Questions S3 (621 questions)

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OCR S3 2012 June Q2
9 marks Standard +0.3
2 Four pairs of randomly chosen twins were each given identical puzzles to solve. The times taken (in seconds) are shown in the following table.
Twin pair1234
Time for first-born46384449
Time for second-born40413746
Stating any necessary assumption, test at the \(10 \%\) significance level whether there is a difference between the population mean times of first-born and second-born twins.
OCR S3 2012 June Q3
6 marks Challenging +1.2
3 A charity raises money by sending letters asking for donations. Because of recent poor responses, the charity's fund-raiser, Anna, decides to alter the letter's appearance and designs two possible alternatives, one colourful and the other plain. She believes that the colourful letter will be more successful. Anna sends 60 colourful letters and 40 plain letters to 100 people randomly chosen from the charity's database. There were 39 positive responses to the colourful letter and 12 positive responses to the plain letter. The population proportions of positive responses to the colourful and plain letters are denoted by \(p _ { C }\) and \(p _ { P }\) respectively. Test the null hypothesis \(p _ { C } - p _ { P } = 0.15\) against the alternative hypothesis \(p _ { C } - p _ { P } > 0.15\) at the \(2 \frac { 1 } { 2 } \%\) significance level and state what Anna could report to her manager.
OCR S3 2012 June Q4
11 marks Standard +0.3
4 The time interval, \(T\) minutes, between consecutive stoppages of a particular grinding machine is regularly measured. \(T\) is normally distributed with mean \(\mu\).
24 randomly chosen values of \(T\) are summarised by $$\sum _ { i = 1 } ^ { 24 } t _ { i } = 348.0 \text { and } \sum _ { i = 1 } ^ { 24 } t _ { i } ^ { 2 } = 5195.5 .$$
  1. Calculate a symmetric \(95 \%\) confidence interval for \(\mu\).
  2. For the machine to be working acceptably, \(\mu\) should be at least 15.0 . Using a test at the 10\% significance level, decide whether the machine is working acceptably.
OCR S3 2012 June Q5
10 marks Moderate -0.3
5 The discrete random variables \(X\) and \(Y\) are independent with \(X \sim \mathrm {~B} \left( 32 , \frac { 1 } { 2 } \right)\) and \(Y \sim \operatorname { Po } ( 28 )\).
  1. Find the values of \(\mathrm { E } ( Y - X )\) and \(\operatorname { Var } ( Y - X )\).
  2. State, with justification, an approximate distribution for \(Y - X\).
  3. Hence find \(\mathrm { P } ( | Y - X | \geqslant 3 )\).
OCR S3 2012 June Q6
13 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{054e0081-afce-4a87-93f5-650dad40b313-3_508_611_262_719} The diagram shows the probability density function f of the continuous random variable \(T\), given by $$f ( t ) = \begin{cases} a t & 0 \leqslant t \leqslant 1 \\ a & 1 < t \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) is a constant.
  1. Find the value of \(a\).
  2. Obtain the cumulative distribution function of \(T\).
  3. Find the cumulative distribution of \(Y\), where \(Y = T ^ { \frac { 1 } { 2 } }\), and hence find the probability density function of \(Y\).
OCR S3 2012 June Q7
16 marks Standard +0.3
7 A study was carried out into whether patients suffering from a certain respiratory disorder would benefit from particular treatments. Each of 90 patients who agreed to take part was given one of three treatments \(A\), \(B\) or \(C\) as shown in the table.
Treatment\(A\)\(B\)\(C\)
Number in group312534
  1. It is claimed that each patient was equally likely to have been given any of the treatments. Test at the \(5 \%\) significance level whether the numbers given each treatment are consistent with this claim.
  2. After 3 months the numbers of patients showing improvement for treatments \(A , B\) and \(C\) were 14, 18 and 25 respectively. By setting up a \(2 \times 3\) contingency table, test whether the outcome is dependent on the treatment. Use a \(5 \%\) significance level.
  3. If one of the treatments is abandoned, explain briefly which it should be. \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE}
OCR S3 2013 June Q1
6 marks Standard +0.3
1 The blood-test procedure at a clinic is that a person arrives, takes a numbered ticket and waits for that number to be called. The waiting times between the numbers called have independent normal distributions with mean 3.5 minutes and standard deviation 0.9 minutes. My ticket is number 39 and as I take my ticket number 1 is being called, so that I have to wait for 38 numbers to be called. Find the probability that I will have to wait between 120 minutes and 140 minutes.
OCR S3 2013 June Q2
9 marks Standard +0.3
2 In order to estimate the total number of rabbits in a certain region, a random sample of 500 rabbits is captured, marked and released. After two days a random sample of 250 rabbits is captured and 24 are found to be marked. It may be assumed that there is no change in the population during the two days.
  1. Estimate the total number of rabbits in the region.
  2. Calculate an approximate \(95 \%\) confidence interval for the population proportion of marked rabbits.
  3. Using your answer to part (ii), estimate a 95\% confidence interval for the total number of rabbits in the region.
OCR S3 2013 June Q3
8 marks Standard +0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{c4adc528-ae3f-4ea7-9420-d3e1068a85fe-2_524_796_1105_623} The continuous random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} a x & 0 < x \leqslant 1 \\ b ( 2 - x ) ^ { 2 } & 1 < x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) and \(b\) are constants. The graph is shown in the above diagram.
  1. Find the values of \(a\) and \(b\).
  2. Find the value of \(\mathrm { E } \left( \frac { 1 } { X } \right)\).
OCR S3 2013 June Q4
8 marks Standard +0.3
4 A new computer was bought by a local council to search council records and was tested by an employee. She searched a random sample of 500 records and the sample mean search time was found to be 2.18 milliseconds and an unbiased estimate of variance was \(1.58 ^ { 2 }\) milliseconds \({ } ^ { 2 }\).
  1. Calculate a \(98 \%\) confidence interval for the population mean search time \(\mu\) milliseconds.
  2. It is required to obtain a sample mean time that differs from \(\mu\) by less than 0.05 milliseconds with probability 0.95 . Estimate the sample size required.
  3. State why it is unnecessary for the validity of your calculations that search time has a normal distribution.
OCR S3 2013 June Q5
13 marks Moderate -0.3
5 The continuous random variable \(Y\) has probability density function given by $$\mathrm { f } ( y ) = \begin{cases} \ln ( y ) & 1 \leqslant y \leqslant \mathrm { e } \\ 0 & \text { otherwise } \end{cases}$$
  1. Verify that this is a valid probability density function.
  2. Show that the (cumulative) distribution function of \(Y\) is given by $$\mathrm { F } ( y ) = \begin{cases} 0 & y < 1 \\ y \ln y - y + 1 & 1 \leqslant y \leqslant \mathrm { e } \\ 1 & \text { otherwise } \end{cases}$$
  3. Verify that the upper quartile of \(Y\) lies in the interval [2.45, 2.46].
  4. Find the (cumulative) distribution function of \(X\) where \(X = \ln Y\).
OCR S3 2013 June Q6
13 marks Standard +0.3
6 A random sample of 80 students who had all studied Biology, Chemistry and Art at a college was each asked which they enjoyed most. The results, classified according to gender, are given in the table.
Subject
\cline { 2 - 5 }BiologyChemistryArt
\cline { 2 - 5 } GenderMale13411
\cline { 2 - 5 }Female3787
\cline { 2 - 5 }
\cline { 2 - 5 }
It is required to carry out a test of independence between subject most enjoyed and gender at the \(2 \frac { 1 } { 2 } \%\) significance level.
  1. Calculate the expected values for the cells.
  2. Explain why it is necessary to combine cells, and choose a suitable combination.
  3. Carry out the test.
OCR S3 2013 June Q7
15 marks Standard +0.3
7 Two machines \(A\) and \(B\) both pack cartons in a factory. The mean packing times are compared by timing the packing of 10 randomly chosen cartons from machine \(A\) and 8 randomly chosen cartons from machine \(B\). The times, \(t\) seconds, taken to pack these cartons are summarised below.
Sample size\(\sum t\)\(\sum t ^ { 2 }\)
Machine \(A\)10221.44920.9
Machine \(B\)8199.24980.3
The packing times have independent normal distributions.
  1. Stating a necessary assumption, carry out a test, at the \(1 \%\) significance level, of whether the population mean packing times differ for the two machines.
  2. Find the largest possible value of the constant \(c\) for which there is evidence at the \(1 \%\) significance level that \(\mu _ { B } - \mu _ { A } > c\), where \(\mu _ { B }\) and \(\mu _ { A }\) denote the respective population mean packing times in seconds.
OCR S3 2016 June Q1
4 marks Standard +0.3
1 On a motorway, lorries pass an observation point independently and at random times. The mean number of lorries travelling north is 6 per minute and the mean number travelling south is 8 per minute. Find the probability that at least 16 lorries pass the observation point in a given 1 -minute period.
OCR S3 2016 June Q2
7 marks Standard +0.3
2 A random sample of 200 American voters were asked about which political party they supported and their attitude to a proposed new form of taxation. The voters' responses are summarised in the table. Attitude
\cline { 2 - 5 }In favourNeutralAgainst
\cline { 2 - 5 }Democrat581616
\cline { 2 - 5 } PartyIndependent25411
\cline { 2 - 5 }Republican172033
\cline { 2 - 5 }
\cline { 2 - 5 }
Carry out a \(\chi ^ { 2 }\) test, at the \(1 \%\) level of significance, to investigate whether there is an association between party supported and attitude to the proposed form of taxation.
OCR S3 2016 June Q3
8 marks Moderate -0.3
3
  1. A company packages butter. Of 50 randomly selected packs, 8 were found to have damaged wrappers. Find an approximate \(95 \%\) confidence interval for the proportion of packs with damaged wrappers.
  2. The mass of a pack has a normal distribution with standard deviation 8.5 g . In a random sample of 10 packs the masses, in g , are as follows. $$\begin{array} { l l l l l l l l l l } 220 & 225 & 218 & 223 & 224 & 220 & 229 & 228 & 226 & 228 \end{array}$$ Find a 99\% confidence interval for the mean mass of a pack.
OCR S3 2016 June Q4
9 marks Standard +0.3
4 A group of students were tested in geography before and after a fieldwork course. The marks of 10 randomly selected students are shown in the table.
Student\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)\(I\)\(J\)
Mark before fieldwork19848499591929495469
Mark after fieldwork23988388683328535888
  1. Use a suitable \(t\)-test, at the \(5 \%\) level of significance, to test whether the students' performance has improved.
  2. State the necessary assumption in applying the test.
OCR S3 2016 June Q5
11 marks Standard +0.8
5 The independent random variables \(X\) and \(Y\) have distributions \(\mathrm { N } \left( 30 , \sigma ^ { 2 } \right)\) and \(\mathrm { N } \left( 20 , \sigma ^ { 2 } \right)\) respectively. The random variable \(a X + b Y\), where \(a\) and \(b\) are constants, has the distribution \(\mathrm { N } \left( 410,130 \sigma ^ { 2 } \right)\).
  1. Given that \(a\) and \(b\) are integers, find the value of \(a\) and the value of \(b\).
  2. Given that \(\mathrm { P } ( X > Y ) = 0.966\), find \(\sigma ^ { 2 }\).
OCR S3 2016 June Q6
11 marks Challenging +1.2
6 The masses at birth, in kg, of random samples of babies were recorded for each of the years 1970 and 2010. The table shows the sample mean and an unbiased estimate of the population variance for each year.
YearNo. of babies
Sample
mean
Unbiased estimate of
population variance
19702853.3030.2043
20102603.3520.2323
  1. A researcher tests the null hypothesis that babies born in 2010 are 0.04 kg heavier, on average, than babies born in 1970, against the alternative hypothesis that they are more than 0.04 kg heavier on average. Show that, at the \(5 \%\) level of significance, the null hypothesis is not rejected.
  2. Another researcher chooses samples of equal size, \(n\), for the two years. Using the same hypothesis as in part (i), she finds that the null hypothesis is rejected at the \(5 \%\) level of significance. Assuming that the sample means and unbiased estimates of population variance for the two years are as given in the table, find the smallest possible value of \(n\).
OCR S3 2016 June Q7
12 marks Standard +0.8
7 A continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} a x ^ { 3 } & 0 \leqslant x \leqslant 1 \\ a x ^ { 2 } & 1 < x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) is a constant.
  1. Show that \(a = \frac { 12 } { 31 }\).
  2. Find \(\mathrm { E } ( X )\). It is thought that the time taken by a student to complete a task can be well modelled by \(X\). The times taken by 992 randomly chosen students are summarised in the table, together with some of the expected frequencies.
    Time\(0 \leqslant x < 0.5\)\(0.5 \leqslant x < 1\)\(1 \leqslant x < 1.5\)\(1.5 \leqslant x \leqslant 2\)
    Observed frequency892279613
    Expected frequency690
  3. Find the other expected frequencies and test, at the \(5 \%\) level of significance, whether the data can be well modelled by \(X\).
OCR S3 2016 June Q8
10 marks Challenging +1.2
8 The radius, \(R\), of a sphere is a random variable with a continuous uniform distribution between 0 and 10 .
  1. Find the cumulative distribution function and probability density function of \(A\), the surface area of the sphere.
  2. Find \(\mathrm { P } ( \mathrm { A } \leqslant 200 \pi )\). \section*{END OF QUESTION PAPER}
OCR MEI S3 2009 January Q1
18 marks Standard +0.3
1
  1. A continuous random variable \(X\) has probability density function $$\mathrm { f } ( x ) = \lambda x ^ { c } , \quad 0 \leqslant x \leqslant 1 ,$$ where \(c\) is a constant and the parameter \(\lambda\) is greater than 1 .
    1. Find \(c\) in terms of \(\lambda\).
    2. Find \(\mathrm { E } ( X )\) in terms of \(\lambda\).
    3. Show that \(\operatorname { Var } ( X ) = \frac { \lambda } { ( \lambda + 2 ) ( \lambda + 1 ) ^ { 2 } }\).
  2. Every day, Godfrey does a puzzle from the newspaper and records the time taken in minutes. Last year, his median time was 32 minutes. His times for a random sample of 12 puzzles this year are as follows. $$\begin{array} { l l l l l l l l l l l l } 40 & 20 & 18 & 11 & 47 & 36 & 38 & 35 & 22 & 14 & 12 & 21 \end{array}$$ Use an appropriate test, with a 5\% significance level, to examine whether Godfrey's times this year have decreased on the whole.
OCR MEI S3 2009 January Q2
18 marks Standard +0.3
2 A factory manufactures paperweights consisting of glass mounted on a wooden base. The volume of glass, in \(\mathrm { cm } ^ { 3 }\), in a paperweight has a Normal distribution with mean 56.5 and standard deviation 2.9. The volume of wood, in \(\mathrm { cm } ^ { 3 }\), also has a Normal distribution with mean 38.4 and standard deviation 1.1. These volumes are independent of each other. For the purpose of quality control, paperweights for testing are chosen at random from the factory's output.
  1. Find the probability that the volume of glass in a randomly chosen paperweight is less than \(60 \mathrm {~cm} ^ { 3 }\).
  2. Find the probability that the total volume of a randomly chosen paperweight is more than \(100 \mathrm {~cm} ^ { 3 }\). The glass has a mass of 3.1 grams per \(\mathrm { cm } ^ { 3 }\) and the wood has a mass of 0.8 grams per \(\mathrm { cm } ^ { 3 }\).
  3. Find the probability that the total mass of a randomly chosen paperweight is between 200 and 220 grams.
  4. The factory manager introduces some modifications intended to reduce the mean mass of the paperweights to 200 grams or less. The variance is also affected but not the Normality. Subsequently, for a random sample of 10 paperweights, the sample mean mass is 205.6 grams and the sample standard deviation is 8.51 grams. Is there evidence, at the \(5 \%\) level of significance, that the intended reduction of the mean mass has not been achieved?
OCR MEI S3 2009 January Q3
18 marks Standard +0.3
3 Pathology departments in hospitals routinely analyse blood specimens. Ideally the analysis should be done while the specimens are fresh to avoid any deterioration, but this is not always possible. A researcher decides to study the effect of freezing specimens for later analysis by measuring the concentrations of a particular hormone before and after freezing. He collects and divides a sample of 15 specimens. One half of each specimen is analysed immediately, the other half is frozen and analysed a month later. The concentrations of the particular hormone (in suitable units) are as follows.
Immediately15.2113.3615.9721.0712.8210.8011.5012.05
After freezing15.9610.6513.3815.0012.1112.6512.488.49
Immediately10.9018.4813.4313.1616.6214.9117.08
After freezing9.1315.5311.848.9916.2414.0316.13
A \(t\) test is to be used in order to see if, on average, there is a reduction in hormone concentration as a result of being frozen.
  1. Explain why a paired test is appropriate in this situation.
  2. State the hypotheses that should be used, together with any necessary assumptions.
  3. Carry out the test using a \(1 \%\) significance level.
  4. A \(p \%\) confidence interval for the true mean reduction in hormone concentration is found to be ( \(0.4869,2.8131\) ). Determine the value of \(p\).
OCR MEI S3 2009 January Q4
18 marks Standard +0.3
4
  1. Explain the meaning of 'opportunity sampling'. Give one reason why it might be used and state one disadvantage of using it. A market researcher is conducting an 'on-street' survey in a busy city centre, for which he needs to stop and interview 100 people. For each interview the researcher counts the number of people he has to ask until one agrees to be interviewed. The data collected are as follows.
    No. of people asked1234567 or more
    Frequency261917131186
    A model for these data is proposed as follows, where \(p\) (assumed constant throughout) is the probability that a person asked agrees to be interviewed, and \(q = 1 - p\).
    No. of people asked1234567 or more
    Probability\(p\)\(p q\)\(p q ^ { 2 }\)\(p q ^ { 3 }\)\(p q ^ { 4 }\)\(p q ^ { 5 }\)\(q ^ { 6 }\)
  2. Verify that these probabilities add to 1 whatever the value of \(p\).
  3. Initially it is thought that on average 1 in 4 people asked agree to be interviewed. Test at the \(10 \%\) level of significance whether it is reasonable to suppose that the model applies with \(p = 0.25\).
  4. Later an estimate of \(p\) obtained from the data is used in the analysis. The value of the test statistic (with no combining of cells) is found to be 9.124 . What is the outcome of this new test? Comment on your answer in relation to the outcome of the test in part (iii).