Questions — OCR MEI S1 (292 questions)

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OCR MEI S1 2010 January Q7
7 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]{2f39c509-5429-4193-9526-15fb45b18a38-4_837_1651_466_246}
  1. Calculate the number of pears which are between 90 and 100 mm long.
  2. Calculate an estimate of the mean length of the pears. Explain why your answer is only an estimate.
  3. Calculate an estimate of the standard deviation.
  4. Use your answers to parts (ii) and (iii) to investigate whether there are any outliers.
  5. Name the type of skewness of the distribution.
  6. Illustrate the data using a cumulative frequency diagram.
OCR MEI S1 2010 January Q8
8 An environmental health officer monitors the air pollution level in a city street. Each day the level of pollution is classified as low, medium or high. The probabilities of each level of pollution on a randomly chosen day are as given in the table.
Pollution levelLowMediumHigh
Probability0.50.350.15
  1. Three days are chosen at random. Find the probability that the pollution level is
    (A) low on all 3 days,
    (B) low on at least one day,
    (C) low on one day, medium on another day, and high on the other day.
  2. Ten days are chosen at random. Find the probability that
    (A) there are no days when the pollution level is high,
    (B) there is exactly one day when the pollution level is high. The environmental health officer believes that pollution levels will be low more frequently in a different street. On 20 randomly selected days she monitors the pollution level in this street and finds that it is low on 15 occasions.
  3. Carry out a test at the \(5 \%\) level to determine if there is evidence to suggest that she is correct. Use hypotheses \(\mathrm { H } _ { 0 } : p = 0.5 , \mathrm { H } _ { 1 } : p > 0.5\), where \(p\) represents the probability that the pollution level in this street is low. Explain why \(\mathrm { H } _ { 1 }\) has this form.
OCR MEI S1 2011 January Q1
1 The stem and leaf diagram shows the weights, rounded to the nearest 10 grams, of 25 female iguanas.
839
935666899
100223469
112478
12345
132
Key: 11 | 2 represents 1120 grams
  1. Find the mode and the median of the data.
  2. Identify the type of skewness of the distribution.
OCR MEI S1 2011 January Q2
2 The table shows all the possible products of the scores on two fair four-sided dice.
\multirow{2}{*}{}Score on second die
1234
\multirow{4}{*}{Score on first die}11234
22468
336912
4481216
  1. Find the probability that the product of the two scores is less than 10 .
  2. Show that the events 'the score on the first die is even' and 'the product of the scores on the two dice is less than 10' are not independent.
OCR MEI S1 2011 January Q3
3 There are 13 men and 10 women in a running club. A committee of 3 men and 3 women is to be selected.
  1. In how many different ways can the three men be selected?
  2. In how many different ways can the whole committee be selected?
  3. A random sample of 6 people is selected from the running club. Find the probability that this sample consists of 3 men and 3 women.
OCR MEI S1 2011 January Q5
5 Andy can walk to work, travel by bike or travel by bus. The tree diagram shows the probabilities of any day being dry or wet and the corresponding probabilities for each of Andy's methods of travel.
\includegraphics[max width=\textwidth, alt={}, center]{7df6dcad-790d-4d0e-b1a5-3371103997d9-3_732_1141_792_500} A day is selected at random. Find the probability that
  1. the weather is wet and Andy travels by bus,
  2. Andy walks or travels by bike,
  3. the weather is dry given that Andy walks or travels by bike.
OCR MEI S1 2011 January Q6
6 A survey is being carried out into the carbon footprint of individual citizens. As part of the survey, 100 citizens are asked whether they have attempted to reduce their carbon footprint by any of the following methods.
  • Reducing car use
  • Insulating their homes
  • Avoiding air travel
The numbers of citizens who have used each of these methods are shown in the Venn diagram.
\includegraphics[max width=\textwidth, alt={}, center]{7df6dcad-790d-4d0e-b1a5-3371103997d9-4_703_1087_712_529} One of the citizens is selected at random.
  1. Find the probability that this citizen
    (A) has avoided air travel,
    (B) has used at least two of the three methods.
  2. Given that the citizen has avoided air travel, find the probability that this citizen has reduced car use. Three of the citizens are selected at random.
  3. Find the probability that none of them have avoided air travel. Section B (36 marks)
OCR MEI S1 2011 January Q7
7 The incomes of a sample of 918 households on an island are given in the table below.
Income
\(( x\) thousand pounds \()\)
\(0 \leqslant x \leqslant 20\)\(20 < x \leqslant 40\)\(40 < x \leqslant 60\)\(60 < x \leqslant 100\)\(100 < x \leqslant 200\)
Frequency23836514212845
  1. Draw a histogram to illustrate the data.
  2. Calculate an estimate of the mean income.
  3. Calculate an estimate of the standard deviation of the incomes.
  4. 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.
  5. 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 2011 January Q8
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
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
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
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
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
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
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
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
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
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
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 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
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
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
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
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
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.