Questions Further Unit 5 (36 questions)

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WJEC Further Unit 5 2023 June Q1
11 marks Standard +0.3
  1. The average time it takes for a new kettle to boil, when full of water, is 305 seconds. An old kettle will take longer, on average, to boil. Alun suspects that a particular kettle is an old kettle. He boils the full kettle on 9 occasions and the times taken, in seconds, are shown below.
    305
    295
    310
    310
    315
    307
    300
    311
    306
You may assume the times taken to boil the full kettle are normally distributed.
  1. Calculate unbiased estimates for the mean and variance of the times taken to boil the full kettle.
  2. Test, at the \(5 \%\) level of significance, whether there is evidence to suggest that this is an old kettle.
  3. State a factor that Alun should control when carrying out this investigation.
WJEC Further Unit 5 2023 June Q2
19 marks Standard +0.3
2. The random variables \(X\) and \(Y\) are independent, with \(X\) having mean \(\mu\) and variance \(\sigma ^ { 2 }\), and \(Y\) having mean \(\mu\) and variance \(k \sigma ^ { 2 }\), where \(k\) is a positive constant. Let \(\bar { X }\) denote the mean of a random sample of 20 observations of \(X\), and let \(\bar { Y }\) denote the mean of a random sample of 25 observations of \(Y\).
  1. Given that \(T _ { 1 } = \frac { 3 \bar { X } + 7 \bar { Y } } { 10 }\), show that \(T _ { 1 }\) is an unbiased estimator for \(\mu\).
  2. Given that \(T _ { 2 } = \frac { \bar { X } + a ^ { 2 } \bar { Y } } { 1 + a } , a > 0\), and \(T _ { 2 }\) is an unbiased estimator for \(\mu\), prove that \(a = 1\).
  3. Find and simplify expressions for the variances of \(T _ { 1 }\) and \(T _ { 2 }\).
  4. Show that the value of \(k\) for which \(T _ { 1 }\) and \(T _ { 2 }\) are equally good estimators is \(\frac { 5 } { 6 }\).
  5. Given that \(T _ { 3 } = ( 1 - \lambda ) \bar { X } + \lambda \bar { Y }\), find an expression for \(\lambda\), in terms of \(k\), for which \(T _ { 3 }\) has the smallest possible variance.
WJEC Further Unit 5 2023 June Q3
11 marks Standard +0.3
3. Athletes who compete in the 400 m event have resting heart rates (RHR), measured in beats per minute, which are normally distributed with known standard deviation \(4 \cdot 7\). A random sample of 90 athletes who compete in the 400 m event is taken. Their resting heart rates are summarised by $$\sum x = 4014 \quad \text { and } \quad \sum x ^ { 2 } = 182257 .$$
  1. Find a \(99 \%\) confidence interval for the mean of the RHR of athletes who compete in the 400 m event. Give the limits of your interval correct to 1 decimal place.
  2. Without doing any further calculation, explain how the width of a \(95 \%\) confidence interval would compare to the width of your interval in part (a). Athletes who compete in the discus event have RHR which are normally distributed with known standard deviation \(\sigma\). A random sample of 100 athletes who compete in the discus event is taken. A 95\% confidence interval for the mean of the RHR is calculated as [49•4, 52•6].
  3. Determine the value of \(\sigma\) that was used to calculate this confidence interval.
  4. Referring to the confidence intervals, state, with a reason, what can be said about the RHR of athletes who compete in the 400 m event compared to the RHR of athletes who compete in the discus event.
WJEC Further Unit 5 2023 June Q4
12 marks Standard +0.3
4. Llŷr believes that he will have more social media followers by appearing on a certain Welsh television show. To investigate his belief, he collects data on 9 randomly selected contestants who have appeared on the show. Llŷr records the number of social media followers one week before and one week after the contestants appeared on the show. The data he collects are shown in the table below.
ContestantABCDEFGH1
Before48010080344351781876741457
After8419987513449545428201011644
    1. Carry out a Wilcoxon signed-rank test on this data set, at a significance level as close to 10\% as possible.
    2. Suggest a possible course of action that Llŷr might take.
  2. Give two reasons why the Wilcoxon signed-rank test is appropriate in this case.
WJEC Further Unit 5 2023 June Q5
13 marks Standard +0.3
5. The masses, \(X\), in kg, of men who work for a large company are normally distributed with mean 75 and standard deviation 10.
  1. Find the probability that the mean mass of a random sample of 5 men is less than 70 kg .
  2. The mean mass, in kg , of a random sample of \(n\) men drawn from this distribution is \(\bar { X }\). Given that \(\mathrm { P } ( \bar { X } > 80 )\) is approximately \(0 \cdot 007\), find \(n\). The masses, in kg, of women who work for the company are normally distributed with mean 68 and standard deviation 6 . A lift in the company building will not move if the total mass in the lift is more than 500 kg .
  3. A random sample of 3 men and 4 women get in the lift. Find the probability that the lift will not move.
  4. State a modelling assumption you have made in calculating your answer for part (c).
WJEC Further Unit 5 2023 June Q6
7 marks Standard +0.3
6. A triathlon race organiser wishes to know whether competitors who are members of a triathlon club race more frequently than competitors who are not members of a triathlon club. Six competitors from a triathlon club and six competitors who are not members of a triathlon club are selected at random. The table below shows the number of triathlon races they each entered last year.
Club
members
11412537
Not club
members
294086
  1. Use a Mann-Whitney U test at a significance level as close to \(5 \%\) as possible to carry out the race organiser's investigation.
  2. Briefly explain why a Wilcoxon signed-rank test is not appropriate in this case.
WJEC Further Unit 5 2023 June Q7
7 marks Challenging +1.2
7. Branwen intends to buy a new bike, either a Cannotrek or a Bianchondale. If there is evidence that the difference in the mean times on the two bikes over a 10 km time trial is more than 1.25 minutes, she will buy the faster bike. Otherwise, she will base her decision on other factors. She negotiates a test period to try both bikes. The times, in minutes, taken by Branwen to complete a 10 km time trial on the Cannotrek may be modelled by a normal distribution with mean \(\mu _ { C }\) and standard deviation \(0 \cdot 75\). The times, in minutes, taken by Branwen to complete a 10 km time trial on the Bianchondale may be modelled by a normal distribution with mean \(\mu _ { B }\) and standard deviation \(0 \cdot 6\). During the test period, she completes 6 time trials with a mean time of 19.5 minutes on the Cannotrek, and 5 time trials with a mean time of 17.3 minutes on the Bianchondale. She calculates a \(p \%\) confidence interval for \(\mu _ { C } - \mu _ { B }\).
  1. What would be the largest value of \(p\) that would lead Branwen to base her purchasing decision on the time trials, without considering other factors?
  2. State an assumption you have made in part (a).
WJEC Further Unit 5 2022 June Q1
5 marks Moderate -0.5
  1. Rachel records the times taken, in minutes, to cycle into town from her house on a random sample of 10 days. Her results are shown below.
$$\begin{array} { l l l l l l l l l l } 15 \cdot 5 & 14 \cdot 9 & 16 \cdot 2 & 17 \cdot 3 & 14 \cdot 8 & 14 \cdot 2 & 16 \cdot 0 & 14 \cdot 2 & 15 \cdot 5 & 15 \cdot 1 \end{array}$$ Assuming that these data come from a normal distribution with mean \(\mu\) and variance \(0 \cdot 9\), calculate a \(90 \%\) confidence interval for \(\mu\).
WJEC Further Unit 5 2022 June Q2
15 marks Challenging +1.2
2. Geraint is a beekeeper. The amounts of honey, \(X \mathrm {~kg}\), that he collects annually, from each hive are modelled by the normal distribution \(\mathrm { N } \left( 15,5 ^ { 2 } \right)\). At location \(A\), Geraint has three hives and at location \(B\) he has five hives. You may assume that the amounts of honey collected from the eight hives are independent of each other.
    1. Find the probability that Geraint collects more than 14 kg of honey from the first hive at location \(A\).
    2. Find the probability that he collects more than 14 kg of honey from exactly two out of the three hives at location \(A\).
  2. Find the probability that the total amount of honey that Geraint collects from all eight hives is more than 160 kg .
  3. Find the probability that Geraint collects at least twice as much honey from location B as from location A.
WJEC Further Unit 5 2022 June Q3
8 marks Standard +0.3
3. A statistics teacher wants to investigate whether students from the north of a county and students from the south of the same county feel similarly stressed about examinations. The teacher carries out a psychometric test on 10 randomly selected students to give a score between 0 (low stress) and 100 (high stress) to measure their stress levels before a set of examinations. The results are shown in the table below.
StudentAreaStress Level
HeleddNorth67
MairNorth55
HywelSouth26
GwynSouth70
LiamSouth36
MarcinSouth57
GosiaSouth32
KestutasNorth64
EricaNorth60
TomosNorth22
  1. State one reason why a Mann-Whitney test is appropriate.
  2. Conduct a Mann-Whitney test at a significance level as close to \(5 \%\) as possible. State your conclusion clearly.
  3. How could this investigation be improved?
WJEC Further Unit 5 2022 June Q4
12 marks Standard +0.3
4. The Department of Health recommends that adults aged 18 to 65 should take part in at least 150 minutes of aerobic exercise per week. The results of a survey show that 940 out of 2000 randomly selected adults aged 18 to 65 in Wales take part in at least 150 minutes of aerobic exercise per week.
  1. Calculate an approximate \(95 \%\) confidence interval for the proportion of adults aged 18 to 65 in Wales who take part in at least 150 minutes of aerobic exercise per week.
  2. Give two reasons why the interval is approximate.
  3. Suppose that a \(99 \%\) confidence interval is required, and that the width of the interval is to be no greater than \(0 \cdot 04\). Estimate the minimum additional number of adults to be surveyed to satisfy this requirement.
WJEC Further Unit 5 2022 June Q5
13 marks Standard +0.3
5. A laboratory carrying out screening for a certain blood disorder claims that the average time taken for test results to be returned is 38 hours. A reporter for a national newspaper suspects that the results take longer, on average, to be returned than claimed by the laboratory. The reporter finds the time, \(x\) hours, for 50 randomly selected results, in order to conduct a hypothesis test. The following summary statistics were obtained. $$\sum x = 2163 \quad \sum x ^ { 2 } = 98508$$
  1. Calculate the \(p\)-value for the reporter's hypothesis test, and complete the test using a \(5 \%\) level of significance. Hence write a headline for the reporter to use.
  2. Explain the relevance or otherwise of the Central Limit Theorem to your answer in part (a).
  3. Briefly explain why a random sample is preferable to taking a batch of 50 consecutive results.
  4. On another occasion, the reporter took a different random sample of 10 results.
    1. State, with a reason, what type of hypothesis test the reporter should use on this occasion.
    2. State one assumption required to carry out this test.
WJEC Further Unit 5 2022 June Q6
8 marks Standard +0.8
6. A zoologist knows that the median body length of adults in a species of fire-bellied toads is 4.2 cm . The zoologist thinks he has discovered a new subspecies of fire-bellied toads. If there is sufficient evidence to suggest the median body length differs from 4.2 cm , he will continue his studies to confirm whether he has discovered a new subspecies. Otherwise, he will abandon his studies on fire-bellied toads. The lengths of 10 randomly selected adult toads from the group being investigated are given below. $$\begin{array} { l l l l l l l l l l } 5 \cdot 0 & 3 \cdot 2 & 4 \cdot 9 & 4 \cdot 0 & 3 \cdot 3 & 4 \cdot 2 & 6 \cdot 1 & 4 \cdot 3 & 4 \cdot 8 & 5 \cdot 9 \end{array}$$ Carry out a suitable Wilcoxon signed rank test at a significance level as close to \(1 \%\) as possible and give your conclusion in context.
WJEC Further Unit 5 2022 June Q7
19 marks Challenging +1.2
7. \includegraphics[max width=\textwidth, alt={}, center]{65369843-222f-48b2-b8cd-a1c304eac3d9-6_707_718_347_660} The diagram above shows a cyclic quadrilateral \(A B C D\), where \(\widehat { B A D } = \alpha , \widehat { B C D } = \beta\) and \(\alpha + \beta = 180 ^ { \circ }\). These angles are measured.
The random variables \(X\) and \(Y\) denote the measured values, in degrees, of \(\widehat { B A D }\) and \(\widehat { B C D }\) respectively. You are given that \(X\) and \(Y\) are independently normally distributed with standard deviation \(\sigma\) and means \(\alpha\) and \(\beta\) respectively.
  1. Calculate, correct to two decimal places, the probability that \(X + Y\) will differ from \(180 ^ { \circ }\) by less than \(\sigma\).
  2. Show that \(T _ { 1 } = 45 ^ { \circ } + \frac { 1 } { 4 } ( 3 X - Y )\) is an unbiased estimator for \(\alpha\) and verify that it is a better estimator than \(X\) for \(\alpha\).
  3. Now consider \(T _ { 2 } = \lambda X + ( 1 - \lambda ) \left( 180 ^ { \circ } - Y \right)\).
    1. Show that \(T _ { 2 }\) is an unbiased estimator for \(\alpha\) for all values of \(\lambda\).
    2. Find \(\operatorname { Var } \left( T _ { 2 } \right)\) in terms of \(\lambda\) and \(\sigma\).
    3. Hence determine the value of \(\lambda\) which gives the best unbiased estimator for \(\alpha\).
WJEC Further Unit 5 2019 June Q1
8 marks Standard +0.3
A coffee shop produces biscuits to sell. The masses, in grams, of the biscuits follow a normal distribution with mean \(\mu\). Eight biscuits are chosen at random and their masses, in grams, are recorded. The results are given below. 32.1 \quad 29.9 \quad 31.0 \quad 31.1 \quad 32.5 \quad 30.8 \quad 30.7 \quad 31.5
  1. Calculate a 95\% confidence interval for \(\mu\) based on this sample. [7]
  2. Explain the relevance or otherwise of the Central Limit Theorem in your calculations. [1]
WJEC Further Unit 5 2019 June Q2
6 marks Standard +0.3
The continuous random variable \(X\) is uniformly distributed over the interval \((\theta - 1, \theta + 5)\), where \(\theta\) is an unknown constant.
  1. Find the mean and the variance of \(X\). [2]
  2. Let \(\overline{X}\) denote the mean of a random sample of 9 observations of \(X\). Find, in terms of \(\overline{X}\), an unbiased estimator for \(\theta\) and determine its standard error. [4]
WJEC Further Unit 5 2019 June Q3
9 marks Challenging +1.2
The rules for the weight of a cricket ball state: ``The ball, when new, shall weigh not less than 155.9 g, nor more than 163 g.'' A company produces cricket balls whose weights are normally distributed. It wants 99\% of the balls it produces to be an acceptable weight.
  1. What is the largest acceptable standard deviation? [3]
The weights of the cricket balls are in fact normally distributed with mean 159.5 grams and standard deviation 1.2 grams. The company also produces tennis balls. The weights of the tennis balls are normally distributed with mean 58.5 grams and standard deviation 1.3 grams.
  1. Find the probability that the weight of a randomly chosen cricket ball is more than three times the weight of a randomly chosen tennis ball. [6]
WJEC Further Unit 5 2019 June Q4
11 marks Standard +0.8
Rugby players sometimes use protein powder to aid muscle increase. The monthly weight gains of rugby players taking protein powder may be modelled by a normal distribution having a standard deviation of 40 g and a mean which may depend on the type of protein powder they consume. A rugby team coach gives the same amount of protein powder over a trial month to 22 randomly selected players. Protein powder \(A\) was used by 12 players, randomly selected, and their mean weight gain was 900 g. Protein powder \(B\) was used by the other 10 players and their mean weight gain was 870 g. Let \(\mu_A\) and \(\mu_B\) be the mean monthly weight gains, in grams, of the populations of rugby players who use protein powder \(A\) and protein powder \(B\) respectively.
  1. Calculate a 98\% confidence interval for \(\mu_A - \mu_B\). [4]
  2. In the given context, what can you conclude from your answer to part (a)? Give a reason for your answer. [2]
  3. Find the confidence level of the largest confidence interval that would lead the coach to favour protein powder \(A\) over protein powder \(B\). [4]
  4. State one non-statistical assumption you have made in order to reach these conclusions. [1]
WJEC Further Unit 5 2019 June Q5
11 marks Standard +0.3
To qualify as a music examiner, a trainee must listen to a series of performances by 8 randomly chosen students. An experienced examiner and the trainee must both award scores for each of the 8 performances. In order for the trainee to qualify, there must not be a significant difference between the average scores given by the experienced examiner and the trainee.
  1. Explain why the Wilcoxon signed rank test is appropriate. [2]
The scores awarded are shown below.
StudentABCDEFGH
Experienced Examiner1081099295145148134120
Trainee1141169593137144133110
    1. Carry out an appropriate Wilcoxon signed rank test on this dataset, using a 5\% significance level.
    2. What conclusion should be reached about the suitability of the trainee to qualify? [9]
WJEC Further Unit 5 2019 June Q6
10 marks Standard +0.3
A manufacturer of batteries for electric cars claims that an hour of charge can power a certain model of car to travel for an average of 123 miles. An electric car company and a consumer, Hopcyn, both wish to test the validity of the manufacturer's claim.
  1. Explain why Hopcyn may want to use a one-sided test and why the car company may want to use a two-sided test. [2]
To test the validity of this claim, Hopcyn collects data from a random sample of 90 drivers of this model of car to see how far they travelled, \(X\) miles, on an hour of charge. He produced the following summary statistics. $$\sum x = 11007 \quad \sum x^2 = 1361913$$
    1. Assuming Hopcyn uses a one-sided test, state the hypotheses.
    2. Test at the 5\% significance level whether the manufacturer's claim is correct. [8]
WJEC Further Unit 5 2019 June Q7
7 marks Standard +0.3
Nathan believes that shearers from Wales can shear more sheep, on average, in a given time than shearers from New Zealand. He takes a random sample of 8 shearers from Wales and 7 shearers from New Zealand. The numbers below indicate how many sheep were sheared in 45 minutes by the 15 shearers. Wales: \quad 60 \quad 53 \quad 42 \quad 38 \quad 37 \quad 36 \quad 31 \quad 28 New Zealand: \quad 39 \quad 35 \quad 27 \quad 26 \quad 17 \quad 16 \quad 15 Use a Mann-Whitney U test at the 1\% significance level to test whether Nathan is correct. You must state your hypotheses clearly and state the critical region. [7]
WJEC Further Unit 5 2019 June Q8
18 marks Challenging +1.2
The random variable \(X\) has probability density function $$f(x) = 1 + \frac{3\lambda x}{2} \quad \text{for } -\frac{1}{2} \leqslant x \leqslant \frac{1}{2},$$ $$f(x) = 0 \quad \text{otherwise,}$$ where \(\lambda\) is an unknown parameter such that \(-1 \leqslant \lambda \leqslant 1\).
    1. Find E\((X)\) in terms of \(\lambda\).
    2. Show that \(\text{Var}(X) = \frac{16 - 3\lambda^2}{192}\). [6]
  2. Show that P\((X > 0) = \frac{8 + 3\lambda}{16}\). [2]
In order to estimate \(\lambda\), \(n\) independent observations of \(X\) are made. The number of positive observations obtained is denoted by \(Y\) and the sample mean is denoted by \(\overline{X}\).
    1. Identify the distribution of \(Y\).
    2. Show that \(T_1\) is an unbiased estimator for \(\lambda\), where $$T_1 = \frac{16Y}{3n} - \frac{8}{3}.$$ [4]
    1. Show that \(\text{Var}(T_1) = \frac{64 - 9\lambda^2}{9n}\).
    2. Given that \(T_2\) is also an unbiased estimator for \(\lambda\), where $$T_2 = 8\overline{X},$$ find an expression for Var\((T_2)\) in terms of \(\lambda\) and \(n\).
    3. Hence, giving a reason, determine which is the better estimator, \(T_1\) or \(T_2\). [6]
WJEC Further Unit 5 2024 June Q1
9 marks Moderate -0.3
During practice sessions, a basketball coach makes his players run several 'line drills'.
  1. He records the times taken, in seconds, by one of his players to run the first 'line drill' on a random sample of 8 practice sessions. The results are shown below. 29.4 \quad 31.1 \quad 28.9 \quad 30.0 \quad 29.9 \quad 30.4 \quad 29.7 \quad 30.2 Assuming that these data come from a normal distribution with mean \(\mu\) and variance 0.6, calculate a 95\% confidence interval for \(\mu\). [5]
  2. State the two ways in which the method used to calculate the confidence interval in part (a) would change if the variance were unknown. [2]
  3. During a practice session, a player recorded a mean time of 35.6 seconds for 'line drills'.
    1. Give a reason why this player may not be the same as the player in part (a).
    2. Give a reason why this player could be the same as the player in part (a). [2]
WJEC Further Unit 5 2024 June Q2
9 marks Standard +0.8
In country A, the median daily caffeine intake per student who drinks coffee is 120 mg. A university professor who oversees a foreign exchange programme believes that students visiting from country B drink more coffee and therefore have a greater daily caffeine intake from coffee. On a randomly chosen day, the caffeine intake, in mg, from coffee consumption by each of 15 randomly selected students from country B is given below. 136 \quad 149 \quad 202 \quad 0 \quad 110 \quad 0 \quad 100 \quad 180 0 \quad 187 \quad 0 \quad 0 \quad 138 \quad 197 \quad 115 The professor suspects that the students with zero caffeine intake do not drink coffee, and decides to ignore those students and instead focus on the coffee-drinking students.
  1. Conduct an appropriate Wilcoxon test at a significance level as close to 5\% as possible. State your conclusion in context. [8]
  2. State one limitation of this investigation. [1]
WJEC Further Unit 5 2024 June Q3
7 marks Moderate -0.8
Tony runs a pie stand that sells two types of pie outside a football ground. He wants to estimate the probability that a customer will buy a steak pie rather than a vegetable pie. He conducts a survey by randomly selecting customers and recording their choice of pie. When he feels he has enough data, he notes that 55 customers bought steak pies and 25 bought vegetable pies.
  1. Calculate an approximate 90\% confidence interval for \(p\), the probability that a randomly selected customer buys a steak pie. [6]
  2. Suppose that Tony carries out 50 such surveys and calculates 90\% confidence intervals for each survey. Determine the expected number of these confidence intervals that would contain the true value of \(p\). [1]