Questions S3 (621 questions)

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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 S3 2009 January Q1
    4 marks Moderate -0.3
    1 At a particular hospital, admissions of patients as a result of visits to the Accident and Emergency Department occur randomly at a uniform average rate of 0.75 per day. Independently, admissions that result from G.P. referrals occur randomly at a uniform average rate of 6.4 per week. The total number of admissions from these two causes over a randomly chosen period of four weeks is denoted by \(T\). State the distribution of \(T\) and obtain its expectation and variance.
    OCR S3 2009 January Q2
    5 marks Standard +0.3
    2 The continuous random variable \(U\) has (cumulative) distribution function given by $$\mathrm { F } ( u ) = \begin{cases} \frac { 1 } { 5 } \mathrm { e } ^ { u } & u < 0 \\ 1 - \frac { 4 } { 5 } \mathrm { e } ^ { - \frac { 1 } { 4 } u } & u \geqslant 0 \end{cases}$$
    1. Find the upper quartile of \(U\).
    2. Find the probability density function of \(U\).
    OCR S3 2009 January Q3
    8 marks Moderate -0.3
    3 In a random sample of credit card holders, it was found that \(28 \%\) of them used their card for internet purchases.
    1. Given that the sample size is 1200 , find a \(98 \%\) confidence interval for the percentage of all credit card holders who use their card for internet purchases.
    2. Estimate the smallest sample size for which a \(98 \%\) confidence interval would have a width of at most \(5 \%\), and state why the value found is only an estimate.
    OCR S3 2009 January Q4
    7 marks Standard +0.3
    4 The weekly sales of petrol, \(X\) thousand litres, at a garage may be modelled by a continuous random variable with probability density function given by $$f ( x ) = \begin{cases} c & 25 \leqslant x \leqslant 45 \\ 0 & \text { otherwise } \end{cases}$$ where \(c\) is a constant. The weekly profit, in \(\pounds\), is given by \(( 400 \sqrt { X } - 240 )\).
    1. Obtain the value of \(c\).
    2. Find the expected weekly profit.
    3. Find the probability that the weekly profit exceeds \(\pounds 2000\).
    OCR S3 2009 January Q5
    10 marks Standard +0.8
    5 The concentration level of mercury in a large lake is known to have a normal distribution with standard deviation 0.24 in suitable units. At the beginning of June 2008, the mercury level was measured at five randomly chosen places on the lake, and the sample mean is denoted by \(\bar { x } _ { 1 }\). Towards the end of June 2008 there was a spillage in the lake which may have caused the mercury level to rise. Because of this the level was then measured at six randomly chosen points of the lake, and the mean of this sample is denoted by \(\bar { x } _ { 2 }\).
    1. State hypotheses for a test based on the two samples for whether, on average, the level of mercury had increased. Define any parameters that you use.
    2. Find the set of values of \(\bar { x } _ { 2 } - \bar { x } _ { 1 }\) for which there would be evidence at the 5\% significance level that, on average, the level of mercury had increased.
    3. Given that the average level had actually increased by 0.3 units, find the probability of making a Type II error in your test, and comment on its value.
    OCR S3 2009 January Q6
    13 marks Standard +0.3
    6 A mathematics examination is taken by 29 boys and 26 girls. Experience has shown that the probability that any boy forgets to bring a calculator to the examination is 0.3 , and that any girl forgets is 0.2 . Whether or not any student forgets to bring a calculator is independent of all other students. The numbers of boys and girls who forget to bring a calculator are denoted by \(B\) and \(G\) respectively, and \(F = B + G\).
    1. Find \(\mathrm { E } ( F )\) and \(\operatorname { Var } ( F )\).
    2. Using suitable approximations to the distributions of \(B\) and \(G\), which should be justified, find the smallest number of spare calculators that should be available in order to be at least \(99 \%\) certain that all 55 students will have a calculator.
    OCR S3 2009 January Q7
    11 marks Standard +0.3
    7 A tutor gives a randomly selected group of 8 students an English Literature test, and after a term's further teaching, she gives the group a similar test. The marks for the two tests are given in the table.
    Student\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)
    First test3827554332245146
    Second test3726574330265448
    1. Stating a necessary condition, show by carrying out a suitable \(t\)-test, at the \(1 \%\) significance level, that the marks do not give evidence of an improvement.
    2. The tutor later found that she had marked the second test too severely, and she decided to add a constant amount \(k\) to each mark. Find the least integer value of \(k\) for which the increased marks would give evidence of improvement at the \(1 \%\) significance level.
    OCR S3 2009 January Q8
    14 marks Standard +0.3
    8 A soft drinks factory produces lemonade which is sold in packs of 6 bottles. As part of the factory's quality control, random samples of 75 packs are examined at regular intervals. The number of underfilled bottles in a pack of 6 bottles is denoted by the random variable \(X\). The results of one quality control check are shown in the following table.
    Number of underfilled bottles0123
    Number of packs442083
    A researcher assumes that \(X \sim \mathrm {~B} ( 3 , p )\).
    1. By finding the sample mean, show that an estimate of \(p\) is 0.2 .
    2. Show that, at the \(5 \%\) significance level, there is evidence that this binomial distribution does not fit the data.
    3. Another researcher suggests that the goodness of fit test should be for \(\mathrm { B } ( 6 , p )\). She finds that the corresponding value of \(\chi ^ { 2 }\) is 2.74 , correct to 3 significant figures. Given that the number of degrees of freedom is the same as in part (ii), state the conclusion of the test at the same significance level.
    OCR S3 2010 January Q1
    8 marks Moderate -0.3
    1 The continuous random variable \(X\) has probability density function given by $$\mathrm { f } ( x ) = \begin{cases} \frac { 2 } { 5 } & - a \leqslant x < 0 \\ \frac { 2 } { 5 } \mathrm { e } ^ { - 2 x } & x \geqslant 0 \end{cases}$$ Find
    1. the value of the constant \(a\),
    2. \(\mathrm { E } ( X )\).
    OCR S3 2010 January Q2
    8 marks Moderate -0.3
    2 The amount of tomato juice, \(X \mathrm { ml }\), dispensed into cartons of a particular brand has a normal distribution with mean 504 and standard deviation 3 . The juice is sold in packs of 4 cartons, filled independently. The total amount of juice in one pack is \(Y \mathrm { ml }\).
    1. Find \(\mathrm { P } ( Y < 2000 )\). The random variable \(V\) is defined as \(Y - 4 X\).
    2. Find \(\mathrm { E } ( V )\) and \(\operatorname { Var } ( V )\).
    3. What is the probability that the amount of juice in a randomly chosen pack is more than 4 times the amount of juice in a randomly chosen carton?
    OCR S3 2010 January Q3
    10 marks Standard +0.3
    3 It is given that \(X _ { 1 }\) and \(X _ { 2 }\) are independent random variables with \(X _ { 1 } \sim \mathrm {~N} \left( \mu _ { 1 } , 2.47 \right)\) and \(X _ { 2 } \sim \mathrm {~N} \left( \mu _ { 2 } , 4.23 \right)\). Random samples of \(n _ { 1 }\) observations of \(X _ { 1 }\) and \(n _ { 2 }\) observations of \(X _ { 2 }\) are taken. The sample means are denoted by \(\bar { X } _ { 1 }\) and \(\bar { X } _ { 2 }\).
    1. State the distribution of \(\bar { X } _ { 1 } - \bar { X } _ { 2 }\), giving its parameters. For two particular samples, \(n _ { 1 } = 5 , \Sigma x _ { 1 } = 48.25 , n _ { 2 } = 10\) and \(\Sigma x _ { 2 } = 72.30\).
    2. Test at the \(2 \%\) significance level whether \(\mu _ { 1 }\) differs from \(\mu _ { 2 }\). A student stated that because of the Central Limit Theorem the sample means will have normal distributions so it is unnecessary for \(X _ { 1 }\) and \(X _ { 2 }\) to have normal distributions.
    3. Comment on the student's statement.
    OCR S3 2010 January Q4
    9 marks Standard +0.8
    4 The continuous random variable \(V\) has (cumulative) distribution function given by $$\mathrm { F } ( v ) = \begin{cases} 0 & v < 1 \\ 1 - \frac { 8 } { ( 1 + v ) ^ { 3 } } & v \geqslant 1 \end{cases}$$ The random variable \(Y\) is given by \(Y = \frac { 1 } { 1 + V }\).
    1. Show that the (cumulative) distribution function of \(Y\) is \(8 y ^ { 3 }\), over an interval to be stated, and find the probability density function of \(Y\).
    2. Find \(\mathrm { E } \left( \frac { 1 } { Y ^ { 2 } } \right)\).
    OCR S3 2010 January Q5
    11 marks Standard +0.3
    5 Each of a random sample of 200 steel bars taken from a production line was examined and 27 were found to be faulty.
    1. Find an approximate \(90 \%\) confidence interval for the proportion of faulty bars produced. A change in the production method was introduced which, it was claimed, would reduce the proportion of faulty bars. After the change, each of a further random sample of 100 bars was examined and 8 were found to be faulty.
    2. Test the claim, at the \(10 \%\) significance level.
    OCR S3 2010 January Q6
    12 marks Standard +0.3
    6 The deterioration of a certain drug over time was investigated as follows. The drug strength was measured in each of a random sample of 8 bottles containing the drug. These were stored for two years and the strengths were then re-measured. The original and final strengths, in suitable units, are shown in the following table.
    Bottle12345678
    Original strength8.79.49.28.99.68.29.98.8
    Final strength8.19.09.08.89.38.09.58.5
    1. Stating any required assumption, test at the \(5 \%\) significance level whether the mean strength has decreased by more than 0.2 over the two years.
    2. Calculate a 95\% confidence interval for the mean reduction in strength over the two years.