Questions — OCR MEI (4301 questions)

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OCR MEI S1 Q5
5 Yasmin has 5 coins. One of these coins is biased with P (heads) \(= 0.6\). The other 4 coins are fair. She tosses all 5 coins once and records the number of heads, \(X\).
  1. Show that \(\mathrm { P } ( X = 0 ) = 0.025\).
  2. Show that \(\mathrm { P } ( X = 1 ) = 0.1375\). The table shows the probability distribution of \(X\).
    \(r\)012345
    \(\mathrm { P } ( X = r )\)0.0250.13750.30.3250.1750.0375
  3. Draw a vertical line chart to illustrate the probability distribution.
  4. Comment on the skewness of the distribution.
  5. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
  6. Yasmin tosses the 5 coins three times. Find the probability that the total number of heads is 3 .
OCR MEI S1 Q1
1 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 Q2
2 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 Q3
3 The numbers of eggs laid by a sample of 70 female herring gulls are shown in the table.
Number of eggs1234
Frequency1040155
  1. Find the mean and standard deviation of the number of eggs laid per gull.
  2. The sample did not include female herring gulls that laid no eggs. How would the mean and standard deviation change if these gulls were included?
OCR MEI S1 Q4
4 The probability distribution of the random variable \(X\) is given by the formula $$\mathrm { P } ( X = r ) = k r ( r + 1 ) \quad \text { for } r = 1,2,3,4,5 .$$
  1. Show that \(k = \frac { 1 } { 70 }\).
  2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
OCR MEI S1 Q5
5 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 Q6
6 A retail analyst records the numbers of loaves of bread of a particular type bought by a sample of shoppers in a supermarket.
Number of loaves012345
Frequency372311301
  1. Calculate the mean and standard deviation of the numbers of loaves bought per person.
  2. Each loaf costs \(\pounds 1.04\). Calculate the mean and standard deviation of the amount spent on loaves per person.
OCR MEI S1 Q1
1 In her purse, Katharine has two \(\pounds 5\) notes, two \(\pounds 10\) notes and one \(\pounds 20\) note. She decides to select two of these notes at random to donate to a charity. The total value of these two notes is denoted by the random variable \(\pounds X\).
  1. (A) Show that \(\mathrm { P } ( X = 10 ) = 0.1\).
    (B) Show that \(\mathrm { P } ( X = 30 ) = 0.2\). The table shows the probability distribution of \(X\).
    \(r\)1015202530
    \(\mathrm { P } ( X = r )\)0.10.40.10.20.2
  2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
OCR MEI S1 Q2
2 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 Q3
3 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 Q4
4 A zoologist is studying the feeding behaviour of a group of 4 gorillas. The random variable \(X\) represents the number of gorillas that are feeding at a randomly chosen moment. The probability distribution of \(X\) is shown in the table below.
\(r\)01234
\(\mathrm { P } ( X = r )\)\(p\)0.10.050.050.25
  1. Find the value of \(p\).
  2. Find the expectation and variance of \(X\).
  3. The zoologist observes the gorillas on two further occasions. Find the probability that there are at least two gorillas feeding on both occasions.
OCR MEI S1 Q5
5 A pottery manufacturer makes teapots in batches of 50. On average 3\% of teapots are faulty.
  1. Find the probability that in a batch of 50 there is
    (A) exactly one faulty teapot,
    (B) more than one faulty teapot.
  2. The manufacturer produces 240 batches of 50 teapots during one month. Find the expected number of batches which contain exactly one faulty teapot.
OCR MEI S1 Q6
6 In a survey, a sample of 44 fields is selected. Their areas ( \(x\) hectares) are summarised in the grouped frequency table.
Area \(( x )\)\(0 < x \leqslant 3\)\(3 < x \leqslant 5\)\(5 < x \leqslant 7\)\(7 < x \leqslant 10\)\(10 < x \leqslant 20\)
Frequency3813146
  1. Calculate an estimate of the sample mean and the sample standard deviation.
  2. Determine whether there could be any outliers at the upper end of the distribution.
OCR MEI S1 Q7
7 A company is searching for oil reserves. The company has purchased the rights to make test drillings at four sites. It investigates these sites one at a time but, if oil is found, it does not proceed to any further sites. At each site, there is probability 0.2 of finding oil, independently of all other sites. The random variable \(X\) represents the number of sites investigated. The probability distribution of \(X\) is shown below.
\(r\)1234
\(\mathrm { P } ( X = r )\)0.20.160.1280.512
  1. Find the expectation and variance of \(X\).
  2. It costs \(\pounds 45000\) to investigate each site. Find the expected total cost of the investigation.
  3. Draw a suitable diagram to illustrate the distribution of \(X\).
OCR MEI S1 Q2
2 The marks \(x\) scored by a sample of 56 students in an examination are summarised by $$n = 56 , \quad \Sigma x = 3026 , \quad \Sigma x ^ { 2 } = 178890 .$$
  1. Calculate the mean and standard deviation of the marks.
  2. The highest mark scored by any of the 56 students in the examination was 93. Show that this result may be considered to be an outlier.
  3. The formula \(y = 1.2 x - 10\) is used to scale the marks. Find the mean and standard deviation of the scaled marks.
OCR MEI S1 Q3
3 In a phone-in competition run by a local radio station, listeners are given the names of 7 local personalities and are told that 4 of them are in the studio. Competitors phone in and guess which 4 are in the studio.
  1. Show that the probability that a randomly selected competitor guesses all 4 correctly is \(\frac { 1 } { 35 }\). Let \(X\) represent the number of correct guesses made by a randomly selected competitor. The probability distribution of \(X\) is shown in the table.
    \(r\)01234
    \(\mathrm { P } ( X = r )\)0\(\frac { 4 } { 35 }\)\(\frac { 18 } { 35 }\)\(\frac { 12 } { 35 }\)\(\frac { 1 } { 35 }\)
  2. Find the expectation and variance of \(X\).
OCR MEI S1 Q4
4 A fair six-sided die is rolled twice. The random variable \(X\) represents the higher of the two scores. The probability distribution of \(X\) is given by the formula $$\mathrm { P } ( X = r ) = k ( 2 r - 1 ) \text { for } r = 1,2,3,4,5,6$$
  1. Copy and complete the following probability table and hence find the exact value of \(k\), giving your answer as a fraction in its simplest form.
    \(r\)123456
    \(\mathrm { P } ( X = r )\)\(k\)\(11 k\)
  2. Find the mean of \(X\). A fair six-sided die is rolled three times.
  3. Find the probability that the total score is 16 .
OCR MEI S1 Q5
5 The score, \(X\), obtained on a given throw of a biased, four-faced die is given by the probability distribution $$\mathrm { P } ( X = r ) = k r ( 8 - r ) \text { for } r = 1,2,3,4 .$$
  1. Show that \(k = \frac { 1 } { 50 }\).
  2. Calculate \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
OCR MEI S1 Q1
1 Four letters are taken out of their envelopes for signing. Unfortunately they are replaced randomly, one in each envelope. The probability distribution for the number of letters, \(X\), which are now in the correct envelope is given in the following table.
\(r\)01
\(\mathrm { P } ( X = r )\)\(\frac { 3 } { 8 }\)\(\frac { 1 } { 3 }\)\(\frac { 1 } { 4 }\)0\(\frac { 1 } { 24 }\)
  1. Explain why the case \(X = 3\) is impossible.
  2. Explain why \(\mathrm { P } ( X = 4 ) = \frac { 1 } { 24 }\).
  3. Calculate \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
OCR MEI S1 Q2
2 A company sells sugar in bags which are labelled as containing 450 grams.
Although the mean weight of sugar in a bag is more than 450 grams, there is concern that too many bags are underweight. The company can adjust the mean or the standard deviation of the weight of sugar in a bag.
  1. State two adjustments the company could make. The weights, \(x\) grams, of a random sample of 25 bags are now recorded.
  2. Given that \(\sum x = 11409\) and \(\sum x ^ { 2 } = 5206937\), calculate the sample mean and sample standard deviation of these weights.
OCR MEI S1 Q4
4 A sprinter runs many 100 -metre trials, and the time, \(x\) seconds, for each is recorded. A sample of eight of these times is taken, as follows. $$\begin{array} { l l l l l l l l } 10.53 & 10.61 & 10.04 & 10.49 & 10.63 & 10.55 & 10.47 & 10.63 \end{array}$$
  1. Calculate the sample mean, \(\bar { x }\), and sample standard deviation, \(s\), of these times.
  2. Show that the time of 10.04 seconds may be regarded as an outlier.
  3. Discuss briefly whether or not the time of 10.04 seconds should be discarded.
OCR MEI S1 Q5
5 The number, \(X\), of children per family in a certain city is modelled by the probability distribution \(\mathrm { P } ( X = r ) = k ( 6 - r ) ( 1 + r )\) for \(r = 0,1,2,3,4\).
  1. Copy and complete the following table and hence show that the value of \(k\) is \(\frac { 1 } { 50 }\).
    \(r\)01234
    \(\mathrm { P } ( X = r )\)\(6 k\)\(10 k\)
  2. Calculate \(\mathrm { E } ( X )\).
  3. Hence write down the probability that a randomly selected family in this city has more than the mean number of children.
OCR MEI S1 Q6
6 The weights, \(w\) grams, of a random sample of 60 carrots of variety A are summarised in the table below.
Weight\(30 \leqslant w < 50\)\(50 \leqslant w < 60\)\(60 \leqslant w < 70\)\(70 \leqslant w < 80\)\(80 \leqslant w < 90\)
Frequency111018147
  1. Draw a histogram to illustrate these data.
  2. Calculate estimates of the mean and standard deviation of \(w\).
  3. 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. $$n = 50 \quad \sum x = 3624.5 \quad \sum x ^ { 2 } = 265416$$
  4. Calculate the mean and standard deviation of \(x\).
  5. Compare the central tendency and variation of the weights of varieties A and B .
OCR MEI S1 Q7
7 A supermarket chain buys a batch of 10000 scratchcard draw tickets for sale in its stores. 50 of these tickets have a \(\pounds 10\) prize, 20 of them have a \(\pounds 100\) prize, one of them has a \(\pounds 5000\) prize and all of the rest have no prize. This information is summarised in the frequency table below.
Prize money\(\pounds 0\)\(\pounds 10\)\(\pounds 100\)\(\pounds 5000\)
Frequency992950201
  1. Find the mean and standard deviation of the prize money per ticket.
  2. I buy two of these tickets at random. Find the probability that I win either two \(\pounds 10\) prizes or two \(\pounds 100\) prizes.
OCR MEI S1 Q1
1 A survey is being carried out into the sports viewing habits of people in a particular area. As part of the survey, 250 people are asked which of the following sports they have watched on television in the past month.
  • Football
  • Cycling
  • Rugby
The numbers of people who have watched these sports are shown in the Venn diagram.
\includegraphics[max width=\textwidth, alt={}, center]{870b6ef1-60f7-42e3-95f8-0544a2a07b15-1_725_921_728_622} One of the people is selected at random.
  1. Find the probability that this person has in the past month
    (A) watched cycling but not football,
    (B) watched either one or two of the three sports.
  2. Given that this person has watched cycling, find the probability that this person has not watched football.