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OCR MEI S1 2011 January Q8
17 marks Standard +0.3
8 Mark is playing solitaire on his computer. The probability that he wins a game is 0.2 , independently of all other games that he plays.
  1. Find the expected number of wins in 12 games.
  2. Find the probability that
    (A) he wins exactly 2 out of the next 12 games that he plays,
    (B) he wins at least 2 out of the next 12 games that he plays.
  3. Mark's friend Ali also plays solitaire. Ali claims that he is better at winning games than Mark. In a random sample of 20 games played by Ali, he wins 7 of them. Write down suitable hypotheses for a test at the \(5 \%\) level to investigate whether Ali is correct. Give a reason for your choice of alternative hypothesis. Carry out the test.
OCR MEI S1 2012 January Q1
7 marks Easy -1.8
1 The mean daily maximum temperatures at a research station over a 12-month period, measured to the nearest degree Celsius, are given below.
JanFebMarAprMayJunJulAugSepOctNovDec
8152529313134363426158
  1. Construct a sorted stem and leaf diagram to represent these data, taking stem values of \(0,10 , \ldots\).
  2. Write down the median of these data.
  3. The mean of these data is 24.3 . Would the mean or the median be a better measure of central tendency of the data? Briefly explain your answer.
OCR MEI S1 2012 January Q2
7 marks Moderate -0.8
2 The hourly wages, \(\pounds x\), of a random sample of 60 employees working for a company are summarised as follows. $$n = 60 \quad \sum x = 759.00 \quad \sum x ^ { 2 } = 11736.59$$
  1. Calculate the mean and standard deviation of \(x\).
  2. The workers are offered a wage increase of \(2 \%\). Use your answers to part (i) to deduce the new mean and standard deviation of the hourly wages after this increase.
  3. As an alternative the workers are offered a wage increase of 25 p per hour. Write down the new mean and standard deviation of the hourly wages after this 25p increase.
OCR MEI S1 2012 January Q3
8 marks Standard +0.3
3 Jimmy and Alan are playing a tennis match against each other. The winner of the match is the first player to win three sets. Jimmy won the first set and Alan won the second set. For each of the remaining sets, the probability that Jimmy wins a set is
  • 0.7 if he won the previous set,
  • 0.4 if Alan won the previous set.
It is not possible to draw a set.
  1. Draw a probability tree diagram to illustrate the possible outcomes for each of the remaining sets.
  2. Find the probability that Alan wins the match.
  3. Find the probability that the match ends after exactly four sets have been played.
OCR MEI S1 2012 January Q4
6 marks Moderate -0.8
4 In a food survey, a large number of people are asked whether they like tomato soup, mushroom soup, both or neither. One of these people is selected at random.
  • \(T\) is the event that this person likes tomato soup.
  • \(M\) is the event that this person likes mushroom soup.
You are given that \(\mathrm { P } ( T ) = 0.55 , \mathrm { P } ( M ) = 0.33\) and \(\mathrm { P } ( T \mid M ) = 0.80\).
  1. Use this information to show that the events \(T\) and \(M\) are not independent.
  2. Find \(\mathrm { P } ( T \cap M )\).
  3. Draw a Venn diagram showing the events \(T\) and \(M\), and fill in the probability corresponding to each of the four regions of your diagram.
OCR MEI S1 2012 January Q5
8 marks Standard +0.3
5 A couple plan to have at least one child of each sex, after which they will have no more children. However, if they have four children of one sex, they will have no more children. You should assume that each child is equally likely to be of either sex, and that the sexes of the children are independent. The random variable \(X\) represents the total number of girls the couple have.
  1. Show that \(\mathrm { P } ( X = 1 ) = \frac { 11 } { 16 }\). The table shows the probability distribution of \(X\).
    \(r\)01234
    \(\mathrm { P } ( X = r )\)\(\frac { 1 } { 16 }\)\(\frac { 11 } { 16 }\)\(\frac { 1 } { 8 }\)\(\frac { 1 } { 16 }\)\(\frac { 1 } { 16 }\)
  2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
OCR MEI S1 2012 January Q6
17 marks Moderate -0.3
6 It is known that \(25 \%\) of students in a particular city are smokers. A random sample of 20 of the students is selected.
  1. (A) Find the probability that there are exactly 4 smokers in the sample.
    (B) Find the probability that there are at least 3 but no more than 6 smokers in the sample.
    (C) Write down the expected number of smokers in the sample. A new health education programme is introduced. This programme aims to reduce the percentage of students in this city who are smokers. After the programme has been running for a year, it is decided to carry out a hypothesis test to assess the effectiveness of the programme. A random sample of 20 students is selected.
  2. (A) Write down suitable null and alternative hypotheses for the test.
    (B) Explain why the alternative hypothesis has the form that it does.
  3. Find the critical region for the test at the \(5 \%\) level, showing all of your calculations.
  4. In fact there are 3 smokers in the sample. Complete the test, stating your conclusion clearly.
OCR MEI S1 2012 January Q7
19 marks Moderate -0.3
7 The birth weights of 200 lambs from crossbred sheep are illustrated by the cumulative frequency diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{4b259fe3-73ef-419f-85ad-1a3b1e6ea56e-4_917_1146_367_447}
  1. Estimate the percentage of lambs with birth weight over 6 kg .
  2. Estimate the median and interquartile range of the data.
  3. 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]{4b259fe3-73ef-419f-85ad-1a3b1e6ea56e-4_328_1616_1749_260}
  4. Use appropriate measures to compare briefly the central tendencies and variations of the weights of the two types of lamb.
  5. 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.
  6. 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 2013 January Q2
8 marks Moderate -0.8
2 The probability distribution of the random variable \(X\) is given by the formula $$\mathrm { P } ( X = r ) = k \left( r ^ { 2 } - 1 \right) \text { for } r = 2,3,4,5 .$$
  1. Show the probability distribution in a table, and find the value of \(k\).
  2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
OCR MEI S1 2013 January Q3
8 marks Moderate -0.8
3 Each weekday Alan drives to work. On his journey, he goes over a level crossing. Sometimes he has to wait at the level crossing for a train to pass.
  • \(W\) is the event that Alan has to wait at the level crossing.
  • \(L\) is the event that Alan is late for work.
You are given that \(\mathrm { P } ( L \mid W ) = 0.4 , \mathrm { P } ( W ) = 0.07\) and \(\mathrm { P } ( L \cup W ) = 0.08\).
  1. Calculate \(\mathrm { P } ( L \cap W )\).
  2. Draw a Venn diagram, showing the events \(L\) and \(W\). Fill in the probability corresponding to each of the four regions of your diagram.
  3. Determine whether the events \(L\) and \(W\) are independent, explaining your method clearly.
OCR MEI S1 2013 January Q4
7 marks Moderate -0.8
4 At a dog show, three out of eleven dogs are to be selected for a national competition.
  1. Find the number of possible selections.
  2. Five of the eleven dogs are terriers. Assuming that the dogs are selected at random, find the probability that at least two of the three dogs selected for the national competition are terriers.
OCR MEI S1 2013 January Q5
5 marks Easy -1.2
5 Malik is playing a game in which he has to throw a 6 on a fair six-sided die to start the game. Find the probability that
  1. Malik throws a 6 for the first time on his third attempt,
  2. Malik needs at most ten attempts to throw a 6.
OCR MEI S1 2013 January Q6
18 marks Standard +0.3
6 The heights \(x \mathrm {~cm}\) of 100 boys in Year 7 at a school are summarised in the table below.
Height\(125 \leqslant x \leqslant 140\)\(140 < x \leqslant 145\)\(145 < x \leqslant 150\)\(150 < x \leqslant 160\)\(160 < x \leqslant 170\)
Frequency252924184
  1. Estimate the number of boys who have heights of at least 155 cm .
  2. Calculate an estimate of the median height of the 100 boys.
  3. 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]{76283206-687f-45d6-9204-952d60843cf1-3_865_1349_1297_349}
  4. How many more girls than boys had heights exceeding 160 cm ?
  5. Calculate an estimate of the mean height of the 100 girls.
OCR MEI S1 2013 January Q7
18 marks Standard +0.3
7 A coffee shop provides free internet access for its customers. It is known that the probability that a randomly selected customer is accessing the internet is 0.35 , independently of all other customers.
  1. 10 customers are selected at random.
    (A) Find the probability that exactly 5 of them are accessing the internet.
    (B) Find the probability that at least 5 of them are accessing the internet.
    (C) Find the expected number of these customers who are accessing the internet. Another coffee shop also provides free internet access. It is suspected that the probability that a randomly selected customer at this coffee shop is accessing the internet may be different from 0.35 . A random sample of 20 customers at this coffee shop is selected. Of these, 10 are accessing the internet.
  2. Carry out a hypothesis test at the \(5 \%\) significance level to investigate whether the probability for this coffee shop is different from 0.35 . Give a reason for your choice of alternative hypothesis.
  3. To get a more reliable result, a much larger random sample of 200 customers is selected over a period of time, and another hypothesis test is carried out. You are given that 90 of the 200 customers were accessing the internet. You are also given that, if \(X\) has the binomial distribution with parameters \(n = 200\) and \(p = 0.35\), then \(\mathrm { P } ( X \geqslant 90 ) = 0.0022\). Using the same hypotheses and significance level which you used in part (ii), complete this test.
OCR MEI S1 2009 June Q1
5 marks Easy -1.8
1 In a traffic survey, the number of people in each car passing the survey point is recorded. The results are given in the following frequency table.
Number of people1234
Frequency5031165
  1. Write down the median and mode of these data.
  2. Draw a vertical line diagram for these data.
  3. State the type of skewness of the distribution.
OCR MEI S1 2009 June Q2
5 marks Moderate -0.8
2 There are 14 girls and 11 boys in a class. A quiz team of 5 students is to be chosen from the class.
  1. How many different teams are possible?
  2. If the team must include 3 girls and 2 boys, find how many different teams are possible.
OCR MEI S1 2009 June Q3
8 marks Moderate -0.8
3 Dwayne is a car salesman. The numbers of cars, \(x\), sold by Dwayne each month during the year 2008 are summarised by $$n = 12 , \quad \Sigma x = 126 , \quad \Sigma x ^ { 2 } = 1582 .$$
  1. Calculate the mean and standard deviation of the monthly numbers of cars sold.
  2. Dwayne earns \(\pounds 500\) each month plus \(\pounds 100\) commission for each car sold. Show that the mean of Dwayne's monthly earnings is \(\pounds 1550\). Find the standard deviation of Dwayne's monthly earnings.
  3. Marlene is a car saleswoman and is paid in the same way as Dwayne. During 2008 her monthly earnings have mean \(\pounds 1625\) and standard deviation \(\pounds 280\). Briefly compare the monthly numbers of cars sold by Marlene and Dwayne during 2008.
OCR MEI S1 2009 June Q4
4 marks Easy -1.2
4 The table shows the probability distribution of the random variable \(X\).
\(r\)10203040
\(\mathrm { P } ( X = r )\)0.20.30.30.2
  1. Explain why \(\mathrm { E } ( X ) = 25\).
  2. Calculate \(\operatorname { Var } ( X )\).
OCR MEI S1 2009 June Q5
8 marks Moderate -0.8
5 The frequency table below shows the distance travelled by 1200 visitors to a particular UK tourist destination in August 2008.
Distance \(( d\) miles \()\)\(0 \leqslant d < 50\)\(50 \leqslant d < 100\)\(100 \leqslant d < 200\)\(200 \leqslant d < 400\)
Frequency360400307133
  1. Draw a histogram on graph paper to illustrate these data.
  2. Calculate an estimate of the median distance.
OCR MEI S1 2009 June Q6
6 marks Moderate -0.8
6 Whitefly, blight and mosaic virus are three problems which can affect tomato plants. 100 tomato plants are examined for these problems. The numbers of plants with each type of problem are shown in the Venn diagram. 47 of the plants have none of the problems.
\includegraphics[max width=\textwidth, alt={}, center]{3a5d18f5-b1fe-4513-ae4e-f37c20f172b5-3_668_812_998_664}
  1. One of the 100 plants is selected at random. Find the probability that this plant has
    (A) at most one of the problems,
    (B) exactly two of the problems.
  2. Three of the 100 plants are selected at random. Find the probability that all of them have at least one of the problems. Section B (36 marks)
OCR MEI S1 2009 June Q7
18 marks Easy -1.2
7 Laura frequently flies to business meetings and often finds that her flights are delayed. A flight may be delayed due to technical problems, weather problems or congestion problems, with probabilities \(0.2,0.15\) and 0.1 respectively. The tree diagram shows this information.
\includegraphics[max width=\textwidth, alt={}, center]{3a5d18f5-b1fe-4513-ae4e-f37c20f172b5-4_608_1651_532_248}
  1. Write down the values of the probabilities \(a , b\) and \(c\) shown in the tree diagram. One of Laura's flights is selected at random.
  2. Find the probability that Laura's flight is not delayed and hence write down the probability that it is delayed.
  3. Find the probability that Laura's flight is delayed due to just one of the three problems.
  4. Given that Laura's flight is delayed, find the probability that the delay is due to just one of the three problems.
  5. Given that Laura's flight has no technical problems, find the probability that it is delayed.
  6. In a particular year, Laura has 110 flights. Find the expected number of flights that are delayed.
OCR MEI S1 2009 June Q8
18 marks Standard +0.3
8 The Department of Health 'eat five a day' advice recommends that people should eat at least five portions of fruit and vegetables per day. In a particular school, \(20 \%\) of pupils eat at least five a day.
  1. 15 children are selected at random.
    (A) Find the probability that exactly 3 of them eat at least five a day.
    (B) Find the probability that at least 3 of them eat at least five a day.
    (C) Find the expected number who eat at least five a day. A programme is introduced to encourage children to eat more portions of fruit and vegetables per day. At the end of this programme, the diets of a random sample of 15 children are analysed. A hypothesis test is carried out to examine whether the proportion of children in the school who eat at least five a day has increased.
  2. (A) Write down suitable null and alternative hypotheses for the test.
    (B) Give a reason for your choice of the alternative hypothesis.
  3. Find the critical region for the test at the \(10 \%\) significance level, showing all of your calculations. Hence complete the test, given that 7 of the 15 children eat at least five a day.
OCR MEI S1 2010 June Q1
8 marks Moderate -0.8
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]{091d6f43-ad01-4849-9f3c-3e58349aa169-2_204_1422_484_363}
  1. Name the type of skewness of the distribution.
  2. 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.
  3. Suggest possible reasons why this may be the case.
OCR MEI S1 2010 June Q2
7 marks Moderate -0.8
2 The probability distribution of the random variable \(X\) is given by the formula $$\mathrm { P } ( X = r ) = k r ( 5 - r ) \text { for } r = 1,2,3,4 .$$
  1. Show that \(k = 0.05\).
  2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
OCR MEI S1 2010 June Q3
7 marks Moderate -0.8
3 The lifetimes in hours of 90 components are summarised in the table.
Lifetime \(( x\) hours \()\)\(0 < x \leqslant 20\)\(20 < x \leqslant 30\)\(30 < x \leqslant 50\)\(50 < x \leqslant 65\)\(65 < x \leqslant 100\)
Frequency2413142118
  1. Draw a histogram to illustrate these data.
  2. In which class interval does the median lie? Justify your answer.