Questions — OCR (4619 questions)

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OCR S3 2006 June Q1
1 The numbers of \(\alpha\)-particles emitted per minute from two types of source, \(A\) and \(B\), have the distributions \(\operatorname { Po } ( 1.5 )\) and \(\operatorname { Po } ( 2 )\) respectively. The total number of \(\alpha\)-particles emitted over a period of 2 minutes from three sources of type \(A\) and two sources of type \(B\), all of which are independent, is denoted by \(X\). Calculate \(\mathrm { P } ( X = 27 )\).
OCR S3 2006 June Q2
2 The manager of a factory with a large number of employees investigated when accidents to employees occurred during 8-hour shifts. An analysis was made of 600 randomly chosen accidents that occurred over a year. The following table shows the numbers of accidents occurring in the four consecutive 2-hour periods of the 8-hour shifts.
Period1234
Number of accidents138127165170
Test, at the \(5 \%\) significance level, whether the proportions of all accidents that occur in the four time periods differ.
OCR S3 2006 June Q3
3 Ten randomly chosen athletes were coached for a 200 m event. For each athlete, the times taken to run 200 m before and after coaching were measured. The sample mean times before and after coaching were 23.43 seconds and 22.84 seconds respectively. For each athlete the difference, \(d\) seconds, in the times before and after coaching was calculated and an unbiased estimate of the population variance of \(d\) was found to be 0.548 . Stating any required assumption, test at the \(5 \%\) significance level whether the population mean time for the 200 m run decreased after coaching.
OCR S3 2006 June Q4
4 The continuous random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { 4 } { 3 x ^ { 3 } } & 1 \leqslant x < 2
\frac { 1 } { 12 } x & 2 \leqslant x \leqslant 4
0 & \text { otherwise } \end{cases}$$
  1. Find the upper quartile of \(X\).
  2. Find the value of \(a\) for which \(\mathrm { E } \left( X ^ { 2 } \right) = a \mathrm { E } ( X )\).
OCR S3 2006 June Q5
5 Gloria is a market trader who sells jeans. She trades on Mondays, Wednesdays and Fridays. Wishing to investigate whether the volume of trade depends on the day of the week, Gloria analysed a random sample of 150 days' sales and classified them by day and volume (low, medium and high). The results are given in the table below.
Day
MondayWednesdayFriday
\multirow{3}{*}{Volume}Low15132
Medium232623
High12927
Gloria asked a statistician to perform a suitable test of independence and, as part of this test, expected frequencies were calculated. These are shown in the table below.
Day
MondayWednesdayFriday
Low10.009.6010.40
VolumeMedium24.0023.0424.96
High16.0015.3616.64
  1. Show how the value 23.04 for medium volume on Wednesday has been obtained.
  2. State, giving a reason, if it is necessary to combine any rows or columns in order to carry out the test. The value of the test statistic is found to be 21.15, correct to 2 decimal places.
  3. Stating suitable hypotheses for the test, give its conclusion using a \(1 \%\) significance level. Gloria wishes to hold a sale and asks the statistician to advise her on which day to hold it in order to sell as much as possible.
  4. State the day that the statistician should advise and give a reason for the choice.
OCR S3 2006 June Q6
6 An anthropologist was studying the inhabitants of two islands, Raloa and Tangi. Part of the study involved the incidence of blood group type A. The blood of 80 randomly chosen inhabitants of Raloa and 85 randomly chosen inhabitants of Tangi was tested. The number of inhabitants with type A blood was 28 for the Raloa sample and 46 for the Tangi sample. The anthropologist calculated \(90 \%\) confidence intervals for the population proportions of inhabitants with type A blood. They were \(( 0.262,0.438 )\) for Raloa and \(( 0.452,0.630 )\) for Tangi, where each figure is correct to 3 decimal places. It is known that \(43 \%\) of the world's population have type A blood.
  1. State, giving your reasons, whether there is evidence for the following assertions about the proportions of people with type A blood.
    (a) The proportion in Raloa is different from the world proportion.
    (b) The proportion in Tangi is different from the world proportion.
  2. Carry out a suitable test, at the \(2 \%\) significance level, of whether the proportions of people with type A blood differ on the two islands.
OCR S3 2006 June Q7
7 A queue of cars has built up at a set of traffic lights which are at red. When the lights turn green, the time for the first car to start to move has a normal distribution with mean 2.2 s and standard deviation 0.75 s . This time is the reaction time for the first car. For each subsequent car the reaction time is the time taken for it to start to move after the car in front starts to move. These reaction times have identical normal distributions with mean 1.8 s and standard deviation 0.70 s . It may be assumed that all reaction times are independent.
  1. Calculate the probability that the reaction time for the second car in the queue is less than half of the reaction time for the first car.
  2. Calculate the probability that the fifth car in the queue starts to move less than 10 seconds after the lights turn green.
  3. State where, in part (i), independence is required.
OCR S3 2006 June Q8
8 Two machines, \(A\) and \(B\), produce metal rods. Machine \(B\) is new and it is required that its accuracy should be checked against that of machine \(A\). The observed variable is the length of a rod. Random samples of rods, 40 from machine \(A\) and 50 from machine \(B\), are taken and their lengths, \(x _ { A } \mathrm {~cm}\) and \(x _ { B } \mathrm {~cm}\), are measured. The results are summarised by $$\Sigma x _ { A } = 136.48 , \quad \Sigma x _ { B } = 176.35 , \quad \Sigma x _ { B } ^ { 2 } = 630.1940 .$$ The variance of the length of the rods produced by machine \(A\) is known to be \(0.0490 \mathrm {~cm} ^ { 2 }\). The mean lengths of the rods produced by the machines are denoted by \(\mu _ { A } \mathrm {~cm}\) and \(\mu _ { B } \mathrm {~cm}\) respectively.
  1. Test, at the \(5 \%\) significance level, the hypothesis \(\mu _ { B } > \mu _ { A }\).
  2. Find the set of values of \(a\) for which the null hypothesis \(\mu _ { B } - \mu _ { A } = 0.025\) would not be rejected in favour of the alternative hypothesis \(\mu _ { B } - \mu _ { A } > 0.025\) at the \(a \%\) significance level.
  3. For the test in part (i) to be valid,
    (a) state whether it is necessary to assume that the two population variances are equal,
    (b) state, giving a reason, whether it is necessary to assume that the lengths of rods are normally distributed.
OCR S3 2007 June Q1
1 The continuous random variable \(X\) has probability density function given by $$\mathrm { f } ( x ) = \begin{cases} a & 0 \leqslant x \leqslant 1 ,
\frac { a } { x ^ { 2 } } & x > 1 ,
0 & \text { otherwise. } \end{cases}$$ Find the value of the constant \(a\).
OCR S3 2007 June Q2
2 Two brands of car battery, ‘Invincible’ and ‘Excelsior’, have lifetimes which are normally distributed. Invincible batteries have a mean lifetime of 5 years with standard deviation 0.7 years. Excelsior batteries have a mean lifetime of 4.5 years with standard deviation 0.5 years. Random samples of 20 Invincible batteries and 25 Excelsior batteries are selected and the sample mean lifetimes are \(\bar { X } _ { I }\) years and \(\bar { X } _ { E }\) years respectively.
  1. State the distributions of \(\bar { X } _ { I }\) and \(\bar { X } _ { E }\).
  2. Calculate \(\mathrm { P } \left( \bar { X } _ { I } - \bar { X } _ { E } \geqslant 1 \right)\).
OCR S3 2007 June Q3
3 A nurse was asked to measure the blood pressure of 12 patients using an aneroid device. The nurse's readings were immediately checked using an accurate electronic device. The differences, \(x\), given by \(x =\) (aneroid reading - electronic reading), in appropriate units, are shown below. $$\begin{array} { c c c c c c c c c c c } - 1.3 & 4.7 & - 0.9 & 3.8 & - 1.5 & 4.0 & - 1.9 & 4.4 & - 0.8 & 5.5 & - 2.9 \end{array} 4.1$$ Stating any assumption you need to make, test, at the \(10 \%\) significance level, whether readings with an aneroid device, on average, overestimate patients’ blood pressure.
OCR S3 2007 June Q4
4 The students in a large university department take a trial examination some time before the proper examination. A random sample of 60 students took both examinations during a particular course. 42 students passed the trial examination, 36 passed the proper examination and 13 failed both examinations.
  1. Copy and complete the following contingency table.
    Proper
    \cline { 2 - 4 } \multicolumn{1}{l}{}PassFailTotal
    \cline { 2 - 5 }Pass42
    \cline { 2 - 5 } TrialFail13
    \cline { 2 - 5 }Total3660
  2. Carry out a test of independence at the \(\frac { 1 } { 2 } \%\) level of significance.
OCR S3 2007 June Q5
5 A music store sells both upright and grand pianos. Grand pianos are sold at random times and at a constant average weekly rate \(\lambda\). The probability that in one week no grand pianos are sold is 0.45 .
  1. Show that \(\lambda = 0.80\), correct to 2 decimal places. Upright pianos are sold, independently, at random times and at a constant average weekly rate \(\mu\). During a period of 100 weeks the store sold 180 upright pianos.
  2. Calculate the probability that the total number of pianos sold in a randomly chosen week will exceed 3.
  3. Calculate the probability that over a period of 3 weeks the store sells a total of 6 pianos during the first week and a total of 4 pianos during the next fortnight.
OCR S3 2007 June Q6
6 Random samples of 200 'Alpha' and 150 'Beta' vacuum cleaners were monitored for reliability. It was found that 62 Alpha and 35 Beta cleaners required repair during the guarantee period of one year. The proportions of all Alpha and Beta cleaners that require repair during the guarantee period are \(p _ { \alpha }\) and \(p _ { \beta }\) respectively.
  1. Find a \(95 \%\) confidence interval for \(p _ { \alpha }\).
  2. Give a reason why, apart from rounding, the interval is approximate.
  3. Test, at the \(5 \%\) significance level, whether \(p _ { \alpha }\) differs from \(p _ { \beta }\).
OCR S3 2007 June Q7
7 The continuous random variable \(X\) has (cumulative) distribution function given by $$\mathrm { F } ( x ) = \begin{cases} 0 & x < 1
1 - \frac { 1 } { x ^ { 4 } } & x \geqslant 1 \end{cases}$$
  1. Find the (cumulative) distribution function, \(\mathrm { G } ( y )\), of the random variable \(Y\), where \(Y = \frac { 1 } { X ^ { 2 } }\).
  2. Hence show that the probability density function of \(Y\) is given by $$g ( y ) = \begin{cases} 2 y & 0 < y \leqslant 1
    0 & \text { otherwise } \end{cases}$$
  3. Find \(\mathrm { E } ( \sqrt [ 3 ] { Y } )\).
OCR S3 2007 June Q8
8 The continuous random variable \(Y\) has a distribution with mean \(\mu\) and variance 20. A random sample of 50 observations of \(Y\) is selected and these observations are summarised in the following grouped frequency table.
Values\(y < 20\)\(20 \leqslant y < 25\)\(25 \leqslant y < 30\)\(y \geqslant 30\)
Frequency327128
  1. Assuming that \(Y \sim \mathrm {~N} ( 25,20 )\), show that the expected frequency for the interval \(20 \leqslant y < 25\) is 18.41, correct to 2 decimal places, and obtain the remaining expected frequencies.
  2. Test, at the \(5 \%\) significance level, whether the distribution \(\mathrm { N } ( 25,20 )\) fits the data.
  3. Given that the sample mean is 24.91 , find a \(98 \%\) confidence interval for \(\mu\).
  4. Does the outcome of the test in part (ii) affect the validity of the confidence interval found in part (iii)? Justify your answer.
OCR S3 2011 June Q1
1 The random variables \(X\) and \(Y\) are independent with \(X \sim \operatorname { Po } ( 5 )\) and \(Y \sim \operatorname { Po } ( 4 )\). \(S\) denotes the sum of 2 observations of \(X\) and 3 observations of \(Y\).
  1. Find \(\mathrm { E } ( S )\) and \(\operatorname { Var } ( S )\).
  2. The random variable \(T\) is defined by \(\frac { 1 } { 2 } X - \frac { 1 } { 4 } Y\). Show that \(\mathrm { E } ( T ) = \operatorname { Var } ( T )\).
  3. State which of \(S\) and \(T\) (if either) does not have a Poisson distribution, giving a reason for your answer.
OCR S3 2011 June Q2
2 The population proportion of all men with red-green colour blindness is denoted by \(p\). Each of a random sample of 80 men was tested and it was found that 6 had red-green colour blindness.
  1. Calculate an approximate \(95 \%\) confidence interval for \(p\).
  2. For a different random sample of men, the proportion with red-green colour blindness is denoted by \(p _ { s }\). Estimate the sample size required in order that \(\left| p _ { s } - p \right| \leqslant 0.05\) with probability \(95 \%\).
  3. Give one reason why the calculated sample size is an estimate.
OCR S3 2011 June Q3
3 The monthly demand for a product, \(X\) thousand units, is modelled by the random variable \(X\) with probability density function given by $$f ( x ) = \begin{cases} a x & 0 \leqslant x \leqslant 1
a ( x - 2 ) ^ { 2 } & 1 < x \leqslant 2
0 & \text { otherwise } \end{cases}$$ where \(a\) is a positive constant. Find
  1. the value of \(a\),
  2. the probability that the monthly demand is at most 1500 units,
  3. the expected monthly demand.
OCR S3 2011 June Q4
4 An experiment by Lord Rutherford at Cambridge in 1909 involved measuring the numbers of \(\alpha\)-particles emitted during radioactive decay. The following table shows emissions during 2608 intervals of 7.5 seconds.
Number of particles emitted, \(x\)012345678910\(\geqslant 11\)
Frequency572033835255324082731394527106
It is given that the mean number of particles emitted per interval, calculated from the data, is 3.87 , correct to 3 significant figures.
  1. Find the contribution to the \(\chi ^ { 2 }\) value of the frequency of 273 corresponding to \(x = 6\) in a goodness of fit test for a Poisson distribution.
  2. Given that no cells need to be combined, state why the number of degrees of freedom is 10 .
  3. Given also that the calculated value of \(\chi ^ { 2 }\) is 13.0 , correct to 3 significant figures, carry out the test at the 10\% significance level.
OCR S3 2011 June Q5
5 The continuous random variable \(X\) has (cumulative) distribution function given by $$\mathrm { F } ( x ) = \begin{cases} 0 & x < 1 ,
\frac { 4 } { 3 } \left( 1 - \frac { 1 } { x ^ { 2 } } \right) & 1 \leqslant x \leqslant 2 ,
1 & x > 2 . \end{cases}$$
  1. Find the median value of \(X\).
  2. Find the (cumulative) distribution function of \(Y\), where \(Y = \frac { 1 } { X ^ { 2 } }\), and hence find the probability density function of \(Y\).
  3. Evaluate \(\mathrm { E } \left( 2 - \frac { 2 } { X ^ { 2 } } \right)\).
OCR S3 2011 June Q6
6 The Body Mass Index (BMI) of each of a random sample of 100 army recruits from a large intake in 2008 was measured. The results are summarised by $$\Sigma x = 2605.0 , \quad \Sigma x ^ { 2 } = 68636.41 .$$ It may be assumed that BMI has a normal distribution.
  1. Find a 98\% confidence interval for the mean BMI of all recruits in 2008.
  2. Estimate the percentage of the intake with a BMI greater than 30.0.
  3. The BMIs of two randomly chosen recruits are denoted by \(\boldsymbol { B } _ { 1 }\) and \(\boldsymbol { B } _ { 2 }\). Estimate \(\mathrm { P } \left( \boldsymbol { B } _ { 1 } - \boldsymbol { B } _ { 2 } < 5 \right)\).
  4. State, giving a reason, for which of the above calculations the normality assumption is unnecessary.
OCR S3 2011 June Q7
7 In order to improve their mathematics results 10 students attended an intensive Summer School course. Each student took a test at the start of the course and a similar test at the end of the course. The table shows the scores achieved in each test.
Student12345678910
First test score37273847542752396223
Second test score47295044723763457632
It is desired to test whether there has been an increase in the population mean score.
  1. Explain why a two-sample \(t\)-test would not be appropriate.
  2. Stating any necessary assumptions, carry out a suitable \(t\)-test at the \(\frac { 1 } { 2 } \%\) significance level.
  3. The Summer School director claims that after taking the course the population mean score increases by more than 5 . Is there sufficient evidence for this claim?
OCR S3 Specimen Q1
1 A car repair firm receives call-outs both as a result of breakdowns and also as a result of accidents. On weekdays (Monday to Friday), call-outs resulting from breakdowns occur at random, at an average rate of 6 per 5 -day week; call-outs resulting from accidents occur at random, at an average rate of 2 per 5 -day week. The two types of call-out occur independently of each other. Find the probability that the total number of call-outs received by the firm on one randomly chosen weekday is more than 3 .
OCR S3 Specimen Q2
7 marks
2 Boxes of matches contain 50 matches. Full boxes have mean mass 20.0 grams and standard deviation 0.4 grams. Empty boxes have mean mass 12.5 grams and standard deviation 0.2 grams. Stating any assumptions that you need to make, calculate the mean and standard deviation of the mass of a match. [7]