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OCR S1 2015 June Q2
10 marks Easy -1.3
2 The masses, in grams, of 400 plums were recorded. The masses were then collected into class intervals of width 5 g and a cumulative frequency graph was drawn, as shown below. \includegraphics[max width=\textwidth, alt={}, center]{e5957185-5fe3-45d9-9ab3-c2aab9cbd8dd-3_1045_1401_358_333}
  1. Find the number of plums with masses in the interval 40 g to 45 g .
  2. Find the percentage of plums with masses greater than 70 g .
  3. Give estimates of the highest and lowest masses in the sample, explaining why their exact values cannot be read from the graph.
  4. On the graph paper in the answer book, draw a box-and-whisker plot to illustrate the masses of the plums in the sample.
  5. Comment briefly on the shape of the distribution of masses.
OCR S1 2015 June Q3
6 marks Moderate -0.8
3 An expert tested the quality of the wines produced by a vineyard in 9 particular years. He placed them in the following order, starting with the best. $$\begin{array} { l l l l l l l l l } 1980 & 1983 & 1981 & 1982 & 1984 & 1985 & 1987 & 1986 & 1988 \end{array}$$
  1. Calculate Spearman's rank correlation coefficient, \(r _ { s }\), between the year of production and the quality of these wines. The years should be ranked from the earliest (1) to the latest (9).
  2. State what this value of \(r _ { s }\) shows in this context.
OCR S1 2015 June Q4
9 marks Moderate -0.3
4 The table shows the load a lorry was carrying, \(x\) tonnes, and the fuel economy, \(y \mathrm {~km}\) per litre, for 8 different journeys. You should assume that neither variable is controlled.
Load
\(( x\) tonnes \()\)
5.15.86.57.17.68.49.510.5
Fuel economy
\(( y \mathrm {~km}\) per litre \()\)
6.26.15.95.65.35.45.35.1
$$n = 8 \quad \sum x = 60.5 \quad \sum y = 44.9 \quad \sum x ^ { 2 } = 481.13 \quad \sum y ^ { 2 } = 253.17 \quad \sum x y = 334.65$$
  1. Calculate the equation of the regression line of \(y\) on \(x\).
  2. Estimate the fuel economy for a load of 9.2 tonnes.
  3. An analyst calculated the equation of the regression line of \(x\) on \(y\). Without calculating this equation, state the coordinates of the point where the two regression lines intersect.
  4. Describe briefly the method required to estimate the load when the fuel economy is 5.8 km per litre.
OCR S1 2015 June Q5
10 marks Standard +0.3
5 Each year Jack enters a ballot for a concert ticket. The probability that Jack will win a ticket in any particular year is 0.27 .
  1. Find the probability that the first time Jack wins a ticket is
    (a) on his 8th attempt,
    (b) after his 8th attempt.
  2. Write down an expression for the probability that Jack wins a ticket on exactly 2 of his first 8 attempts, and evaluate this expression.
  3. Find the probability that Jack wins his 3rd ticket on his 9th attempt and his 4th ticket on his 12th attempt.
OCR S1 2015 June Q6
8 marks Moderate -0.8
6
  1. The seven digits \(1,1,2,3,4,5,6\) are arranged in a random order in a line. Find the probability that they form the number 1452163.
  2. Three of the seven digits \(1,1,2,3,4,5,6\) are chosen at random, without regard to order.
    (a) How many possible groups of three digits contain two 1s?
    (b) How many possible groups of three digits contain exactly one 1?
    (c) How many possible groups of three digits can be formed altogether?
OCR S1 2015 June Q7
8 marks Standard +0.3
7 Froox sweets are packed into tubes of 10 sweets, chosen at random. \(25 \%\) of Froox sweets are yellow.
  1. Find the probability that in a randomly selected tube of Froox sweets there are
    (a) exactly 3 yellow sweets,
    (b) at least 3 yellow sweets.
  2. Find the probability that in a box containing 6 tubes of Froox sweets, there is at least 1 tube that contains at least 3 yellow sweets.
OCR S1 2015 June Q8
9 marks Standard +0.3
8 A game is played with a fair, six-sided die which has 4 red faces and 2 blue faces. One turn consists of throwing the die repeatedly until a blue face is on top or until the die has been thrown 4 times.
  1. In the answer book, complete the probability tree diagram for one turn. \includegraphics[max width=\textwidth, alt={}, center]{e5957185-5fe3-45d9-9ab3-c2aab9cbd8dd-5_314_302_1000_884}
  2. Find the probability that in one particular turn the die is thrown 4 times.
  3. Adnan and Beryl each have one turn. Find the probability that Adnan throws the die more times than Beryl.
  4. State one change that needs to be made to the rules so that the number of throws in one turn will have a geometric distribution.
OCR S1 2015 June Q9
6 marks Moderate -0.3
9 The random variable \(X\) has probability distribution given by $$\mathrm { P } ( X = x ) = a + b x \quad \text { for } x = 1,2 \text { and } 3 ,$$ where \(a\) and \(b\) are constants.
  1. Show that \(3 a + 6 b = 1\).
  2. Given that \(\mathrm { E } ( X ) = \frac { 5 } { 3 }\), find \(a\) and \(b\).
OCR S2 2009 January Q1
4 marks Moderate -0.8
1 A newspaper article consists of 800 words. For each word, the probability that it is misprinted is 0.005 , independently of all other words. Use a suitable approximation to find the probability that the total number of misprinted words in the article is no more than 6 . Give a reason to justify your approximation.
OCR S2 2009 January Q2
4 marks Standard +0.3
2 The continuous random variable \(Y\) has the distribution \(\mathrm { N } \left( 23.0,5.0 ^ { 2 } \right)\). The mean of \(n\) observations of \(Y\) is denoted by \(\bar { Y }\). It is given that \(\mathrm { P } ( \bar { Y } > 23.625 ) = 0.0228\). Find the value of \(n\).
OCR S2 2009 January Q3
8 marks Moderate -0.3
3 The number of incidents of radio interference per hour experienced by a certain listener is modelled by a random variable with distribution \(\operatorname { Po } ( 0.42 )\).
  1. Find the probability that the number of incidents of interference in one randomly chosen hour is
    (a) 0 ,
    (b) exactly 1 .
  2. Find the probability that the number of incidents in a randomly chosen 5-hour period is greater than 3.
  3. One hundred hours of listening are monitored and the numbers of 1 -hour periods in which 0,1 , \(2 , \ldots\) incidents of interference are experienced are noted. A bar chart is drawn to represent the results. Without any further calculations, sketch the shape that you would expect for the bar chart. (There is no need to use an exact numerical scale on the frequency axis.)
OCR S2 2009 January Q4
10 marks Moderate -0.3
4 A television company believes that the proportion of adults who watched a certain programme is 0.14 . Out of a random sample of 22 adults, it is found that 2 watched the programme.
  1. Carry out a significance test, at the \(10 \%\) level, to determine, on the basis of this sample, whether the television company is overestimating the proportion of adults who watched the programme.
  2. The sample was selected randomly. State what properties of this method of sampling are needed to justify the use of the distribution used in your test.
OCR S2 2009 January Q5
9 marks Standard +0.3
5 The continuous random variables \(S\) and \(T\) have probability density functions as follows. $$\begin{array} { l l } S : & \mathrm { f } ( x ) = \begin{cases} \frac { 1 } { 4 } & - 2 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases} \\ T : & \mathrm { g } ( x ) = \begin{cases} \frac { 5 } { 64 } x ^ { 4 } & - 2 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases} \end{array}$$
  1. Sketch, on the same axes, the graphs of f and g .
  2. Describe in everyday terms the difference between the distributions of the random variables \(S\) and \(T\). (Answers that comment only on the shapes of the graphs will receive no credit.)
  3. Calculate the variance of \(T\).
OCR S2 2009 January Q6
11 marks Standard +0.3
6 The weight of a plastic box manufactured by a company is \(W\) grams, where \(W \sim \mathrm {~N} ( \mu , 20.25 )\). A significance test of the null hypothesis \(\mathrm { H } _ { 0 } : \mu = 50.0\), against the alternative hypothesis \(\mathrm { H } _ { 1 } : \mu \neq 50.0\), is carried out at the \(5 \%\) significance level, based on a sample of size \(n\).
  1. Given that \(n = 81\),
    (a) find the critical region for the test, in terms of the sample mean \(\bar { W }\),
    (b) find the probability that the test results in a Type II error when \(\mu = 50.2\).
  2. State how the probability of this Type II error would change if \(n\) were greater than 81 .
OCR S2 2009 January Q7
12 marks Standard +0.3
7 A motorist records the time taken, \(T\) minutes, to drive a particular stretch of road on each of 64 occasions. Her results are summarised by $$\Sigma t = 876.8 , \quad \Sigma t ^ { 2 } = 12657.28$$
  1. Test, at the \(5 \%\) significance level, whether the mean time for the motorist to drive the stretch of road is greater than 13.1 minutes.
  2. Explain whether it is necessary to use the Central Limit Theorem in your test.
OCR S2 2009 January Q8
14 marks Moderate -0.3
8 A sales office employs 21 representatives. Each day, for each representative, the probability that he or she achieves a sale is 0.7 , independently of other representatives. The total number of representatives who achieve a sale on any one day is denoted by \(K\).
  1. Using a suitable approximation (which should be justified), find \(\mathrm { P } ( K \geqslant 16 )\).
  2. Using a suitable approximation (which should be justified), find the probability that the mean of 36 observations of \(K\) is less than or equal to 14.0 . 4
OCR S2 2011 January Q1
4 marks Easy -1.2
1 A random sample of nine observations of a random variable is obtained. The results are summarised as $$\Sigma x = 468 , \quad \Sigma x ^ { 2 } = 24820 .$$ Calculate unbiased estimates of the population mean and variance.
OCR S2 2011 January Q2
6 marks Standard +0.3
2 The random variable \(H\) has the distribution \(\mathrm { N } \left( \mu , 5 ^ { 2 } \right)\). The mean of a sample of \(n\) observations of \(H\) is denoted by \(\bar { H }\). It is given that \(\mathrm { P } ( \bar { H } > 53.28 ) = 0.0250\) and \(\mathrm { P } ( \bar { H } < 51.65 ) = 0.0968\), both correct to 4 decimal places. Find the values of \(\mu\) and \(n\).
OCR S2 2011 January Q3
6 marks Moderate -0.8
3 The probability that a randomly chosen PPhone has a faulty casing is 0.0228 . A random sample of 200 PPhones is obtained. Use a suitable approximation to find the probability that the number of PPhones in the sample with a faulty casing is 2 or fewer. Justify your approximation.
OCR S2 2011 January Q4
7 marks Standard +0.3
4 The continuous random variable \(X\) has mean \(\mu\) and standard deviation 45. A significance test is to be carried out of the null hypothesis \(\mathrm { H } _ { 0 } : \mu = 230\) against the alternative hypothesis \(\mathrm { H } _ { 1 } : \mu \neq 230\), at the \(1 \%\) significance level. A random sample of size 50 is obtained, and the sample mean is found to be 213.4.
  1. Carry out the test.
  2. Explain whether it is necessary to use the Central Limit Theorem in your test.
OCR S2 2011 January Q5
7 marks Standard +0.3
5 A temporary job is advertised annually. The number of applicants for the job is a random variable which is known from many years' experience to have a distribution \(\operatorname { Po } ( 12 )\). In 2010 there were 19 applicants for the job. Test, at the 10\% significance level, whether there is evidence of an increase in the mean number of applicants for the job.
OCR S2 2011 January Q6
10 marks Standard +0.3
6 The number of randomly occurring events in a given time interval is denoted by \(R\). In order that \(R\) is well modelled by a Poisson distribution, it is necessary that events occur independently.
  1. Let \(R\) represent the number of customers dining at a restaurant on a randomly chosen weekday lunchtime. Explain what the condition 'events occur independently' means in this context, and give a reason why it would probably not hold in this context. Let \(D\) represent the number of tables booked at the restaurant on a randomly chosen day. Assume that \(D\) can be well modelled by distribution \(\operatorname { Po } ( 7 )\).
  2. Find \(\mathrm { P } ( D < 5 )\).
  3. Use a suitable approximation to find the probability that, in five randomly chosen days, the total number of tables booked is greater than 40 .
OCR S2 2011 January Q7
10 marks Moderate -0.8
7 Two continuous random variables \(S\) and \(T\) have probability density functions \(\mathrm { f } _ { S }\) and \(\mathrm { f } _ { T }\) given respectively by $$\begin{aligned} & f _ { S } ( x ) = \begin{cases} \frac { a } { x ^ { 2 } } & 1 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases} \\ & f _ { T } ( x ) = \begin{cases} b & 1 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases} \end{aligned}$$ where \(a\) and \(b\) are constants.
  1. Sketch on the same axes the graphs of \(y = \mathrm { f } _ { S } ( x )\) and \(y = \mathrm { f } _ { T } ( x )\).
  2. Find the value of \(a\).
  3. Find \(\mathrm { E } ( S )\).
  4. A student gave the following description of the distribution of \(T\) : "The probability that \(T\) occurs is constant". Give an improved description, in everyday terms.
OCR S2 2011 January Q8
11 marks Moderate -0.3
8 A company has 3600 employees, of whom \(22.5 \%\) live more than 30 miles from their workplace. A random sample of 40 employees is obtained.
  1. Use a suitable approximation, which should be justified, to find the probability that more than 5 of the employees in the sample live more than 30 miles from their workplace.
  2. Describe how to use random numbers to select a sample of 40 from a population of 3600 employees.
OCR S2 2011 January Q9
11 marks Standard +0.3
9 A pharmaceutical company is developing a new drug to treat a certain disease. The company will continue to develop the drug if the proportion \(p\) of those who have the disease and show a substantial improvement after treatment is greater than 0.7 . The company carries out a test, at the \(5 \%\) significance level, on a random sample of 14 patients who suffer from the disease.
  1. Find the critical region for the test.
  2. Given that 12 of the 14 patients in the sample show a substantial improvement, carry out the test.
  3. Find the probability that the test results in a Type II error if in fact \(p = 0.8\). RECOGNISING ACHIEVEMENT