Questions — AQA S2 (141 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 PURE S1 S2 S3 S4 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
Sort by: Default | Easiest first | Hardest first
AQA S2 2015 June Q3
10 marks Moderate -0.3
3 A machine fills bags with frozen peas. Measurements taken over several weeks have shown that the standard deviation of the weights of the filled bags of peas has been 2.2 grams. Following maintenance on the machine, a quality control inspector selected 8 bags of peas. The weights, in grams, of the bags were $$\begin{array} { l l l l l l l l } 910.4 & 908.7 & 907.2 & 913.2 & 905.6 & 911.1 & 909.5 & 907.9 \end{array}$$ It may be assumed that the bags constitute a random sample from a normal distribution.
  1. Giving the limits to four significant figures, calculate a 95\% confidence interval for the mean weight of a bag of frozen peas filled by the machine following the maintenance:
    1. assuming that the standard deviation of the weights of the bags of peas is known to be 2.2 grams;
    2. assuming that the standard deviation of the weights of the bags of peas may no longer be 2.2 grams.
  2. The weight printed on the bags of peas is 907 grams. One of the inspector's concerns is that bags should not be underweight. Make two comments about this concern with regard to the data and your calculated confidence intervals.
    [0pt] [2 marks]
AQA S2 2015 June Q4
11 marks Standard +0.3
4 Wellgrove village has a main road running through it that has a 40 mph speed limit. The villagers were concerned that many vehicles travelled too fast through the village, and so they set up a device for measuring the speed of vehicles on this main road. This device indicated that the mean speed of vehicles travelling through Wellgrove was 44.1 mph . In an attempt to reduce the mean speed of vehicles travelling through Wellgrove, life-size photographs of a police officer were erected next to the road on the approaches to the village. The speed, \(X \mathrm { mph }\), of a sample of 100 vehicles was then measured and the following data obtained. $$\sum x = 4327.0 \quad \sum ( x - \bar { x } ) ^ { 2 } = 925.71$$
  1. State an assumption that must be made about the sample in order to carry out a hypothesis test to investigate whether the desired reduction in mean speed had occurred.
  2. Given that the assumption that you stated in part (a) is valid, carry out such a test, using the \(1 \%\) level of significance.
  3. Explain, in the context of this question, the meaning of:
    1. a Type I error;
    2. a Type II error.
      [0pt] [2 marks]
AQA S2 2015 June Q5
10 marks Standard +0.3
5 In a particular town, a survey was conducted on a sample of 200 residents aged 41 years to 50 years. The survey questioned these residents to discover the age at which they had left full-time education and the greatest rate of income tax that they were paying at the time of the survey. The summarised data obtained from the survey are shown in the table.
\multirow{2}{*}{Greatest rate of income tax paid}Age when leaving education (years)\multirow[b]{2}{*}{Total}
16 or less17 or 1819 or more
Zero323439
Basic1021217131
Higher175830
Total1512029200
  1. Use a \(\chi ^ { 2 }\)-test, at the \(5 \%\) level of significance, to investigate whether there is an association between age when leaving education and greatest rate of income tax paid.
  2. It is believed that residents of this town who had left education at a later age were more likely to be paying the higher rate of income tax. Comment on this belief.
    [0pt] [1 mark]
AQA S2 2015 June Q6
12 marks Moderate -0.3
6 The continuous random variable \(X\) has the cumulative distribution function $$\mathrm { F } ( x ) = \begin{cases} 0 & x < 0 \\ \frac { 1 } { 2 } x - \frac { 1 } { 16 } x ^ { 2 } & 0 \leqslant x \leqslant 4 \\ 1 & x > 4 \end{cases}$$
  1. Find the probability that \(X\) lies between 0.4 and 0.8 .
  2. Show that the probability density function, \(\mathrm { f } ( x )\), of \(X\) is given by $$f ( x ) = \begin{cases} \frac { 1 } { 2 } - \frac { 1 } { 8 } x & 0 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$
    1. Find the value of \(\mathrm { E } ( X )\).
    2. Show that \(\operatorname { Var } ( X ) = \frac { 8 } { 9 }\).
  4. The continuous random variable \(Y\) is defined by $$Y = 3 X - 2$$ Find the values of \(\mathrm { E } ( Y )\) and \(\operatorname { Var } ( Y )\).
AQA S2 2015 June Q7
15 marks Standard +0.3
7 Each week, a newsagent stocks 5 copies of the magazine Statistics Weekly. A regular customer always buys one copy. The demand for additional copies may be modelled by a Poisson distribution with mean 2. The number of copies sold in a week, \(X\), has the probability distribution shown in the table, where probabilities are stated correct to three decimal places.
\(\boldsymbol { x }\)12345
\(\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )\)0.1350.2710.271\(a\)\(b\)
  1. Show that, correct to three decimal places, the values of \(a\) and \(b\) are 0.180 and 0.143 respectively.
  2. Find the values of \(\mathrm { E } ( X )\) and \(\mathrm { E } \left( X ^ { 2 } \right)\), showing the calculations needed to obtain these values, and hence calculate the standard deviation of \(X\).
  3. The newsagent makes a profit of \(\pounds 1\) on each copy of Statistics Weekly that is sold and loses 50 p on each copy that is not sold. Find the mean weekly profit for the newsagent from sales of this magazine.
  4. Assuming that the weekly demand remains the same, show that the mean weekly profit from sales of Statistics Weekly will be greater if the newsagent stocks only 4 copies.
    [0pt] [5 marks]
    \includegraphics[max width=\textwidth, alt={}]{6cdf244b-168a-4be5-8ef8-8125daae8608-24_2488_1728_219_141}
AQA S2 2011 June Q3
10 marks Standard +0.3
  1. State the null hypothesis that Emily used.
  2. Find the value of the test statistic, \(X ^ { 2 }\), giving your answer to one decimal place.
  3. State, in context, the conclusion that Emily should reach based on the results of her \(\chi ^ { 2 }\) test.
  4. Make one comment on the GCSE performances of 16-year-old students attending Bailey Language School.
  5. Emily's friend, Joanna, used the same data to correctly conduct a \(\chi ^ { 2 }\) test using the \(10 \%\) level of significance. State, with justification, the conclusion that Joanna should reach.
AQA S2 2009 January Q1
11 marks Standard +0.3
1 Fortune High School gave its students a wider choice of subjects to study. The table shows the number of students, of each gender, who chose to study each of the additional subjects during the school year 2007/08.
\cline { 2 - 5 } \multicolumn{1}{c|}{}Bulgarian
Climate
Change
FinancePolish
Male7312540
Female2242219
Assuming that these data form a random sample, use a \(\chi ^ { 2 }\) test, at the \(10 \%\) level of significance, to test whether the choice of these subjects is independent of gender.
(11 marks)
AQA S2 2009 January Q2
9 marks Standard +0.3
2 A group of estate agents in a particular area claimed that, after the introduction of a new search procedure at the Land Registry, the mean completion time for the purchase of a house in the area had not changed from 8 weeks.
  1. A random sample of 9 house purchases in the area revealed that their completion times, in weeks, were as follows. $$\begin{array} { l l l l l l l l l } 6 & 7 & 10 & 12 & 9 & 11 & 7 & 8 & 14 \end{array}$$ Assuming that completion times in the area are normally distributed with standard deviation 2.5 weeks, test, at the \(5 \%\) level of significance, the group's claim. (7 marks)
  2. It was subsequently discovered that, after the introduction of the new search procedure at the Land Registry, the mean completion time for the purchase of a house in the area remained at 8 weeks. Indicate whether a Type I error, a Type II error or neither has occurred in carrying out your hypothesis test in part (a). Give a reason for your answer.
    (2 marks)
AQA S2 2009 January Q3
14 marks Moderate -0.3
3 Joe owns two garages, Acefit and Bestjob, each specialising in the fitting of the latest satellite navigation device. The daily demand, \(X\), for the device at Acefit garage may be modelled by a Poisson distribution with mean 3.6. The daily demand, \(Y\), for the device at Bestjob garage may be modelled by a Poisson distribution with mean 4.4.
  1. Calculate:
    1. \(\mathrm { P } ( X \leqslant 3 )\);
    2. \(\quad \mathrm { P } ( Y = 5 )\).
  2. The total daily demand for the device at Joe's two garages is denoted by \(T\).
    1. Write down the distribution of \(T\), stating any assumption that you make.
    2. Determine \(\mathrm { P } ( 6 < T < 12 )\).
    3. Calculate the probability that the total demand for the device will exceed 14 on each of two consecutive days. Give your answer to one significant figure.
    4. Determine the minimum number of devices that Joe should have in stock if he is to meet his total demand on at least \(99 \%\) of days.
AQA S2 2009 January Q4
6 marks Moderate -0.3
4 The continuous random variable \(X\) has the cumulative distribution function $$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x < - c \\ \frac { x + c } { 4 c } & - c \leqslant x \leqslant 3 c \\ 1 & x > 3 c \end{array} \right.$$ where \(c\) is a positive constant.
  1. Determine \(\mathrm { P } \left( - \frac { 3 c } { 4 } < X < \frac { 3 c } { 4 } \right)\).
  2. Show that the probability density function, \(\mathrm { f } ( x )\), of \(X\) is $$f ( x ) = \left\{ \begin{array} { c c } \frac { 1 } { 4 c } & - c \leqslant x \leqslant 3 c \\ 0 & \text { otherwise } \end{array} \right.$$
  3. Hence, or otherwise, find expressions, in terms of \(c\), for:
    1. \(\mathrm { E } ( X )\);
    2. \(\operatorname { Var } ( X )\).
AQA S2 2009 January Q5
13 marks Standard +0.3
5 Jane, who supplies fruit to a jam manufacturer, knows that the weight of fruit in boxes that she sends to the manufacturer can be modelled by a normal distribution with unknown mean, \(\mu\) grams, and unknown standard deviation, \(\sigma\) grams. Jane selects a random sample of 16 boxes and, using the \(t\)-distribution, calculates correctly that a \(98 \%\) confidence interval for \(\mu\) is \(( 70.65,80.35 )\).
    1. Show that the sample mean is 75.5 grams.
    2. Find the width of the confidence interval.
    3. Calculate an estimate of the standard error of the mean.
    4. Hence, or otherwise, show that an unbiased estimate of \(\sigma ^ { 2 }\) is 55.6 , correct to three significant figures.
  2. Jane decides that the width of the \(98 \%\) confidence interval is too large. Construct a \(95 \%\) confidence interval for \(\mu\), based on her sample of 16 boxes.
  3. Jane is informed that the manufacturer would prefer the confidence interval to have a width of at most 5 grams.
    1. Write down a confidence interval for \(\mu\), again based on Jane's sample of 16 boxes, which has a width of 5 grams.
    2. Determine the percentage confidence level for your interval in part (c)(i).
AQA S2 2009 January Q6
10 marks Standard +0.3
6 A small supermarket has a total of four checkouts, at least one of which is always staffed. The probability distribution for \(R\), the number of checkouts that are staffed at any given time, is $$\mathrm { P } ( R = r ) = \left\{ \begin{array} { c l } \frac { 2 } { 3 } \left( \frac { 1 } { 3 } \right) ^ { r - 1 } & r = 1,2,3 \\ k & r = 4 \end{array} \right.$$
  1. Show that \(k = \frac { 1 } { 27 }\).
  2. Find the probability that, at any given time, there will be at least 3 checkouts that are staffed.
  3. It is suggested that the total number of customers, \(C\), that can be served at the checkouts per hour may be modelled by $$C = 27 R + 5$$ Find:
    1. \(\mathrm { E } ( C )\);
    2. the standard deviation of \(C\).
AQA S2 2009 January Q7
12 marks Standard +0.3
7 The continuous random variable \(X\) has the probability density function given by $$f ( x ) = \left\{ \begin{array} { c c } \frac { 1 } { 16 } x ^ { 3 } & 0 \leqslant x \leqslant 2 \\ \frac { 1 } { 6 } ( 5 - x ) & 2 \leqslant x \leqslant 5 \\ 0 & \text { otherwise } \end{array} \right.$$
  1. Sketch the graph of f.
  2. Prove that the cumulative distribution function of \(X\) for \(2 \leqslant x \leqslant 5\) can be written in the form $$\mathrm { F } ( x ) = 1 - \frac { 1 } { 12 } ( 5 - x ) ^ { 2 }$$
  3. Hence, or otherwise, determine \(\mathrm { P } ( X \geqslant 3 \mid X \leqslant 4 )\).
AQA S2 2007 June Q1
10 marks Standard +0.3
1 Two groups of patients, suffering from the same medical condition, took part in a clinical trial of a new drug. One of the groups was given the drug whilst the other group was given a placebo, a drug that has no physical effect on their medical condition. The table shows the number of patients in each group and whether or not their condition improved.
\cline { 2 - 3 } \multicolumn{1}{c|}{}PlaceboDrug
Condition improved2046
Condition did not improve5529
Conduct a \(\chi ^ { 2 }\) test, at the \(5 \%\) level of significance, to determine whether the condition of the patients at the conclusion of the trial is associated with the treatment that they were given.
(10 marks)
AQA S2 2007 June Q2
10 marks Moderate -0.8
2 The number of telephone calls per day, \(X\), received by Candice may be modelled by a Poisson distribution with mean 3.5. The number of e-mails per day, \(Y\), received by Candice may be modelled by a Poisson distribution with mean 6.0.
  1. For any particular day, find:
    1. \(\mathrm { P } ( X = 3 )\);
    2. \(\quad \mathrm { P } ( Y \geqslant 5 )\).
    1. Write down the distribution of \(T\), the total number of telephone calls and e-mails per day received by Candice.
    2. Determine \(\mathrm { P } ( 7 \leqslant T \leqslant 10 )\).
    3. Hence calculate the probability that, on each of three consecutive days, Candice will receive a total of at least 7 but at most 10 telephone calls and e-mails.
      (2 marks)
AQA S2 2007 June Q3
8 marks Standard +0.3
3 David is the professional coach at the golf club where Becki is a member. He claims that, after having a series of lessons with him, the mean number of putts that Becki takes per round of golf will reduce from her present mean of 36 . After having the series of lessons with David, Becki decides to investigate his claim.
She therefore records, for each of a random sample of 50 rounds of golf, the number of putts, \(x\), that she takes to complete the round. Her results are summarised below, where \(\bar { x }\) denotes the sample mean. $$\sum x = 1730 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 784$$ Using a \(z\)-test and the \(1 \%\) level of significance, investigate David's claim.
AQA S2 2007 June Q4
7 marks Moderate -0.8
4 Students are each asked to measure the distance between two points to the nearest tenth of a metre.
  1. Given that the rounding error, \(X\) metres, in these measurements has a rectangular distribution, explain why its probability density function is $$f ( x ) = \left\{ \begin{array} { c c } 10 & - 0.05 < x \leqslant 0.05 \\ 0 & \text { otherwise } \end{array} \right.$$
  2. Calculate \(\mathrm { P } ( - 0.01 < X < 0.02 )\).
  3. Find the mean and the standard deviation of \(X\).
AQA S2 2007 June Q5
10 marks Standard +0.3
5 Members of a residents' association are concerned about the speeds of cars travelling through their village. They decide to record the speed, in mph , of each of a random sample of 10 cars travelling through their village, with the following results: $$\begin{array} { l l l l l l l l l l } 33 & 27 & 34 & 30 & 48 & 35 & 34 & 33 & 43 & 39 \end{array}$$
  1. Construct a \(99 \%\) confidence interval for \(\mu\), the mean speed of cars travelling through the village, stating any assumption that you make.
  2. Comment on the claim that a 30 mph speed limit is being adhered to by most motorists.
    (3 marks)
AQA S2 2007 June Q6
12 marks Standard +0.8
6 The continuous random variable \(X\) has the probability density function given by $$f ( x ) = \left\{ \begin{array} { c c } 3 x ^ { 2 } & 0 < x \leqslant 1 \\ 0 & \text { otherwise } \end{array} \right.$$
  1. Determine:
    1. \(\mathrm { E } \left( \frac { 1 } { X } \right)\);
      (3 marks)
    2. \(\operatorname { Var } \left( \frac { 1 } { X } \right)\).
  2. Hence, or otherwise, find the mean and the variance of \(\left( \frac { 5 + 2 X } { X } \right)\).
AQA S2 2007 June Q7
7 marks Moderate -0.8
7 On a multiple choice examination paper, each question has five alternative answers given, only one of which is correct. For each question, candidates gain 4 marks for a correct answer but lose 1 mark for an incorrect answer.
  1. James guesses the answer to each question.
    1. Copy and complete the following table for the probability distribution of \(X\), the number of marks obtained by James for each question.
      \(\boldsymbol { x }\)4- 1
      \(\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )\)
    2. Hence find \(\mathrm { E } ( X )\).
  2. Karen is able to eliminate two of the incorrect answers from the five alternative answers given for each question before guessing the answer from those remaining. Given that the examination paper contains 24 questions, calculate Karen's expected total mark.
AQA S2 2007 June Q8
11 marks Moderate -0.3
8 A jam producer claims that the mean weight of jam in a jar is 230 grams.
  1. A random sample of 8 jars is selected and the weight of jam in each jar is determined. The results, in grams, are $$\begin{array} { l l l l l l l l } 220 & 228 & 232 & 219 & 221 & 223 & 230 & 229 \end{array}$$ Assuming that the weight of jam in a jar is normally distributed, test, at the \(5 \%\) level of significance, the jam producer's claim.
  2. It is later discovered that the mean weight of jam in a jar is indeed 230 grams. Indicate whether a Type I error, a Type II error or neither has occurred in carrying out the hypothesis test in part (a). Give a reason for your answer.
AQA S2 2009 June Q1
6 marks Moderate -0.3
1 A machine fills bottles with bleach. The volume, in millilitres, of bleach dispensed by the machine into a bottle may be modelled by a normal distribution with mean \(\mu\) and standard deviation 8 . A recent inspection indicated that the value of \(\mu\) was 768 . Yvonne, the machine's operator, claims that this value has not subsequently changed. Zara, the quality control supervisor, records the volume of bleach in each of a random sample of 18 bottles filled by the machine and calculates their mean to be 764.8 ml . Test, at the \(5 \%\) level of significance, Yvonne's claim that the mean volume of bleach dispensed by the machine has not changed from 768 ml .
AQA S2 2009 June Q2
14 marks Moderate -0.3
2 John works from home. The number of business letters, \(X\), that he receives on a weekday may be modelled by a Poisson distribution with mean 5.0. The number of private letters, \(Y\), that he receives on a weekday may be modelled by a Poisson distribution with mean 1.5.
  1. Find, for a given weekday:
    1. \(\mathrm { P } ( X < 4 )\);
    2. \(\quad \mathrm { P } ( Y = 4 )\).
    1. Assuming that \(X\) and \(Y\) are independent random variables, determine the probability that, on a given weekday, John receives a total of more than 5 business and private letters.
    2. Hence calculate the probability that John receives a total of more than 5 business and private letters on at least 7 out of 8 given weekdays.
  3. The numbers of letters received by John's neighbour, Brenda, on 10 consecutive weekdays are $$\begin{array} { l l l l l l l l l l } 15 & 8 & 14 & 7 & 6 & 8 & 2 & 8 & 9 & 3 \end{array}$$
    1. Calculate the mean and the variance of these data.
    2. State, giving a reason based on your answers to part (c)(i), whether or not a Poisson distribution might provide a suitable model for the number of letters received by Brenda on a weekday.
AQA S2 2009 June Q3
12 marks Standard +0.3
3 A sample survey, conducted to determine the attitudes of residents to a proposed reorganisation of local schools, gave the following results.
Against reorganisationNot against reorganisation
\multirow{5}{*}{Age of resident}16-1792
18-211710
22-4911590
50-654134
Over 6534
Use a \(\chi ^ { 2 }\) test, at the \(5 \%\) level of significance, to determine whether there is an association between the ages of residents and their attitudes to the proposed reorganisation of local schools.
AQA S2 2009 June Q4
12 marks Standard +0.3
4 The continuous random variable \(X\) has probability density function given by $$f ( x ) = \left\{ \begin{array} { c c } \frac { 1 } { 2 } & 0 \leqslant x \leqslant 1 \\ \frac { 3 - x } { 4 } & 1 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{array} \right.$$
  1. Sketch the graph of f.
  2. Explain why the value of \(\eta\), the median of \(X\), is 1 .
  3. Show that the value of \(\mu\), the mean of \(X\), is \(\frac { 13 } { 12 }\).
  4. Find \(\mathrm { P } ( X < 3 \mu - \eta )\).