Questions - OCR MEI S1 - A-Level Maths

Calculate the probability that Douglas hits 20 twice.
Douglas aims fifty sets of 3 darts at the number 20. Find the expected number of sets for which Douglas hits 20 twice.
Douglas aims four sets of 3 darts at the number 20. Calculate the probability that he hits 20 twice for two sets out of the four.

OCR MEI S1 Q4

4 A geologist splits rocks to look for fossils. On average 10\% of the rocks selected from a particular area do in fact contain fossils. The geologist selects a random sample of 20 rocks from this area.

Find the probability that
(A) exactly one of the rocks contains fossils,
(B) at least one of the rocks contains fossils.
A random sample of $n$ rocks is selected from this area. The geologist wants to have a probability of 0.8 or greater of finding fossils in at least one of the $n$ rocks. Find the least possible value of $n$.
The geologist explores a new area in which it is claimed that less than $10 \%$ of rocks contain fossils. In order to investigate the claim, a random sample of 30 rocks from this area is selected, and the number which contain fossils is recorded. A hypothesis test is carried out at the 5\% level.
(A) Write down suitable hypotheses for the test.
(B) Show that the critical region consists only of the value 0 .
(C) In fact, 2 of the 30 rocks in the sample contain fossils. Complete the test, stating your conclusions clearly.

OCR MEI S1 Q1

1 Over a long period of time, $20 \%$ of all bowls made by a particular manufacturer are imperfect and cannot be sold.

Find the probability that fewer than 4 bowls from a random sample of 10 made by the manufacturer are imperfect. The manufacturer introduces a new process for producing bowls. To test whether there has been an improvement, each of a random sample of 20 bowls made by the new process is examined. From this sample, 2 bowls are found to be imperfect.
Show that this does not provide evidence, at the $5 \%$ level of significance, of a reduction in the proportion of imperfect bowls. You should show your hypotheses and calculations clearly.

OCR MEI S1 Q2

2 A game requires 15 identical ordinary dice to be thrown in each turn.
Assuming the dice to be fair, find the following probabilities for any given turn.

No sixes are thrown.
Exactly four sixes are thrown.
More than three sixes are thrown. David and Esme are two players who are not convinced that the dice are fair. David believes that the dice are biased against sixes, while Esme believes the dice to be biased in favour of sixes. In his next turn, David throws no sixes. In her next turn, Esme throws 5 sixes.
Writing down your hypotheses carefully in each case, decide whether
(A) David's turn provides sufficient evidence at the $10 \%$ level that the dice are biased against sixes.
(B) Esme's turn provides sufficient evidence at the $10 \%$ level that the dice are biased in favour of sixes.
Comment on your conclusions from part (iv).

OCR MEI S1 Q3

3 At a doctor's surgery, records show that $20 \%$ of patients who make an appointment fail to turn up. During afternoon surgery the doctor has time to see 16 patients. There are 16 appointments to see the doctor one afternoon.

Find the probability that all 16 patients turn up.
Find the probability that more than 3 patients do not turn up. To improve efficiency, the doctor decides to make more than 16 appointments for afternoon surgery, although there will still only be enough time to see 16 patients. There must be a probability of at least 0.9 that the doctor will have enough time to see all the patients who turn up.
The doctor makes 17 appointments for afternoon surgery. Find the probability that at least one patient does not turn up. Hence show that making 17 appointments is satisfactory.
Now find the greatest number of appointments the doctor can make for afternoon surgery and still have a probability of at least 0.9 of having time to see all patients who turn up. A computerised appointment system is introduced at the surgery. It is decided to test, at the $5 \%$ level, whether the proportion of patients failing to turn up for their appointments has changed. There are always 20 appointments to see the doctor at morning surgery. On a randomly chosen morning, 1 patient does not turn up.
Write down suitable hypotheses and carry out the test.

OCR MEI S1 Q1

1 At a tourist information office the numbers of people seeking information each hour over the course of a 12-hour day are shown below. $$\begin{array} { l l l l l l l l l l l l } 6 & 25 & 38 & 39 & 31 & 18 & 35 & 31 & 33 & 15 & 21 & 28 \end{array}$$

Construct a sorted stem and leaf diagram to represent these data.
State the type of skewness suggested by your stem and leaf diagram.
For these data find the median, the mean and the mode. Comment on the usefulness of the mode in this case.

OCR MEI S1 Q2

2 The box and whisker plot below summarises the weights in grams of the 20 chocolates in a box.
\includegraphics[max width=\textwidth, alt={}, center]{452a52c9-b1fa-4b98-a85d-a34ba0f84a9d-1_290_1186_1099_452}

Find the interquartile range of the data and hence determine whether there are any outliers at either end of the distribution. Ben buys a box of these chocolates each weekend. The chocolates all look the same on the outside, but 7 of them have orange centres, 6 have cherry centres, 4 have coffee centres and 3 have lemon centres. One weekend, each of Ben's 3 children eats one of the chocolates, chosen at random.
Calculate the probabilities of the following events. A: all 3 chocolates have orange centres
$B$ : all 3 chocolates have the same centres
Find $\mathrm { P } ( A \mid B )$ and $\mathrm { P } ( B \mid A )$. The following weekend, Ben buys an identical box of chocolates and again each of his 3 children eats one of the chocolates, chosen at random.
Find the probability that, on both weekends, the 3 chocolates that they eat all have orange centres.
Ben likes all of the chocolates except those with cherry centres. On another weekend he is the first of his family to eat some of the chocolates. Find the probability that he has to select more than 2 chocolates before he finds one that he likes.

OCR MEI S1 Q3

3 The ages, $x$ years, of the senior members of a running club are summarised in the table below.

Age $( x )$	$20 \leqslant x < 30$	$30 \leqslant x < 40$	$40 \leqslant x < 50$	$50 \leqslant x < 60$	$60 \leqslant x < 70$	$70 \leqslant x < 80$	$80 \leqslant x < 90$
Frequency	10	30	42	23	9	5	1

Draw a cumulative frequency diagram to illustrate the data.
Use your diagram to estimate the median and interquartile range of the data.

OCR MEI S1 Q4

4 The weights, $w$ grams, of a random sample of 60 carrots of variety A are summarised in the table below.

Weight	$30 \leqslant w < 50$	$50 \leqslant w < 60$	$60 \leqslant w < 70$	$70 \leqslant w < 80$	$80 \leqslant w < 90$
Frequency	11	10	18	14	7

Draw a histogram to illustrate these data.
Calculate estimates of the mean and standard deviation of $w$.
Use your answers to part (ii) to investigate whether there are any outliers. The weights, $x$ grams, of a random sample of 50 carrots of variety B are summarised as follows. $$n = 50 \quad \sum x = 3624.5 \quad \sum x ^ { 2 } = 265416$$
Calculate the mean and standard deviation of $x$.
Compare the central tendency and variation of the weights of varieties A and B .

OCR MEI S1 Q1

1 The heights $x \mathrm {~cm}$ of 100 boys in Year 7 at a school are summarised in the table below.

Height	$125 \leqslant x \leqslant 140$	$140 < x \leqslant 145$	$145 < x \leqslant 150$	$150 < x \leqslant 160$	$160 < x \leqslant 170$
Frequency	25	29	24	18	4

Estimate the number of boys who have heights of at least 155 cm .
Calculate an estimate of the median height of the 100 boys.
Draw a histogram to illustrate the data. The histogram below shows the heights of 100 girls in Year 7 at the same school.
\includegraphics[max width=\textwidth, alt={}, center]{ab4d5ab1-e3b7-495f-9142-d37df7e712de-1_868_1361_1015_381}
How many more girls than boys had heights exceeding 160 cm ?
Calculate an estimate of the mean height of the 100 girls.

OCR MEI S1 Q2

2 The engine sizes $x \mathrm {~cm} ^ { 3 }$ of a sample of 80 cars are summarised in the table below.

Engine size $x$	$500 \leqslant x \leqslant 1000$	$1000 < x \leqslant 1500$	$1500 < x \leqslant 2000$	$2000 < x \leqslant 3000$	$3000 < x \leqslant 5000$
Frequency	7	22	26	18	7

Draw a histogram to illustrate the distribution.
A student claims that the midrange is $2750 \mathrm {~cm} ^ { 3 }$. Discuss briefly whether he is likely to be correct.
Calculate estimates of the mean and standard deviation of the engine sizes. Explain why your answers are only estimates.
Hence investigate whether there are any outliers in the sample.
A vehicle duty of $\pounds 1000$ is proposed for all new cars with engine size greater than $2000 \mathrm {~cm} ^ { 3 }$. Assuming that this sample of cars is representative of all new cars in Britain and that there are 2.5 million new cars registered in Britain each year, calculate an estimate of the total amount of money that this vehicle duty would raise in one year.
Why in practice might your estimate in part (v) turn out to be too high?

OCR MEI S1 Q3

3 The birth weights of 200 lambs from crossbred sheep are illustrated by the cumulative frequency diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{ab4d5ab1-e3b7-495f-9142-d37df7e712de-3_919_1144_430_476}

Estimate the percentage of lambs with birth weight over 6 kg .
Estimate the median and interquartile range of the data.
Use your answers to part (ii) to show that there are very few, if any, outliers. Comment briefly on whether any outliers should be disregarded in analysing these data. The box and whisker plot shows the birth weights of 100 lambs from Welsh Mountain sheep.
\includegraphics[max width=\textwidth, alt={}, center]{ab4d5ab1-e3b7-495f-9142-d37df7e712de-3_321_1610_1818_293}
Use appropriate measures to compare briefly the central tendencies and variations of the weights of the two types of lamb.
The weight of the largest Welsh Mountain lamb was originally recorded as 6.5 kg , but then corrected. If this error had not been corrected, how would this have affected your answers to part (iv)? Briefly explain your answer.
One lamb of each type is selected at random. Estimate the probability that the birth weight of both lambs is at least 3.9 kg .

OCR MEI S1 Q1

1 In the Paris-Roubaix cycling race, there are a number of sections of cobbled road. The lengths of these sections, measured in metres, are illustrated in the histogram.
\includegraphics[max width=\textwidth, alt={}, center]{3aabac69-ead8-40e4-b06f-5e812bb02906-1_897_1398_494_410}

Find the number of sections which are between 1000 and 2000 metres in length.
Name the type of skewness suggested by the histogram.
State the minimum and maximum possible values of the midrange.

OCR MEI S1 Q2

2 Two fair six-sided dice are thrown. The random variable $X$ denotes the difference between the scores on the two dice. The table shows the probability distribution of $X$.

$r$	0	1	2	3	4	5
$\mathrm { P } ( X = r )$	$\frac { 1 } { 6 }$	$\frac { 5 } { 18 }$	$\frac { 2 } { 9 }$	$\frac { 1 } { 6 }$	$\frac { 1 } { 9 }$	$\frac { 1 } { 18 }$

Draw a vertical line chart to illustrate the probability distribution.
Use a probability argument to show that
(A) $\mathrm { P } ( X = 1 ) = \frac { 5 } { 18 }$,
(B) $\mathrm { P } ( X = 0 ) = \frac { 1 } { 6 }$.
Find the mean value of $X$.

OCR MEI S1 Q3

3 The heating quality of the coal in a sample of 50 sacks is measured in suitable units. The data are summarised below.

Heating quality $( x )$	$9.1 \leqslant x \leqslant 9.3$	$9.3 < x \leqslant 9.5$	$9.5 < x \leqslant 9.7$	$9.7 < x \leqslant 9.9$	$9.9 < x \leqslant 10.1$
Frequency	5	7	15	16	7

Draw a cumulative frequency diagram to illustrate these data.
Use the diagram to estimate the median and interquartile range of the data.
Show that there are no outliers in the sample.
Three of these 50 sacks are selected at random. Find the probability that
(A) in all three, the heating quality $x$ is more than 9.5 ,
$( B )$ in at least two, the heating quality $x$ is more than 9.5.

OCR MEI S1 Q4

4 The incomes of a sample of 918 households on an island are given in the table below.

Income

$( x$ thousand pounds $)$

$0 \leqslant x \leqslant 20$

$20 < x \leqslant 40$

$40 < x \leqslant 60$

$60 < x \leqslant 100$

$100 < x \leqslant 200$

Draw a histogram to illustrate the data.
Calculate an estimate of the mean income.
Calculate an estimate of the standard deviation of the incomes.
Use your answers to parts (ii) and (iii) to show there are almost certainly some outliers in the sample. Explain whether or not it would be appropriate to exclude the outliers from the calculation of the mean and the standard deviation.
The incomes were converted into another currency using the formula $y = 1.15 x$. Calculate estimates of the mean and variance of the incomes in the new currency.

OCR MEI S1 Q1

1 A business analyst collects data about the distribution of hourly wages, in $\pounds$, of shop-floor workers at a factory. These data are illustrated in the box and whisker plot.
\includegraphics[max width=\textwidth, alt={}, center]{56f1bd5c-4b45-4e36-a324-e7e0edbb5bdd-1_206_1420_505_397}

Name the type of skewness of the distribution.
Find the interquartile range and hence show that there are no outliers at the lower end of the distribution, but there is at least one outlier at the upper end.
Suggest possible reasons why this may be the case.

OCR MEI S1 Q2

2 The lifetimes in hours of 90 components are summarised in the table.

Lifetime $( x$ hours $)$	$0 < x \leqslant 20$	$20 < x \leqslant 30$	$30 < x \leqslant 50$	$50 < x \leqslant 65$	$65 < x \leqslant 100$
Frequency	24	13	14	21	18

Draw a histogram to illustrate these data.
In which class interval does the median lie? Justify your answer.

OCR MEI S1 Q3

3 A pear grower collects a random sample of 120 pears from his orchard. The histogram below shows the lengths, in mm, of these pears.
\includegraphics[max width=\textwidth, alt={}, center]{56f1bd5c-4b45-4e36-a324-e7e0edbb5bdd-2_825_1634_467_295}

Calculate the number of pears which are between 90 and 100 mm long.
Calculate an estimate of the mean length of the pears. Explain why your answer is only an estimate.
Calculate an estimate of the standard deviation.
Use your answers to parts (ii) and (iii) to investigate whether there are any outliers.
Name the type of skewness of the distribution.
Illustrate the data using a cumulative frequency diagram.

OCR MEI S1 Q4

4 The frequency table below shows the distance travelled by 1200 visitors to a particular UK tourist destination in August 2008.

Distance $( d$ miles $)$	$0 \leqslant d < 50$	$50 \leqslant d < 100$	$100 \leqslant d < 200$	$200 \leqslant d < 400$
Frequency	360	400	307	133

Draw a histogram on graph paper to illustrate these data.
Calculate an estimate of the median distance.

OCR MEI S1 Q1

1 The temperature of a supermarket fridge is regularly checked to ensure that it is working correctly. Over a period of three months the temperature (measured in degrees Celsius) is checked 600 times. These temperatures are displayed in the cumulative frequency diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{c7cb0f6b-7b6b-4c52-8287-7efc6bd70247-1_1052_1647_549_289}

Use the diagram to estimate the median and interquartile range of the data.
Use your answers to part (i) to show that there are very few, if any, outliers in the sample.
Suppose that an outlier is identified in these data. Discuss whether it should be excluded from any further analysis.
Copy and complete the frequency table below for these data.
Temperature
$( t$ degrees Celsius $)$
$3.0 \leqslant t \leqslant 3.4$ $3.4 < t \leqslant 3.8$ $3.8 < t \leqslant 4.2$ $4.2 < t \leqslant 4.6$ $4.6 < t \leqslant 5.0$
Frequency 243 157
Use your table to calculate an estimate of the mean.
The standard deviation of the temperatures in degrees Celsius is 0.379 . The temperatures are converted from degrees Celsius into degrees Fahrenheit using the formula $F = 1.8 C + 32$. Hence estimate the mean and find the standard deviation of the temperatures in degrees Fahrenheit.

OCR MEI S1 Q2

2 The histogram shows the amount of money, in pounds, spent by the customers at a supermarket on a particular day.
\includegraphics[max width=\textwidth, alt={}, center]{c7cb0f6b-7b6b-4c52-8287-7efc6bd70247-2_985_1473_470_379}

Express the data in the form of a grouped frequency table.
Use your table to estimate the total amount of money spent by customers on that day.

OCR MEI S1 Q3

3 A GCSE geography student is investigating a claim that global warming is causing summers in Britain to have more rainfall. He collects rainfall data from a local weather station for 2001 and 2006. The vertical line chart shows the number of days per week on which some rainfall was recorded during the 22 weeks of summer 2001.
\includegraphics[max width=\textwidth, alt={}, center]{c7cb0f6b-7b6b-4c52-8287-7efc6bd70247-3_804_1557_547_337}

Show that the median of the data is 4 , and find the interquartile range.
For summer 2006 the median is 3 and the interquartile range is also 3. The student concludes that the data demonstrate that global warming is causing summer rainfall to decrease rather than increase. Is this a valid conclusion from the data? Give two brief reasons to justify your answer.

OCR MEI S1 Q4

4 The numbers of absentees per day from Mrs Smith's reception class over a period of 50 days are summarised below.

Number of absentees	0	1	2	3	4	5	6	$> 6$
Frequency	8	15	11	8	3	4	1	0

Illustrate these data by means of a vertical line chart.
Calculate the mean and root mean square deviation of these data.
There are 30 children in Mrs Smith's class altogether. Find the mean and root mean square deviation of the number of children who are present during the 50 days.

OCR MEI S1 Q5

5 The times taken for 480 university students to travel from their accommodation to lectures are summarised below.

Time $( t$ minutes $)$	$0 \leqslant t < 5$	$5 \leqslant t < 10$	$10 \leqslant t < 20$	$20 \leqslant t < 30$	$30 \leqslant t < 40$	$40 \leqslant t < 60$
Frequency	34	153	188	73	27	5

Illustrate these data by means of a histogram.
Identify the type of skewness of the distribution.

Age \(( x )\)	\(20 \leqslant x < 30\)	\(30 \leqslant x < 40\)	\(40 \leqslant x < 50\)	\(50 \leqslant x < 60\)	\(60 \leqslant x < 70\)	\(70 \leqslant x < 80\)	\(80 \leqslant x < 90\)
Frequency	10	30	42	23	9	5	1

Engine size \(x\)	\(500 \leqslant x \leqslant 1000\)	\(1000 < x \leqslant 1500\)	\(1500 < x \leqslant 2000\)	\(2000 < x \leqslant 3000\)	\(3000 < x \leqslant 5000\)
Frequency	7	22	26	18	7

\(r\)	0	1	2	3	4	5
\(\mathrm { P } ( X = r )\)	\(\frac { 1 } { 6 }\)	\(\frac { 5 } { 18 }\)	\(\frac { 2 } { 9 }\)	\(\frac { 1 } { 6 }\)	\(\frac { 1 } { 9 }\)	\(\frac { 1 } { 18 }\)

Heating quality \(( x )\)	\(9.1 \leqslant x \leqslant 9.3\)	\(9.3 < x \leqslant 9.5\)	\(9.5 < x \leqslant 9.7\)	\(9.7 < x \leqslant 9.9\)	\(9.9 < x \leqslant 10.1\)
Frequency	5	7	15	16	7

Time \(( t\) minutes \()\)	\(0 \leqslant t < 5\)	\(5 \leqslant t < 10\)	\(10 \leqslant t < 20\)	\(20 \leqslant t < 30\)	\(30 \leqslant t < 40\)	\(40 \leqslant t < 60\)
Frequency	34	153	188	73	27	5

Weight	\(30 \leqslant w < 50\)	\(50 \leqslant w < 60\)	\(60 \leqslant w < 70\)	\(70 \leqslant w < 80\)	\(80 \leqslant w < 90\)
Frequency	11	10	18	14	7

Height	\(125 \leqslant x \leqslant 140\)	\(140 < x \leqslant 145\)	\(145 < x \leqslant 150\)	\(150 < x \leqslant 160\)	\(160 < x \leqslant 170\)
Frequency	25	29	24	18	4

Lifetime \(( x\) hours \()\)	\(0 < x \leqslant 20\)	\(20 < x \leqslant 30\)	\(30 < x \leqslant 50\)	\(50 < x \leqslant 65\)	\(65 < x \leqslant 100\)
Frequency	24	13	14	21	18

Distance \(( d\) miles \()\)	\(0 \leqslant d < 50\)	\(50 \leqslant d < 100\)	\(100 \leqslant d < 200\)	\(200 \leqslant d < 400\)
Frequency	360	400	307	133

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