Questions Further Unit 2 (35 questions)

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WJEC Further Unit 2 2018 June Q1
  1. The random variable \(X\) has the binomial distribution \(\mathrm { B } ( 12,0 \cdot 3 )\). The independent random variable \(Y\) has the Poisson distribution \(\mathrm { Po } ( 4 )\). Find
    1. \(\mathrm { E } ( X Y )\),
    2. \(\quad \operatorname { Var } ( X Y )\).
    3. The length of time a battery works, in tens of hours, is modelled by a random variable \(X\) with cumulative distribution function
    $$F ( x ) = \begin{cases} 0 & \text { for } x < 0
    \frac { x ^ { 3 } } { 432 } ( 8 - x ) & \text { for } 0 \leqslant x \leqslant 6
    1 & \text { for } x > 6 \end{cases}$$
  2. Find \(P ( X > 5 )\).
  3. A head torch uses three of these batteries. All three batteries must work for the torch to operate. Find the probability that the head torch will operate for more than 50 hours.
  4. Show that the upper quartile of the distribution lies between \(4 \cdot 5\) and \(4 \cdot 6\).
  5. Find \(f ( x )\), the probability density function for \(X\).
  6. Find the mean lifetime of the batteries in hours.
  7. The graph of \(f ( x )\) is given below.
    \includegraphics[max width=\textwidth, alt={}, center]{3cf25b54-d4a1-4d30-b632-3f6d3182a930-2_695_1463_1896_299}
WJEC Further Unit 2 2018 June Q3
3. A game at a school fete is played with a fair coin and a random number generator which generates random integers between 1 and 52 inclusive. It costs 50 pence to play the game. First, the player tosses the coin. If it lands on tails, the player loses. If it lands on heads, the player is allowed to generate a random number. If the number is 1 , the player wins \(\pounds 5\). If the number is between 2 and 13 inclusive, the player wins \(\pounds 1\). If the number is greater than 13 , the player loses.
  1. Find the probability distribution of the player's profit.
  2. Find the mean and standard deviation of the player's profit.
  3. Given that 200 people play the game, calculate
    1. the expected number of players who win some money,
    2. the expected profit for the fete.
WJEC Further Unit 2 2018 June Q4
4. On a Welsh television game show, contestants are asked to guess the weights of a random sample of seven cows. The game show judges want to investigate whether there is positive correlation between the actual weights and the estimated weights. The results are shown below for one contestant.
CowABCDEFG
Actual weight, kg61411057181001889770682
Estimated weight, kg70015008501400750900800
  1. Calculate Spearman's rank correlation coefficient for this data set.
  2. Stating your hypotheses clearly, determine whether or not there is evidence at the \(5 \%\) significance level of a positive association between the actual weights and the weights as estimated by this contestant.
  3. One of the game show judges says, "This contestant was good at guessing the weights of the cows." Comment on this statement.
WJEC Further Unit 2 2018 June Q5
5. A life insurance saleswoman investigates the number of policies she sells per day. The results for a random sample of 50 days are shown in the table below.
Number of
policies sold
0123456
Number of days229121591
She sees the same fixed number of clients each day. She would like to know whether the binomial distribution with parameters 6 and 0.6 is a suitable model for the number of policies she sells per day.
  1. State suitable hypotheses for a goodness of fit test.
  2. Here is part of the table for a \(\chi ^ { 2 }\) goodness of fit test on the data.
    Number of policies sold0123456
    Observed229121591
    Expected0.2051.8436.912\(d\)\(e\)9.3312.333
    1. Calculate the values of \(d\) and \(e\).
    2. Carry out the test using a 10\% level of significance and draw a conclusion in context.
  3. What do the parameters 6 and 0.6 mean in this context?
WJEC Further Unit 2 2018 June Q6
6. A student, considering options for the future, collects data on education and salary. The table below shows the highest level of education attained and the salary bracket of a random sample of 664 people.
Fewer than 5 GCSE5 or more GCSE3 A LevelsUniversity degreePost graduate qualificationTotal
Less than £200001832202810108
£20000 to £60000509511215550462
More than £600003222935594
Total7114916121865664
By conducting a chi-squared test for independence, the student investigates the relationship between the highest level of education attained and the salary earned.
  1. State the null and alternative hypotheses.
  2. The table below shows the expected values. Calculate the value of \(k\).
    Expected valuesFewer than 5 GCSE5 or more GCSE3 A LevelsUniversity degreePost graduate qualification
    Less than £20000\(k\)24.2326•1935.4610.57
    £20000 to £6000049.40103.67112.02151.68\(45 \cdot 23\)
    More than £6000010.0521.09\(22 \cdot 79\)30.869.20
  3. The following computer output is obtained. Calculate the values of \(m\) and \(n\).
    Chi Squared ContributionsFewer than 5 GCSE5 or more GCSE3 A LevelsUniversity degreePost graduate qualification
    Less than £200003.604530799\(m\)1.461651.56860.03098
    £20000 to £600000.0072727350.725354E-060.072640.50396
    More than £60000\(4 \cdot 946619863\)0.03897169081\(0 \cdot 55498\)\(n\)
    X-squared \(= 19 \cdot 61301 , d f = 8 , p\)-value \(= 0 \cdot 0119\)
    1. Without carrying out any further calculations, explain how X-squared \(= 19 \cdot 61301\) (the \(\chi ^ { 2 }\) test statistic) was calculated.
    2. Comment on the values in the "Fewer than 5 GCSE" column of the table in part (c).
  5. The student says that the highest levels of education lead to the highest paying jobs. Comment on the accuracy of the student's statement.
WJEC Further Unit 2 2018 June Q7
7. A university professor conducted some research into factors that affect job satisfaction. The four factors considered were Interesting work, Good wages, Job security and Appreciation of work done. The professor interviewed workers at 14 different companies and asked them to rate their companies on each of the factors. The workers' ratings were averaged to give each company a score out of 5 on each factor. Each company was also given a score out of 100 for Job satisfaction.
The following graph shows the part of the research concerning Job Satisfaction versus Interesting work. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Job satisfaction versus Interesting work} \includegraphics[alt={},max width=\textwidth]{3cf25b54-d4a1-4d30-b632-3f6d3182a930-6_647_1330_959_340}
\end{figure}
  1. Calculate the equation of the least squares regression line of Job satisfaction ( \(y\) ) on Interesting work ( \(x\) ), given the following summary statistics. $$\begin{array} { l l l } \sum x = 46 \cdot 2 , & \sum y = 898 , & S _ { x x } = 3 \cdot 48
    S _ { x y } = 49 \cdot 45 , & S _ { y y } = 1437 \cdot 714 , & n = 14 \end{array}$$
  2. Give two reasons why it would be inappropriate for the professor to use this equation to calculate the score for Interesting work from a Job satisfaction score of 90.
WJEC Further Unit 2 2019 June Q1
  1. (a) Sketch a scatter diagram of a dataset for which Spearman's rank correlation coefficient is + 1 , but the product moment correlation coefficient is less than 1 .
Two judges were judging cheese at the UK Cheese Festival. There were 8 blue cheeses in a particular category. The rankings are shown below.
CheeseABCDEFGH
Judge 115876432
Judge 213852467
(b) Calculate Spearman's rank correlation coefficient for this dataset.
(c) By sketching a scatter diagram of the rankings, or otherwise, comment on the extent to which the judges agree.
WJEC Further Unit 2 2019 June Q2
2. The probability of winning a certain game at a funfair is \(p\). Aman plays the game 5 times and Boaz plays the game 8 times. The independent random variables \(X\) and \(Y\) denote the number of wins for Aman and Boaz respectively.
  1. Given that \(\mathrm { E } ( X Y ) = 6 \cdot 4\), calculate \(p\).
  2. Find \(\operatorname { Var } ( X Y )\).
WJEC Further Unit 2 2019 June Q3
3. The number of claims made to the home insurance department of an insurance company follows a Poisson distribution with mean 4 per day.
  1. Find the probability that more than 11 claims are made in a 2 -day period. The number of claims made in a day to the pet insurance department of the same company follows a Poisson distribution with parameter \(\lambda\). An insurance company worker notices that the probability of two claims being made in a day is three times the probability of four claims being made in a day.
  2. Determine the value of \(\lambda\). The car insurance department models the length of time between claims for drivers aged 17 to 21 as an exponential distribution with mean 10 months. Rachel is 17 years old and has just passed her test. Her father says he will give her the car that they share if she does not make a claim in the first 12 months.
  3. What is the probability that her father gives her the car?
WJEC Further Unit 2 2019 June Q4
4. The continuous random variable, \(X\), has the following probability density function $$f ( x ) = \begin{cases} k x & \text { for } 0 \leqslant x < 1
k x ^ { 3 } & \text { for } 1 \leqslant x \leqslant 2
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
\includegraphics[max width=\textwidth, alt={}, center]{4ecf99c5-c4b3-41b7-a8df-a7c2ca7fcd6a-3_851_935_826_678}
  1. Show that \(k = \frac { 4 } { 17 }\).
  2. Determine \(\mathrm { E } ( X )\).
  3. Calculate \(\mathrm { E } ( 3 X - 1 )\) and \(\operatorname { Var } ( 3 X - 1 )\).
WJEC Further Unit 2 2019 June Q5
5. Chris is investigating the distribution of birth months for ice hockey players. He collects data for 869 randomly chosen National Hockey League (NHL) players. He decides to carry out a chi-squared test. Using a spreadsheet, he produces the following output.
ABcD
1Birth MonthObservedExpectedChi-Squared Contributions
2Jan-Mar259217.258.023302647
3Apr-June232217.251.001438435
4Jul-Sept200217.251.369677791
5Oct-Dec178217.257.091196778
6Total86986917.48561565
7
8p value
90.000561458
  1. By considering the output, state the null hypothesis that Chris is testing. State what conclusion Chris should reach and explain why. Chris now wonders if Premier League football players' birth months are distributed uniformly throughout the year. He collects the birth months of 75 randomly selected Premier League footballers. This information is shown in the table below.
    JanFebMarAprMayJunJulAugSepOctNovDec
    37114122665856
  2. Carry out the chi-squared goodness of fit test at the 10\% significance level that Chris should use to conduct his investigation.
WJEC Further Unit 2 2019 June Q6
6. The University of Arizona surveyed a large number of households. One purpose of the survey was to determine if annual household income could be predicted from size of family home. The graph of Annual household income, \(y\), versus Size of family home, \(x\), is shown below.
\includegraphics[max width=\textwidth, alt={}, center]{4ecf99c5-c4b3-41b7-a8df-a7c2ca7fcd6a-5_616_1257_566_365}
  1. State the limitations of using the regression line above with reference to the scatter diagram. The data for size of family homes between 2000 and 3000 square feet are shown in the diagram below.
    \includegraphics[max width=\textwidth, alt={}, center]{4ecf99c5-c4b3-41b7-a8df-a7c2ca7fcd6a-5_652_1244_1516_360} Summary statistics for these data are as follows. $$\begin{array} { r c c } \sum x = 93160 & \sum y = 3907142 & n = 37
    S _ { x x } = 2869673.03 & S _ { y y } = 44312797167 & S _ { x y } = 348512820 \cdot 6 \end{array}$$
  2. Calculate the equation of the least squares regression line to predict Annual household income from Size of family home for these data.
WJEC Further Unit 2 2019 June Q7
7. An article published in a medical journal investigated sports injuries in adolescents' ball games: football, handball and basketball. In a study of 906 randomly selected adolescent players in the three ball games, 379 players incurred injuries over the course of one year of playing the sport. Rhian wants to test whether there is an association between the site of injury and the sport played. A summary of the injuries is shown in the table below.
\multirow{2}{*}{}Site of injury
Observed valuesShoulder/ ArmHand/ FingersThigh/ LegKneeAnkleFootOtherTotal
\multirow{3}{*}{Sport}Football834536513612191
Handball14266154266115
Basketball428442211073
Total265755551154328379
  1. Calculate the values of \(A , B , C\) in the tables below.
    \multirow{2}{*}{}Site of injury
    Expected valuesShoulder/ ArmHand/ FingersThigh/ LegKneeAnkleFootOther
    \multirow{3}{*}{sodod}Football13.102928.725627.717727.717757.955121.670214.1108
    Handball7.889217.295516.688716.6887A13.04758.4960
    Basketball5.007910.978910.593710.593722.15048.28235.3931
    \multirow{2}{*}{}\multirow[b]{2}{*}{Chi-Squared Contributions}Site of injury
    Shoulder/ ArmHand/ FingersThigh/ LegKneeAnkleFootOther
    \multirow{3}{*}{sodoct}Football1.9873223.03890\(10 \cdot 77575\)2.47484\(B\)9.475860.31575
    Handball4.733334.38079C0.170871.446903.806640.73331
    Basketball0.2028626.388654.104004.104000.001026.403063.93521
  2. Given that the test statistic, \(X ^ { 2 }\), is 116.16, carry out the significance test at the \(5 \%\) level.
  3. Which site of injury most affects the conclusion of this test? Comment on your answer. Rhian also analyses the data on the type of contact that caused the injuries and the sport in which they occur, shown in the table below.
    Observed valuesBallOpponentSurfaceNoneTotal
    Football17681792194
    Handball23341938114
    Basketball2817121471
    Total6811948144379
    The chi-squared test statistic is 46.0937 . Rhian notes that this value is smaller than 116.16 , the test statistic in part (b). She concludes that there is weaker evidence for association in this case than there was in part (b).
  4. State Rhian's misconception and explain what she should consider instead. \section*{END OF PAPER}
WJEC Further Unit 2 2022 June Q1
  1. The probability distribution for the prize money, \(\pounds X\) per ticket, in a local fundraising lottery is shown below.
\(x\)021001000
\(\mathrm { P } ( X = x )\)0.90.09\(p\)0.0001
  1. Calculate the value of \(p\).
  2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
    1. What is the minimum lottery ticket price that the organiser should set in order to make a profit in the long run?
    2. Suggest why, in practice, people would be prepared to pay more than this minimum price.
WJEC Further Unit 2 2022 June Q2
2. An economist suggested the rate of unemployment and the rate of wage inflation are independent. Amy sets about investigating this suggestion. She collects unemployment data and wage inflation data from a random sample of regions in the UK and decides that it is appropriate to carry out a significance test on Pearson's product moment correlation coefficient. Amy's summary statistics for percentage unemployment, \(x\), and percentage wage inflation, \(y\), are shown below. $$\begin{array} { l l l } \sum x = 62 \cdot 8 & \sum y = 19 \cdot 4 & n = 10
\sum x ^ { 2 } = 413 \cdot 44 & \sum y ^ { 2 } = 46 \cdot 16 & \sum x y = 113 \cdot 16 \end{array}$$
  1. Calculate Pearson's product moment correlation coefficient for these data.
  2. Carry out Amy's test at the \(5 \%\) level of significance and state whether the economist's suggestion is reasonable. Amy also collects unemployment data and wage inflation data from a random sample of 10 regions in Spain and calculates Pearson's product moment correlation coefficient to be - 0.2525 .
  3. Should this change Amy's opinion on the economist's suggestion above? What could she do to improve her investigation?
  4. What assumption has Amy made in deciding that it is appropriate to carry out a significance test on Pearson's product moment correlation coefficient?
WJEC Further Unit 2 2022 June Q3
3. Two basketball players, Steph and Klay, score baskets at random at a rate of \(2 \cdot 1\) and \(1 \cdot 9\) respectively per quarter of a game. Assume that baskets are scored independently, and that Steph and Klay each play all four quarters of the game.
  1. Stating the model that you are using, find the probability that they will score a combined total of exactly 20 baskets in a randomly selected game.
  2. A quarter of a game lasts 12 minutes.
    1. State the distribution of the time between baskets for Steph. Give the mean and standard deviation of this distribution.
    2. Given that Klay scores at the end of the third minute in a quarter of a game, find the probability that Klay doesn't score for the rest of the quarter.
  3. When practising, Klay misses \(4 \%\) of the free throws he takes. One week he takes 530 free throws. Calculate the probability that he misses more than 25 free throws.
WJEC Further Unit 2 2022 June Q4
4. The continuous random variable \(R\) has probability density function \(f ( r )\) given by $$f ( r ) = \begin{cases} k r ( b - r ) & \text { for } 1 \leqslant r \leqslant 4 ,
0 & \text { otherwise } , \end{cases}$$ where \(k\) and \(b\) are positive constants.
  1. Explain why \(b \geqslant 4\).
  2. Given that \(b = 4\),
    1. show that \(k = \frac { 1 } { 9 }\),
    2. find an expression for \(F ( r )\), valid for \(1 \leqslant r \leqslant 4\), where \(F\) denotes the cumulative distribution function of \(R\),
    3. find the probability that \(R\) lies between 2 and 3 .
WJEC Further Unit 2 2022 June Q5
5. John has a game that involves throwing a set of three identical, cubical dice with faces numbered 1 to 6 . He wishes to investigate whether these dice are fair in terms of the number of sixes obtained when they are thrown. John throws the set of three dice 1100 times and records the number of sixes obtained for each throw. The results are shown in the table below.
Number of sixes0123
Frequency6253848110
Using these results, conduct a goodness of fit test and draw an appropriate conclusion.
WJEC Further Unit 2 2022 June Q6
6. An online survey on the use of social media asked the following question: \begin{displayquote} "Do you use any form of social media?" \end{displayquote} The results for a total of 1953 respondents are shown in the table below.
Age in years
Use social media18-2930-4950-6465 or olderTotal
Yes3104123481961266
No42116196333687
Total3525285445291953
To test whether there is a relationship between social media use and age, a significance test is carried out at the \(5 \%\) level.
  1. State the null and alternative hypotheses.
  2. Show how the expected frequency \(228 \cdot 18\) is calculated in the table below.
    Expected valuesAge in years
    Use social media18-2930-4950-6465 or older
    Yes\(228 \cdot 18\)\(342 \cdot 27\)352.64342.92
    No123.82185.73191.36186.08
  3. Determine the value of \(s\) in the table below.
    Chi-squared contributionsAge in years
    Use social media18-2930-4950-6465 or older
    Yes29.34\(s\)0.0662.94
    No54.0726-180.11115.99
  4. Complete the significance test, showing all your working.
  5. A student, analysing these data on a spreadsheet, obtains the following output.
    \includegraphics[max width=\textwidth, alt={}, center]{77fd7ad7-f5a3-4947-afc6-e5ef45bef7a8-5_202_1271_445_415} Explain why the student must have made an error in calculating the \(p\)-value.
WJEC Further Unit 2 2022 June Q7
7. Data from a large dataset shows the percentage of children enrolled in secondary education and the percentage of the adult population who are literate. The following graphs show data from 30 randomly selected regions from each of the Arab World, Africa and Asia. In each case, the least squares regression line of '\% Literacy' on '\% Enrolled in Secondary Education' is shown.
\includegraphics[max width=\textwidth, alt={}, center]{77fd7ad7-f5a3-4947-afc6-e5ef45bef7a8-6_682_1200_584_395} \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Africa} \includegraphics[alt={},max width=\textwidth]{77fd7ad7-f5a3-4947-afc6-e5ef45bef7a8-6_623_1191_1548_397}
\end{figure} \includegraphics[max width=\textwidth, alt={}, center]{77fd7ad7-f5a3-4947-afc6-e5ef45bef7a8-7_665_1200_331_434}
  1. Calculate the equation of the least squares regression line of '\% Literacy' ( \(y\) ) on '\% Enrolled in Secondary Education' ( \(x\) ) for Asia, given the following summary statistics. $$\begin{array} { l l l } \sum x = 2850.836 & \sum y = 2738.656 & S _ { x x } = 88.42142
    S _ { y y } = 204.733 & S _ { x y } = 96.60984 & n = 30 \end{array}$$
  2. The Arab World, Africa and Asia each contain a region where \(70 \%\) are enrolled in secondary education. The three regression lines are used to estimate the corresponding \% Literacy. Which of these estimates is likely to be the most reliable? Clearly explain your reasoning. \section*{END OF PAPER}
WJEC Further Unit 2 2023 June Q1
  1. The random variable \(X\) has mean 17 and variance 64 . The independent random variable \(Y\) has mean 10 and variance 16 . Find the value of
    1. \(\mathrm { E } ( 4 Y - 2 X + 1 )\),
    2. \(\quad \operatorname { Var } ( 4 Y - 5 X + 3 )\),
    3. \(\mathrm { E } \left( X ^ { 2 } Y \right)\).
    4. For a set of 30 pairs of observations of the variables \(x\) and \(y\), it is known that \(\sum x = 420\) and \(\sum y = 240\). The least squares regression line of \(y\) on \(x\) passes through the point with coordinates \(( 19,20 )\).
    5. Show that the equation of the regression line of \(y\) on \(x\) is \(y = 2 \cdot 4 x - 25 \cdot 6\) and use it to predict the value of \(y\) when \(x = 26\).
    6. State two reasons why your prediction in part (a) may not be reliable.
    7. It is known that the average lifetime of hair dryers from a certain manufacturer is 2 years. The lifetimes are exponentially distributed.
    8. Find the probability that the lifetime of a randomly selected hair dryer is between 1.8 and \(2 \cdot 5\) years.
    9. Given that \(20 \%\) of hair dryers have a lifetime of at least \(k\) years, find the value of \(k\).
    10. Jon buys his first hair dryer from the manufacturer today. He will replace his hair dryer with another from the same manufacturer immediately when it stops working. Find the probability that, in the next 5 years, Jon will have to replace more than 3 hair dryers.
    11. State one assumption that you have made in part (c).
    12. A continuous random variable \(X\) has cumulative distribution function \(F\) given by
    $$F ( x ) = \begin{cases} 0 & \text { for } x < 0
    \frac { 1 } { 4 } x & \text { for } 0 \leqslant x \leqslant 2
    \frac { 1 } { 480 } x ^ { 4 } + \frac { 7 } { 15 } & \text { for } 2 < x \leqslant b
    1 & \text { for } x > b \end{cases}$$
  2. Show that \(b = 4\).
  3. Find \(\mathrm { P } ( X \leqslant 2 \cdot 5 )\).
  4. Write down the value of the lower quartile of \(X\).
  5. Find the value of the upper quartile of \(X\).
  6. Find, correct to three significant figures, the value of \(k\) that satisfies the equation \(\mathrm { P } ( X > 3 \cdot 5 ) = \mathrm { P } ( X < k )\).
WJEC Further Unit 2 2023 June Q5
5. (a) Give two circumstances where it may be more appropriate to use Spearman's rank correlation coefficient rather than Pearson's product moment correlation coefficient.
(b) A farmer needs a new tractor. The tractor salesman selects 6 tractors at random to show the farmer. The farmer ranks these tractors, in order of preference, according to their ability to meet his needs on the farm. The tractor salesman makes a note of the price and power take-off (PTO) of the tractors.
TractorFarmer's rankPTO (horsepower)Price ( \(\boldsymbol { \pounds } \mathbf { 1 0 0 0 s }\) )
A177.580
B687.945
C5\(53 \cdot 0\)47
D4\(41 \cdot 0\)53
E2\(112 \cdot 0\)60
F3\(90 \cdot 0\)61
Spearman's rank correlation coefficient between the farmer's ranks and the price is 0.9429 .
  1. Test at the \(5 \%\) significance level whether there is an association between the price of a tractor and the farmer's judgement of the ability of the tractor to meet his needs on the farm.
  2. Calculate Spearman's rank correlation coefficient between the farmer's rank and PTO.
  3. How should the tractor salesman interpret the results in (i) and (ii)?
WJEC Further Unit 2 2023 June Q6
6. A company has 20 boats to hire out. Payment is always taken in advance and all 20 boats are hired out each day. A manager at the company notices that \(10 \%\) of groups do not turn up to take the boats, despite having already paid to hire them. The manager wishes to investigate whether the numbers of boats that do not get taken each day can be modelled by the binomial distribution \(B ( 20,0 \cdot 1 )\). The numbers of boats that were not taken for 110 randomly selected days are given below.
Number of boats not taken01234
5 or
more
Frequency1035292583
  1. State suitable hypotheses to carry out a goodness of fit test.
  2. Here is part of the table for a \(\chi ^ { 2 }\) goodness of fit test on the data.
    Number of boats not taken012345 or more
    Observed1035292583
    Expected\(f\)29.72\(g\)20.919.88\(4 \cdot 75\)
    1. Calculate the values of \(f\) and \(g\).
    2. By completing the test, give the conclusion the manager should reach. The cost of hiring a boat is \(\pounds 15\). Since demand is high and the proportion of groups that do not turn up is also relatively high, the manager decides to take payment for 22 boats each day. She would give \(\pounds 20\) (a full refund and some compensation) to any group that has paid and turned up, but cannot take a boat out due to the overselling. Assume that the proportion of groups not turning up stays the same.
    1. Suggest a binomial model that the manager could use for the number of groups arriving expecting to hire a boat.
    2. Hence calculate the expected daily net income for the company following the manager's decision.
  4. Is the manager justified in her decision? Give a reason for your answer.
WJEC Further Unit 2 2024 June Q1
6 marks
  1. Dave and Llinos like to go fishing. When they go fishing, on average, Dave catches 4.3 fish per day and Llinos catches 3.8 fish per day. A day of fishing is assumed to be 8 hours.
    1. (i) Calculate the probability that they will catch fewer than 2 fish in total on a randomly selected half-day of fishing.
      (ii) Justify any distribution you have used in answering (a)(i).
    2. On a randomly selected day, Dave starts fishing at 7 am. Given that Dave has not caught a fish by 11 am,
      1. find the expected time he catches his first fish,
      2. calculate the probability that he will not catch a fish by 3 pm .
    3. On average, only \(2 \%\) of the fish that Llinos catches are trout. Over the course of a year, she catches 950 fish. Calculate the probability that at least 30 of these fish are trout. [3]
      [0pt] she catches 950 fish. Calculate the probability that at least 30 of these fish are trout. [3]
    4. State, with a reason, a distribution, including any parameters, that could approximate the distribution used in part (c).
    PLEASE DO NOT WRITE ON THIS PAGE
WJEC Further Unit 2 2024 June Q2
2. Emlyn aims to produce podcast episodes that are a standard length of time, which he calls the 'target time'. The time, \(X\) minutes, above or below the target time, which he calls the 'allowed time', can be modelled by the following cumulative distribution function. $$F ( x ) = \begin{cases} 0 & x < - 2
\frac { x + 2 } { 5 } & - 2 \leqslant x < 1
\frac { x ^ { 2 } - x + 3 } { 5 } & 1 \leqslant x \leqslant 2
1 & x > 2 \end{cases}$$
  1. Calculate the upper quartile for the 'allowed time'.
  2. Find \(f ( x )\), the probability density function, for all values of \(x\).
    1. Calculate the mean 'allowed time'.
    2. Interpret your answer in context.