Questions S3 (597 questions)

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OCR MEI S3 2007 June Q2
2 The operator of a section of motorway toll road records its weekly takings according to the types of vehicles using the motorway. For purposes of charging, there are three types of vehicle: cars, coaches, lorries. The weekly takings (in thousands of pounds) for each type are assumed to be Normally distributed. These distributions are independent of each other and are summarised in the table.
Vehicle typeMeanStandard deviation
Cars60.25.2
Coaches33.96.3
Lorries52.44.9
  1. Find the probability that the weekly takings for coaches are less than \(\pounds 40000\).
  2. Find the probability that the weekly takings for lorries exceed the weekly takings for cars.
  3. Find the probability that over a 4 -week period the total takings for cars exceed \(\pounds 225000\). What assumption must be made about the four weeks?
  4. Each week the operator allocates part of the takings for repairs. This is determined for each type of vehicle according to estimates of the long-term damage caused. It is calculated as follows: \(5 \%\) of takings for cars, \(10 \%\) for coaches and \(20 \%\) for lorries. Find the probability that in any given week the total amount allocated for repairs will exceed \(\pounds 20000\).
OCR MEI S3 2007 June Q3
3 The management of a large chain of shops aims to reduce the level of absenteeism among its workforce by means of an incentive bonus scheme. In order to evaluate the effectiveness of the scheme, the management measures the percentage of working days lost before and after its introduction for each of a random sample of 11 shops. The results are shown below.
ShopABCDEFGHIJK
\% days lost before3.55.03.53.24.54.94.16.06.88.16.0
\% days lost after1.84.32.94.54.45.83.56.76.45.45.1
  1. The management decides to carry out a \(t\) test to investigate whether there has been a reduction in absenteeism.
    1. State clearly the hypotheses that should be used together with any necessary assumptions.
    2. Carry out the test using a \(5 \%\) significance level.
  2. Find a 95\% confidence interval for the true mean percentage of days lost after the introduction of the incentive scheme and state any assumption needed. The management has set a target that the mean percentage should be 3.5. Do you think this has been achieved? Explain your answer.
OCR MEI S3 2007 June Q4
4 A machine produces plastic strip in a continuous process. Occasionally there is a flaw at some point along the strip. The length of strip (in hundreds of metres) between successive flaws is modelled by a continuous random variable \(X\) with probability density function \(\mathrm { f } ( x ) = \frac { 18 } { ( 3 + x ) ^ { 3 } }\) for \(x > 0\). The table below gives the frequencies for 100 randomly chosen observations of \(X\). It also gives the probabilities for the class intervals using the model.
Length \(x\) (hundreds of metres)Observed frequencyProbability
\(0 < x \leqslant 0.5\)210.2653
\(0.5 < x \leqslant 1\)240.1722
\(1 < x \leqslant 2\)120.2025
\(2 < x \leqslant 3\)150.1100
\(3 < x \leqslant 5\)130.1094
\(5 < x \leqslant 10\)90.0874
\(x > 10\)60.0532
  1. Examine the fit of this model to the data at the \(5 \%\) level of significance. You are given that the median length between successive flaws is 124 metres. At a later date the following random sample of ten lengths (in metres) between flaws is obtained. $$\begin{array} { l l l l l l l l l l } 239 & 77 & 179 & 221 & 100 & 312 & 52 & 129 & 236 & 42 \end{array}$$
  2. Test at the \(10 \%\) level of significance whether the median length may still be assumed to be 124 metres.
OCR MEI S3 2008 June Q1
1
  1. Sarah travels home from work each evening by bus; there is a bus every 20 minutes. The time at which Sarah arrives at the bus stop varies randomly in such a way that the probability density function of \(X\), the length of time in minutes she has to wait for the next bus, is given by $$\mathrm { f } ( x ) = k ( 20 - x ) \text { for } 0 \leqslant x \leqslant 20 \text {, where } k \text { is a constant. }$$
    1. Find \(k\). Sketch the graph of \(\mathrm { f } ( x )\) and use its shape to explain what can be deduced about how long Sarah has to wait.
    2. Find the cumulative distribution function of \(X\) and hence, or otherwise, find the probability that Sarah has to wait more than 10 minutes for the bus.
    3. Find the median length of time that Sarah has to wait.
    1. Define the term 'simple random sample'.
    2. Explain briefly how to carry out cluster sampling.
    3. A researcher wishes to investigate the attitudes of secondary school pupils to pollution. Explain why he might prefer to collect his data using a cluster sample rather than a simple random sample. An electronics company purchases two types of resistor from a manufacturer. The resistances of the resistors (in ohms) are known to be Normally distributed. Type A have a mean of 100 ohms and standard deviation of 1.9 ohms. Type B have a mean of 50 ohms and standard deviation of 1.3 ohms.
    4. Find the probability that the resistance of a randomly chosen resistor of type A is less than 103 ohms.
    5. Three resistors of type A are chosen at random. Find the probability that their total resistance is more than 306 ohms.
    6. One resistor of type A and one resistor of type B are chosen at random. Find the probability that their total resistance is more than 147 ohms.
    7. Find the probability that the total resistance of two randomly chosen type B resistors is within 3 ohms of one randomly chosen type A resistor.
    8. The manufacturer now offers type C resistors which are specified as having a mean resistance of 300 ohms. The resistances of a random sample of 100 resistors from the first batch supplied have sample mean 302.3 ohms and sample standard deviation 3.7 ohms. Find a \(95 \%\) confidence interval for the true mean resistance of the resistors in the batch. Hence explain whether the batch appears to be as specified.
OCR MEI S3 2008 June Q3
3
  1. A tea grower is testing two types of plant for the weight of tea they produce. A trial is set up in which each type of plant is grown at each of 8 sites. The total weight, in grams, of tea leaves harvested from each plant is measured and shown below.
    SiteABCDEFGH
    Type I225.2268.9303.6244.1230.6202.7242.1247.5
    Type II215.2242.1260.9241.7245.5204.7225.8236.0
    1. The grower intends to perform a \(t\) test to examine whether there is any difference in the mean yield of the two types of plant. State the hypotheses he should use and also any necessary assumption.
    2. Carry out the test using a \(5 \%\) significance level.
  2. The tea grower deals with many types of tea and employs tasters to rate them. The tasters do this by giving each tea a score out of 100. The tea grower wishes to compare the scores given by two of the tasters. Their scores for a random selection of 10 teas are as follows.
    TeaQRSTUVWXYZ
    Taster 169798563816585868977
    Taster 274759966756496949686
    Use a Wilcoxon test to examine, at the \(5 \%\) level of significance, whether it appears that, on the whole, the scores given to teas by these two tasters differ.
OCR MEI S3 2008 June Q4
4
  1. A researcher is investigating the feeding habits of bees. She sets up a feeding station some distance from a beehive and, over a long period of time, records the numbers of bees arriving each minute. For a random sample of 100 one-minute intervals she obtains the following results.
    Number of bees01234567\(\geqslant 8\)
    Number of intervals61619181714640
    1. Show that the sample mean is 3.1 and find the sample variance. Do these values support the possibility of a Poisson model for the number of bees arriving each minute? Explain your answer.
    2. Use the mean in part (i) to carry out a test of the goodness of fit of a Poisson model to the data.
  2. The researcher notes the length of time, in minutes, that each bee spends at the feeding station. The times spent are assumed to be Normally distributed. For a random sample of 10 bees, the mean is found to be 1.465 minutes and the standard deviation is 0.3288 minutes. Find a \(95 \%\) confidence interval for the overall mean time.
OCR S3 2014 June Q1
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
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
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
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
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
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
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
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
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*{\(\mathrm { OCR } ^ { \text {勾 } }\)}
OCR S3 2015 June Q1
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
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
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
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
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
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
    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
    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
    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
    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.