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OCR S2 2015 June Q3
10 marks Standard +0.3
3 A continuous random variable \(X\) has probability density function $$f ( x ) = \left\{ \begin{array} { c l } \frac { 3 } { 2 a ^ { 3 } } x ^ { 2 } & - a \leqslant x \leqslant a \\ 0 & \text { otherwise } \end{array} \right.$$ where \(a\) is a constant.
  1. It is given that \(\mathrm { P } ( - 3 \leqslant X \leqslant 3 ) = 0.125\). Find the value of \(a\) in this case.
  2. It is given instead that \(\operatorname { Var } ( X ) = 1.35\). Find the value of \(a\) in this case.
  3. Explain the relationship between \(x\) and \(X\) in this question.
OCR S2 2015 June Q4
10 marks Standard +0.3
4 A continuous random variable is normally distributed with mean \(\mu\). A significance test for \(\mu\) is carried out, at the \(5 \%\) significance level, on 90 independent occasions.
  1. Given that the null hypothesis is correct on all 90 occasions, use a suitable approximation to find the probability that on 6 or fewer occasions the test results in a Type I error. Justify your approximation.
  2. Given instead that on all 90 occasions the probability of a Type II error is 0.35 , use a suitable approximation to find the probability that on fewer than 29 occasions the test results in a Type II error.
OCR S2 2015 June Q5
8 marks Standard +0.3
5
  1. State an advantage of using random numbers in selecting samples.
  2. It is known that in analysing the digits in large sets of financial records, the probability that the leading digit is 1 is 0.25 . A random sample of 18 leading digits from a certain large set of financial records is obtained and it is found that 8 of the leading digits are 1 s . Test, at the \(5 \%\) significance level, whether the probability that the leading digit is 1 in this set of records is greater than 0.25 .
OCR S2 2015 June Q6
12 marks Standard +0.3
6 Records for a doctors' surgery over a long period suggest that the time taken for a consultation, \(T\) minutes, has a mean of 11.0. Following the introduction of new regulations, a doctor believes that the average time has changed. She finds that, with new regulations, the consultation times for a random sample of 120 patients can be summarised as $$n = 120 , \Sigma t = 1411.20 , \Sigma t ^ { 2 } = 18737.712 .$$
  1. Test, at the \(10 \%\) significance level, whether the doctor's belief is correct.
  2. Explain whether, in answering part (i), it was necessary to assume that the consultation times were normally distributed.
OCR S2 2015 June Q7
13 marks Standard +0.3
7 A large railway network suffers points failures at an average rate of 1 every 3 days. Assume that the number of points failures can be modelled by a Poisson distribution. The network employs a new firm of engineers. After the new engineers have become established, it is found that in a randomly chosen period of 15 days there are 2 instances of points failures.
  1. Test, at the \(5 \%\) significance level, whether there is evidence that the mean number of points failures has been reduced.
  2. A new test is carried out over a period of 150 days. Use a suitable approximation to find the greatest number of points failures there could be in 150 days that would lead to a \(5 \%\) significance test concluding that the average number of points failures had been reduced.
OCR S2 2015 June Q8
7 marks Standard +0.8
8 The random variable \(S\) has the distribution \(\mathrm { B } ( 14 , p )\). A significance test is carried out of the null hypothesis \(\mathrm { H } _ { 0 } : p = 0.3\) against the alternative hypothesis \(\mathrm { H } _ { 1 } : p > 0.3\). The critical region for the test is \(S \geqslant 8\).
  1. Find the significance level of the test, correct to 3 significant figures.
  2. It is given that, on each occasion that the test is carried out, the true value of \(p\) is equally likely to be \(0.3,0.5\) or 0.7 , independently of any other test. Four independent tests are carried out. Find the probability that at least one of the tests results in a Type II error.
OCR S3 2014 June Q1
5 marks Standard +0.3
1 The independent random variables \(X\) and \(Y\) have Poisson distributions with parameters 16 and 2 respectively, and \(Z = \frac { 1 } { 2 } X - Y\).
  1. Find \(\mathrm { E } ( Z )\) and \(\operatorname { Var } ( Z )\).
  2. State whether \(Z\) has a Poisson distribution, giving a reason for your answer.
OCR S3 2014 June Q2
6 marks Standard +0.3
2 In a study of the inheritance of skin colouration in corn snakes, a researcher found 865 snakes with black and orange bodies, 320 snakes with black bodies, 335 snakes with orange bodies and 112 snakes with bodies of other colours. Theory predicts that snakes of these colours should occur in the ratios \(9 : 3 : 3 : 1\). Test, at the \(5 \%\) significance level, whether these experimental results are compatible with theory.
OCR S3 2014 June Q3
7 marks Standard +0.3
3 An athlete finds that her times for running 100 m are normally distributed. Before a period of intensive training, her mean time is 11.8 s . After the period of intensive training, five randomly selected times, in seconds, are as follows. $$\begin{array} { l l l l l } 11.70 & 11.65 & 11.80 & 11.75 & 11.60 \end{array}$$ Carry out a suitable test, at the \(5 \%\) significance level, to investigate whether times after the training are less, on average, than times before the training.
OCR S3 2014 June Q4
7 marks Standard +0.3
4 Cola is sold in bottles and cans. The volume of cola in a bottle is normally distributed with mean 500 ml and standard deviation 10 ml . The volume of cola in a can is normally distributed with mean 330 ml and standard deviation 8 ml . Find the probability that the total volume of cola in 2 randomly selected bottles is greater than 3 times the volume of cola in a randomly selected can.
OCR S3 2014 June Q5
9 marks Standard +0.3
5 The day before the 1992 General Election, an opinion poll showed that \(37.6 \%\) of a random sample of 1731 voters intended to vote for the Conservative party.
  1. Calculate an approximate \(99.9 \%\) confidence interval for the proportion of voters intending to vote Conservative. The actual proportion voting Conservative was above the upper limit of the confidence interval.
  2. Give two possible reasons for this occurrence.
  3. What sample size would be required to produce a \(99.9 \%\) confidence interval of width 0.05 ?
OCR S3 2014 June Q6
8 marks Standard +0.3
6 The continuous random variable \(X\) has probability density function given by $$\mathrm { f } ( x ) = \left\{ \begin{array} { c l } k \sin x & 0 \leqslant x \leqslant \frac { 1 } { 2 } \pi , \\ k \left( 2 - \frac { 2 x } { \pi } \right) & \frac { 1 } { 2 } \pi \leqslant x \leqslant \pi , \\ 0 & \text { otherwise, } \end{array} \right.$$ where \(k\) is a constant.
  1. Show that \(k = \frac { 4 } { 4 + \pi }\).
  2. Find \(\mathrm { E } ( X )\), correct to 3 significant figures, showing all necessary working.
OCR S3 2014 June Q7
9 marks Standard +0.3
7 A random sample of 100 adults with a chronic disease was chosen. Each adult was randomly assigned to one of three different treatments. After six months of treatment, each adult was then assessed and classified as 'much improved', 'improved', 'slightly improved' or 'no change'. The results are summarised in Table 1. \begin{table}[h]
Treatment \(A\)Treatment \(B\)Treatment \(C\)
Much improved12164
Improved13126
Slightly improved767
No change539
\captionsetup{labelformat=empty} \caption{Table 1}
\end{table} A \(\chi ^ { 2 }\) test, at the \(5 \%\) significance level, is to be carried out.
  1. State suitable hypotheses. Combining the last two rows of Table 1 gives Table 2. \begin{table}[h]
    Treatment \(A\)Treatment \(B\)Treatment \(C\)
    Much improved12164
    Improved13126
    Slightly improved/ No change12916
    \captionsetup{labelformat=empty} \caption{Table 2}
    \end{table}
  2. By considering the expected frequencies for Treatment \(C\) in Table 1, explain why it was necessary to combine rows.
  3. Show that the contribution to the \(\chi ^ { 2 }\) value for the cell 'slightly improved/no change, Treatment \(C\) ' is 4.231 , correct to 3 decimal places. You are given that the \(\chi ^ { 2 }\) test statistic is 10.51 , correct to 2 decimal places.
  4. Carry out the test.
OCR S3 2014 June Q8
10 marks Standard +0.3
8 A random sample of 20 plots of land, each of equal area, was used to test whether the addition of phosphorus would increase the yield of corn. 10 plots were treated with phosphorus and 10 plots were untreated. The yields of corn, in litres, on a treated plot and on an untreated plot are denoted by \(X\) and \(Y\) respectively. You are given that $$\sum x = 2112 , \quad \sum y = 2008$$ You are also given that an unbiased estimate for the variance of treated plots is 87.96 and an unbiased estimate for the variance of untreated plots is 31.96 , both correct to 4 significant figures.
  1. You may assume that the population variance estimates are sufficiently similar for the assumption of common variance to be made. What other assumption needs to be made for a \(t\)-test to be valid?
  2. Carry out a suitable \(t\)-test at the \(1 \%\) significance level, to test whether the use of phosphorus increases the yield of corn.
OCR S3 2014 June Q9
11 marks Challenging +1.2
9 A rectangle of area \(A \mathrm {~m} ^ { 2 }\) has a perimeter of 20 m and each of the two shorter sides are of length \(X \mathrm {~m}\), where \(X\) is uniformly distributed between 0 and 2 .
  1. Write down an expression for \(A\) in terms of \(X\), and hence show that \(A = 25 - ( X - 5 ) ^ { 2 }\).
  2. Write down the probability density function of \(X\).
  3. Show that the cumulative distribution function of \(A\) is $$\mathrm { F } ( a ) = \left\{ \begin{array} { l r } 0 & a < 0 , \\ \frac { 1 } { 2 } ( 5 - \sqrt { 25 - a } ) & 0 \leqslant a \leqslant 16 , \\ 1 & a > 16 . \end{array} \right.$$
  4. Find the probability density function of \(A\). \section*{END OF QUESTION PAPER} \section*{OCR}
OCR S3 2015 June Q1
6 marks Moderate -0.8
1 A laminate consists of 4 layers of material \(C\) and 3 layers of material \(D\). The thickness of a layer of material \(C\) has a normal distribution with mean 1 mm and standard deviation 0.1 mm , and the thickness of a layer of material \(D\) has a normal distribution with mean 8 mm and standard deviation 0.2 mm . The layers are independent of one another.
  1. Find the mean and variance of the total thickness of the laminate.
  2. What total thickness is exceeded by \(1 \%\) of the laminates?
OCR S3 2015 June Q2
7 marks Standard +0.3
2 In a poll of people aged 18-21, 46 out of 200 randomly chosen university students agreed with a proposition. 51 out of 300 randomly chosen others who were not university students agreed with it. Test, at the \(5 \%\) significance level, whether the proportion of university students who agree with the proposition differs from the proportion of those who are not university students.
OCR S3 2015 June Q3
12 marks Standard +0.3
3 A tutor gave an assessment to 6 randomly chosen new eleven-year-old students. After each student had received 30 hours of tuition, they were given a second assessment. The scores are shown in the table.
Student\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)
1st assessment124121111113118119
2nd assessment127119114110120122
  1. Show that, at the \(5 \%\) significance level, there is insufficient evidence that students' scores are higher, on average, after tuition than before tuition. State a necessary assumption.
  2. Disappointed by this result, the tutor looked again at the first assessment. She discovered that the first assessment was too easy, in fact being a test for ten-year-olds, not eleven-year-olds. She decided to reduce each score for the first assessment by a constant integer \(k\). Find the least value of \(k\) for which there is evidence at the \(5 \%\) significance level that the students' scores have, on average, improved.
OCR S3 2015 June Q4
9 marks Standard +0.3
4 A set of bathroom scales is known to operate with an error which is normally distributed. One morning a man weighs himself 4 times. The 4 values for his mass, in kg , which can be considered to be a random sample are as follows. $$\begin{array} { l l l l } 62.6 & 62.8 & 62.1 & 62.5 \end{array}$$
  1. Find a \(95 \%\) confidence interval for his mass. Give the end-points of the interval correct to 3 decimal places.
  2. Based on these results, a \(y \%\) confidence interval has width 0.482 . Find \(y\).
OCR S3 2015 June Q5
11 marks Challenging +1.2
5 Two guesthouses, the Albion and the Blighty, have 8 and 6 rooms respectively. The demand for rooms at the Albion has a Poisson distribution with mean 6.5 and the demand for rooms at the Blighty has an independent Poisson distribution with mean 5.5. The owners have agreed that if their guesthouse is full, they will re-direct guests to the other.
  1. Find the probability that, on any particular night, the two guesthouses together do not have enough rooms to meet demand.
  2. The Albion charges \(\pounds 60\) per room per night, and the Blighty \(\pounds 80\). Find the probability, that on a particular night, the total income of the two guesthouses is exactly \(\pounds 400\).
  3. If \(A\) is the number of rooms demanded at the Albion each night, and \(B\) the number of rooms demanded at the Blighty each night, find the mean and variance of the variable \(C = 60 A + 80 B\). State whether \(C\) has a Poisson distribution, giving a reason for your answer.
OCR S3 2015 June Q6
13 marks Standard +0.3
6 In each of 38 randomly selected weeks of the English Premier Football League there were 10 matches. Table 1 summarises the number of home wins in 10 matches, \(X\), and the corresponding number of weeks. \begin{table}[h]
Number of home wins012345678910
Number of weeks01288971200
\captionsetup{labelformat=empty} \caption{Table 1}
\end{table} A researcher investigates whether \(X\) can be modelled by the distribution \(\mathrm { B } ( 10 , p )\). He calculates the expected frequencies using a value of \(p\) obtained from the sample mean.
  1. Show that \(p = 0.45\). Table 2 shows the observed and expected number of weeks. \begin{table}[h]
    Number of home wins012345678910Totals
    Observed number of weeks0128897120038
    Expected number of weeks0.0960.7882.8996.3269.0588.8936.0642.8350.8700.1580.01338
    \captionsetup{labelformat=empty} \caption{Table 2
  2. Show how the value of 2.835 for 7 home wins is obtained.}
\end{table} The researcher carries out a test, at the \(5 \%\) significance level, of whether the distribution \(\mathrm { B } ( 10 , p )\) fits the data.
  • Explain why it is necessary to combine classes.
  • Carry out the test.
    • OCR S3 2015 June Q7
    14 marks Standard +0.3
    7 A continuous random variable \(X\) has probability density function $$f ( x ) = \left\{ \begin{array} { c c } k x & 0 \leqslant x < 2 \\ \frac { k ( 4 - x ) ^ { 2 } } { 2 } & 2 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{array} \right.$$ where \(k\) is a constant.
    1. Show that \(k = \frac { 3 } { 10 }\).
    2. Find \(\mathrm { E } ( X )\).
    3. Find the cumulative distribution function of \(X\).
    4. Find the upper quartile of \(X\), correct to 3 significant figures. \section*{END OF QUESTION PAPER}
    OCR S4 2016 June Q1
    8 marks Moderate -0.8
    1 Ten archers shot at targets with two types of bow. Their scores out of 100 are shown in the table.
    Archer\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)\(I\)\(J\)
    Bow type \(P\)95979285879290899877
    Bow type \(Q\)91918890808893859484
    1. Use the sign test, at the \(5 \%\) level of significance, to test the hypothesis that bow type \(P\) is better than bow type \(Q\).
    2. Why would a Wilcoxon signed rank test, if valid, be a better test than the sign test?
    OCR S4 2016 June Q2
    8 marks Standard +0.3
    2 Low density lipoprotein (LDL) cholesterol is known as 'bad' cholesterol.
    15 randomly chosen patients, each with an LDL level of 190 mg per decilitre of blood, were given one of two treatments, chosen at random. After twelve weeks their LDL levels, in mg per decilitre, were as follows.
    Treatment \(A\)189168176186183187188
    Treatment \(B\)177179173180178170175174
    Use a Wilcoxon rank sum test, at the \(5 \%\) level of significance, to test whether the LDL levels of patients given treatment \(B\) are lower than the LDL levels of patients given treatment \(A\).
    OCR S4 2016 June Q3
    9 marks Standard +0.3
    3 The table shows the joint probability distribution of two random variables \(X\) and \(Y\).
    \cline { 2 - 5 } \multicolumn{2}{c|}{}\(Y\)
    \cline { 2 - 5 } \multicolumn{2}{c|}{}012
    \multirow{3}{*}{\(X\)}00.070.070.16
    \cline { 2 - 5 }10.060.090.15
    \cline { 2 - 5 }20.070.140.19
    1. Find \(\operatorname { Cov } ( X , Y )\).
    2. Are \(X\) and \(Y\) independent? Give a reason for your answer.
    3. Find \(\mathrm { P } ( X = 1 \mid X Y = 2 )\).