Questions S3 (597 questions)

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OCR S3 2009 June Q6
6 The function \(\mathrm { F } ( t )\) is defined as follows. $$\mathrm { F } ( t ) = \begin{cases} 0 & t < 0
\sin ^ { 4 } t & 0 \leqslant t \leqslant \frac { 1 } { 2 } \pi
1 & t > \frac { 1 } { 2 } \pi \end{cases}$$
  1. Verify that F is a (cumulative) distribution function. The continuous random variable \(T\) has (cumulative) distribution function \(\mathrm { F } ( t )\).
  2. Find the lower quartile of \(T\).
  3. Find the (cumulative) distribution function of \(Y\), where \(Y = \sin T\), and obtain the probability density function of \(Y\).
  4. Find the expected value of \(\frac { 1 } { Y ^ { 3 } + 2 Y ^ { 4 } }\).
OCR S3 2009 June Q7
7 In 1761, James Short took measurements of the parallax of the sun based on the transit of Venus. The mean and standard deviation of a random sample of 50 of these measurements are 8.592 and 0.7534 respectively, in suitable units.
  1. Show that if \(X \sim \mathrm {~N} \left( 8.592,0.7534 ^ { 2 } \right)\), then $$\mathrm { P } ( X \leqslant 8.084 ) = \mathrm { P } ( 8.084 < X \leqslant 8.592 ) = \mathrm { P } ( 8.592 < X \leqslant 9.100 ) = \mathrm { P } ( X > 9.100 ) = 0.25 \text {. }$$ The following table summarises the 50 measurements using these intervals.
    Measurement \(( x )\)\(x \leqslant 8.084\)\(8.084 < x \leqslant 8.592\)\(8.592 < x \leqslant 9.100\)\(x > 9.100\)
    Frequency822119
  2. Carry out a test, at the \(\frac { 1 } { 2 } \%\) significance level, of whether a normal distribution fits the data.
  3. Obtain a 99\% confidence interval for the mean of all similar parallax measurements.
OCR S3 2010 June Q1
1 The numbers of minor flaws that occur on reels of copper wire and reels of steel wire have Poisson distributions with means 0.21 per metre and 0.24 per metre respectively. One length of 5 m is cut from each reel.
  1. Calculate the probability that the total number of flaws on these two lengths of wire is at least 2 .
  2. State one assumption needed in the calculation.
OCR S3 2010 June Q2
2 A coffee machine used in a bar is claimed by the manager to dispense 170 ml of coffee per cup on average. A customer believes that the average amount of coffee dispensed is less than 170 ml . She measures the amount of coffee in 6 randomly chosen cups. The results, in ml , are as follows. $$\begin{array} { l l l l l l } 167 & 171 & 164 & 169 & 168 & 166 \end{array}$$ Assuming a relevant normal distribution, test the manager's claim at the 5\% significance level.
OCR S3 2010 June Q3
3 The developers of a shopping mall sponsored a study of the shopping habits of its users. Each of a random sample of 100 users was asked whether their weekend shopping was mainly on Saturday or mainly on Sunday. The results, classified according to whether the user lived in the city or the country, are shown in the table.
City dwellerCountry dweller
Saturday shopper2319
Sunday shopper4216
  1. Test, at the \(10 \%\) significance level, whether there is an association between the area in which shoppers live and the day on which they shop at the weekend.
  2. State, with a reason, whether the conclusion of the test would be different at the \(3 \%\) significance level.
OCR S3 2010 June Q4
4 Part of an ecological study involved measuring the heights of trees in a young forest. In order to obtain an estimate of the mean height of all the trees in the forest, a random sample of 70 trees was selected and their heights measured. These heights, \(x\) metres, are summarised by \(\Sigma x = 246.6\) and \(\Sigma x ^ { 2 } = 1183.65\). The mean height of all trees in the forest is denoted by \(\mu\) metres.
  1. Calculate a symmetric \(90 \%\) confidence interval for \(\mu\).
  2. A student was asked to interpret the interval and said,
    "If 100 independent \(90 \%\) confidence intervals were calculated then 90 of them would contain \(\mu\)." Explain briefly what is wrong with this statement.
  3. Four independent \(90 \%\) confidence intervals for \(\mu\) are obtained. Calculate the probability that at least three of the intervals contain \(\mu\).
OCR S3 2010 June Q5
5 A random variable \(X\) is believed to have (cumulative) distribution function given by $$\mathrm { F } ( x ) = \begin{cases} 0 & x < 0 ,
1 - \mathrm { e } ^ { - x ^ { 2 } } & x \geqslant 0 . \end{cases}$$ In order to test this, a random sample of 150 observations of \(X\) were taken, and their values are summarised in the following grouped frequency table.
Values\(0 \leqslant x < 0.5\)\(0.5 \leqslant x < 1\)\(1 \leqslant x < 1.5\)\(1.5 \leqslant x < 2\)\(x \geqslant 2\)
Frequency415032234
The expected frequencies, correct to 1 decimal place, corresponding to the above distribution, are 33.2, 61.6 and 39.4 respectively for the first 3 cells.
  1. Find the expected frequencies for the last 2 cells.
  2. Carry out a goodness of fit test at the \(2 \frac { 1 } { 2 } \%\) significance level.
OCR S3 2010 June Q6
6 It has been suggested that people who suffer anxiety when they are about to undergo surgery can have their anxiety reduced by listening to relaxation tapes. A study was carried out on 18 experimental subjects who listened to relaxation tapes, and 13 control subjects who listened to neutral tapes. After listening to the tapes, the subjects were given a test which produced an anxiety score, \(X\). Higher scores indicated higher anxiety. The results are summarised in the table.
Sample size\(\bar { x }\)\(\Sigma ( x - \bar { x } ) ^ { 2 }\)
Experimental subjects1832.161923.56
Control subjects1338.211147.58
  1. Use a two-sample \(t\)-test, at the \(5 \%\) significance level, to test whether anxiety is reduced by listening to relaxation tapes. State two necessary assumptions for the validity of your test.
  2. State why a test using a normal distribution would not have been appropriate.
OCR S3 2010 June Q7
7 The employees of a certain company have masses which are normally distributed. Female employees have a mean of 66.7 kg and standard deviation 9.3 kg , and male employees have a mean of 78.3 kg and standard deviation 8.5 kg . It may be assumed that all employees' masses are independent. On the ground floor 6 women and 9 men enter the empty staff lift for which it is stated that the maximum load is 1150 kg .
  1. Calculate the probability that the maximum load is exceeded. At the first floor all 15 passengers leave and 6 women, 8 men and an unknown employee enter.
  2. Assuming that the unknown employee is equally likely to be a woman or a man, calculate the probability that the maximum load is now exceeded.
OCR S3 2010 June Q8
8 The continuous random variable \(S\) has probability density function given by $$f ( s ) = \begin{cases} \frac { 8 } { 3 s ^ { 3 } } & 1 \leqslant s \leqslant 2
0 & \text { otherwise } \end{cases}$$ An isosceles triangle has equal sides of length \(S\), and the angle between them is \(30 ^ { \circ }\) (see diagram).
  1. Find the (cumulative) distribution function of the area \(X\) of the triangle, and hence show that the probability density function of \(X\) is \(\frac { 1 } { 3 x ^ { 2 } }\) over an interval to be stated.
  2. Find the median value of \(X\). {www.ocr.org.uk}) after the live examination series.
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OCR S3 2012 June Q1
1 A machine fills packets of flour whose nominal weights are 500 g . Each of a random sample of 100 packets was weighed and 14 packets weighed less than 500 g . The population proportion of packets that weigh less than 500 g is denoted by \(p\).
  1. Calculate an approximate \(95 \%\) confidence interval for \(p\).
  2. The weights of the packets, in grams, are normally distributed with mean \(\mu\) and variance 50 . Assuming that \(p = 0.14\), calculate the value of \(\mu\).
OCR S3 2012 June Q2
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
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
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
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
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
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
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
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
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
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
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
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
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
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.