Questions S3 (597 questions)

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OCR S3 2009 January Q4
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
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
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
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
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
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
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
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
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
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
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.
OCR S3 2010 January Q7
7 A chef wished to ascertain her customers' preference for certain vegetables. She asked a random sample of 120 customers for their preferred vegetable from asparagus, broad beans and cauliflower. The responses, classified according to the gender of the customer, are shown in the table.
  1. Test, at the \(5 \%\) significance level, whether vegetable preference and gender are independent.
  2. Determine whether, at the \(10 \%\) significance level, the vegetables are equally preferred.
OCR S3 2013 January Q1
1 The independent random variables \(X\) and \(Y\) have the distributions \(\mathrm { N } \left( 10 , \sigma ^ { 2 } \right)\) and \(\operatorname { Po } ( 2 )\) respectively. The random variable \(S\) is given by \(S = 5 X - 2 Y + c\), where \(c\) is a constant.
It is given that \(\mathrm { E } ( S ) = \operatorname { Var } ( S ) = 408\).
  1. Find the value of \(c\) and show that \(\sigma ^ { 2 } = 16\).
  2. Find \(\mathrm { P } ( X \geqslant \mathrm { E } ( Y ) )\).
OCR S3 2013 January Q2
2 A new running track has been developed and part of the testing procedure involves 7 randomly chosen athletes. They each run 100 m on both the old and new tracks.
The results are as follows.
Athlete1234567
Time on old track \(( s )\)12.210.311.513.011.811.711.9
Time on new track \(( s )\)11.110.511.012.611.010.912.0
The population mean times on the old and new tracks are denoted by \(\mu _ { \mathrm { O } }\) seconds and \(\mu _ { \mathrm { N } }\) seconds respectively. Stating any necessary assumption, carry out a suitable \(t\)-test of the null hypothesis \(\mu _ { \mathrm { O } } - \mu _ { \mathrm { N } } = 0\) against the alternative hypothesis \(\mu _ { \mathrm { O } } - \mu _ { \mathrm { N } } > 0\). Use a \(2 \frac { 1 } { 2 } \%\) significance level .
OCR S3 2013 January Q3
3 Two reading schemes, \(A\) and \(B\), are compared by using them with a random sample of 9 five-year-old children. The children are divided into two groups, 5 allotted to scheme \(A\) and 4 to scheme \(B\), and the schemes are taught under similar conditions.
After one year the children are given the same test and their scores, \(x _ { A }\) and \(x _ { B }\), are summarised below. With the usual notation, $$\begin{aligned} & n _ { A } = 5 , \bar { x } _ { A } = 52.0 , \sum \left( x _ { A } - \bar { x } _ { A } \right) ^ { 2 } = 248 ,
& n _ { B } = 4 , \bar { x } _ { B } = 56.5 , \sum \left( x _ { B } - \bar { x } _ { B } \right) ^ { 2 } = 381 . \end{aligned}$$ It may be assumed that scores have normal distributions.
  1. Calculate an \(80 \%\) confidence interval for the difference in population mean scores for the two methods.
  2. State a further assumption required for the validity of the interval.
OCR S3 2013 January Q4
4 The continuous random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { 3 } { 2 } \sqrt { x } & 0 < x \leqslant 1
0 & \text { otherwise } \end{cases}$$ The random variable \(Y\) is given by \(Y = \frac { 1 } { \sqrt { X } }\).
  1. Find the (cumulative) distribution function of \(Y\), and hence show that its probability density function is given by $$\mathrm { g } ( y ) = \frac { 3 } { y ^ { 4 } }$$ for a set of values of \(y\) to be stated.
  2. Find the value of \(\mathrm { E } \left( Y ^ { 2 } \right)\).
OCR S3 2013 January Q5
5 A constitutional change was proposed for a Golf Club with a large membership. This was to be voted on at the Annual General Meeting. A month before this meeting the secretary asked a random sample of 50 members for their opinions. Out of the 50 members \(70 \%\) said they approved.
  1. Calculate an approximate \(90 \%\) confidence interval for the proportion \(p\) of all members who would approve the proposal.
  2. Explain what is meant by a \(90 \%\) confidence interval in this context.
  3. Nearer the date of the meeting the secretary asked a random sample of \(n\) members, and, as before, \(70 \%\) said they approved. This time the secretary calculated an approximate \(99 \%\) confidence interval for \(p\). It is given that the confidence interval does not include 0.85 . Find the smallest possible value of \(n\).
OCR S3 2013 January Q6
6 A large population of plants consists of five species \(A , B , C , D\) and \(E\) in the proportions \(p _ { A } , p _ { B } , p _ { C } , p _ { D }\) and \(p _ { E }\) respectively. A random sample of 120 plants consisted of \(23,14,24,27\) and 32 of \(A , B , C , D\) and \(E\) respectively. Carry out a test at the \(10 \%\) significance level of the null hypothesis that the proportions are \(p _ { \mathrm { A } } = p _ { \mathrm { B } } = 0.15 , p _ { \mathrm { C } } = p _ { \mathrm { D } } = 0.25\) and \(p _ { \mathrm { E } } = 0.2\).
OCR S3 2013 January Q7
7 The random variable \(X\) has distribution \(\mathrm { N } ( \mu , 1 )\). A random sample of 4 observations of \(X\) is taken. The sample mean is denoted by \(\bar { X }\).
  1. Find the value of the constant \(a\) for which ( \(\bar { X } - a , \bar { X } + a\) ) is a \(98 \%\) confidence interval for \(\mu\). The independent random variable \(Y\) has distribution \(\mathrm { N } ( \mu , 9 )\). A random sample of 16 observations of \(Y\) is taken. The sample mean is denoted by \(\bar { Y }\).
  2. Write down the distribution of \(\bar { X } - \bar { Y }\).
  3. A \(90 \%\) confidence interval for \(\mu\) based on \(\bar { Y }\) is given by ( \(\bar { Y } - 1.234 , \bar { Y } + 1.234\) ). Find the probability that this interval does not overlap with the interval in part (i).
OCR S3 2013 January Q8
8 After contracting a particular disease, patients from a hospital are advised to have their blood tested monthly for a year. In order to test whether patients comply with this advice the hospital management commissioned a survey of 100 patients. A hospital statistician selected the patients randomly from records and asked the patients whether or not they had complied with the advice. The results classified by gender are as follows.
Gender
\cline { 2 - 4 }FemaleMale
\cline { 2 - 4 } ComplyYes3430
\cline { 2 - 4 }No1125
\cline { 2 - 4 }
\cline { 2 - 4 }
  1. Test at the \(5 \%\) significance level whether compliance with the advice is independent of gender.
  2. A manager believed that a greater proportion of female patients than male patients comply with the advice. Carry out an appropriate test of proportions at the \(10 \%\) significance level.
OCR S3 2009 June Q1
1 A continuous random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { 2 x } { 5 } & 0 \leqslant x \leqslant 1
\frac { 2 } { 5 \sqrt { x } } & 1 < x \leqslant 4
0 & \text { otherwise } \end{cases}$$ Find
  1. \(\mathrm { E } ( X )\),
  2. \(\mathrm { P } ( X \geqslant \mathrm { E } ( X ) )\).
OCR S3 2009 June Q2
2 The number of bacteria in 1 ml of drug \(A\) has a Poisson distribution with mean 0.5. The number of the same bacteria in 1 ml of drug \(B\) has a Poisson distribution with mean 0.75 . A mixture of these drugs used to treat a particular disease consists of 1.4 ml of drug \(A\) and 1.2 ml of drug \(B\). Bacteria in the drugs will cause infection in a patient if 5 or more bacteria are injected.
  1. Calculate the probability that, in a sample of 20 patients treated with the mixture, infection will occur in no more than one patient.
  2. State an assumption required for the validity of the calculation.
OCR S3 2009 June Q3
3 A machine produces circular metal discs whose radii have a normal distribution with mean \(\mu \mathrm { cm }\). A random sample of five discs is selected and their radii, in cm, are as follows. $$\begin{array} { l l l l l } 6.47 & 6.52 & 6.46 & 6.47 & 6.51 \end{array}$$
  1. Calculate a \(95 \%\) confidence interval for \(\mu\).
  2. Hence state a 95\% confidence interval for the mean circumference of a disc.
OCR S3 2009 June Q4
4 In order to compare the difficulty of two Su Doku puzzles, two random samples of 40 fans were selected. One sample was given Puzzle 1 and the other sample was given Puzzle 2. Of those given Puzzle 1, 24 could solve it within ten minutes. Of those given Puzzle 2, 15 could solve it within ten minutes.
  1. Using proportions, test at the \(5 \%\) significance level whether there is a difference in the standard of difficulty of the two puzzles.
  2. The setter believed that Puzzle 2 was more difficult than Puzzle 1. Obtain the smallest significance level at which this belief is supported.
OCR S3 2009 June Q5
5 Each person in a random sample of 15 men and 17 women from a university campus was asked how many days in a month they took exercise. The numbers of days for men and women, \(x _ { M }\) and \(x _ { W }\) respectively, are summarised by $$\Sigma x _ { M } = 221 , \quad \Sigma x _ { M } ^ { 2 } = 3992 , \quad \Sigma x _ { W } = 276 , \quad \Sigma x _ { W } ^ { 2 } = 5538 .$$
  1. State conditions for the validity of a suitable test of the difference in the mean numbers of days for men and women on the campus.
  2. Given that these conditions hold, carry out the test at the \(5 \%\) significance level.
  3. If in fact the random sample was drawn entirely from the university Mathematics Department, state with a reason whether the validity of the test is in doubt.