watt low energy light bulbs is 1720 lumens. A consumer organisation suspects that the true figure may be lower than this. The intensities of a random sample of 20 of these bulbs are measured. A hypothesis test is then carried out to check the claim.
Write down a suitable null hypothesis and explain briefly why the alternative hypothesis should be $\mathrm { H } _ { 1 } : \mu < 1720$. State the meaning of $\mu$.
Given that the standard deviation of the intensity of such bulbs is 90 lumens and that the mean intensity of the sample of 20 bulbs is 1703 lumens, carry out the test at the $5 \%$ significance level.

Contribution		Location
\cline { 3 - 5 }	Exposed	Sheltered	Pool
\multirow{3}{*}{Species}	Limpet	0.0009	0.2585	0.4450
\cline { 2 - 5 }	Mussel	10.3472	1.2756	4.8773
\cline { 2 - 5 }	Other	8.0719	0.1402	7.4298

\(x\)	18	43	52	94	98	206	784	1530
\(y\)	1.15	0.97	1.26	1.35	1.28	1.42	1.32	1.64

\multirow{2}{*}{}		Growth			\multirow[t]{2}{*}{Row totals}
		Good	Average	Poor
\multirow{3}{*}{Type of plant}	Coriander	12	28	15	55
	Aster	7	18	23	48
	Fennel	14	22	11	47
Column totals		33	68	49	150

\multirow{2}{*}{Contribution}		Place
		Farm	Garden	Woodland
\multirow{3}{*}{Species}	Thrushes	11.6539	60.7489	19.2192
	Tits	2.0017	20.5201	9.5969
	Finches	6.3332	6.1108	0.0000

Thickness \(( t \mathrm {~mm} )\)	20	40	60	80	100
Maximum height \(( h \mathrm {~m} )\)	0.72	1.09	1.62	1.97	2.34

		Hours worked
\cline { 3 - 4 } \multicolumn{2}{\|c\|}{}	Less than 5	At least 5
\multirow{2}{*}{Grade}	A or B	20	11
\cline { 2 - 4 }	C or lower	13	16

\(t\)	20	30	40	50	60	70
\(w\)	2.49	2.41	2.38	2.14	1.97	2.03

Questions — OCR MEI S2 (80 questions)

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OCR MEI S2 2008 June Q3

OCR MEI S2 2008 June Q4

OCR MEI S2 2009 January Q1

OCR MEI S2 2009 January Q2

OCR MEI S2 2009 January Q3

OCR MEI S2 2009 January Q4

OCR MEI S2 2010 January Q1

OCR MEI S2 2010 January Q2

OCR MEI S2 2010 January Q3

OCR MEI S2 2010 January Q4

OCR MEI S2 2011 January Q1

OCR MEI S2 2011 January Q2

OCR MEI S2 2011 January Q3

OCR MEI S2 2011 January Q4

OCR MEI S2 2012 January Q1

OCR MEI S2 2012 January Q2

OCR MEI S2 2012 January Q3

OCR MEI S2 2012 January Q4

OCR MEI S2 2013 January Q1

OCR MEI S2 2013 January Q2

OCR MEI S2 2013 January Q3

OCR MEI S2 2013 January Q4

OCR MEI S2 2009 June Q1

OCR MEI S2 2009 June Q2

OCR MEI S2 2009 June Q3