WJEC Unit 2 (Unit 2) 2024 June

1. An exercise gym opens at 6:00 a.m. every day. The manager decides to use a questionnaire to gather the opinions of the gym members. The first 30 members arriving at the gym on a particular morning are asked to complete the questionnaire.
   1. What is the intended population in this context?
   2. What type of sampling is this?
   3. How could the sampling process be improved?
      
   4. A baker sells \(3 \cdot 5\) birthday cakes per hour on average.
   5. State, in context, two assumptions you would have to make in order to model the number of birthday cakes sold using a Poisson distribution.
   6. Using a Poisson distribution and showing your calculation, find the probability that exactly 2 birthday cakes are sold in a randomly selected 1-hour period.
      （c）Calculate the probability that，during a randomly selected 3 －hour period，the baker sells more than 10 birthday cakes．
      （d）The baker sells a birthday cake at 9：30 a．m．Calculate the probability that the baker will sell the next birthday cake before 10：00 a．m．
   7. Select one of the assumptions in part (a) and comment on its reasonableness.
      
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1. The following Venn diagram shows the participation of 100 students in three activities, \(A , B\), and \(C\), which represent athletics, baseball and climbing respectively.
   \includegraphics[max width=\textwidth, alt={}, center]{d9ef2033-bf8b-4aec-bc88-34dbc8b9c208-08_641_1050_477_511}

For these 100 students, participation in athletics and participation in climbing are independent events.

1. Show that \(x = 10\) and find the value of \(y\).
2. Two students are selected at random, one after the other without replacement. Find the probability that the first student does athletics and the second student does only climbing.

4. A company produces sweets of varying colours. The company claims that the proportion of blue sweets is \(13.6 \%\). A consumer believes that the true proportion is less than this. In order to test this belief, the consumer collects a random sample of 80 sweets.

1. State suitable hypotheses for the test.
2. 1. Determine the critical region if the test is to be carried out at a significance level as close as possible to, but not exceeding, \(5 \%\).
   2. Given that there are 6 blue sweets in the sample of 80, complete the significance test.
3. Suppose the proportion of blue sweets claimed by the company is correct. The consumer conducts the sampling and testing process on a further 20 occasions, using the sample size of 80 each time. What is the expected number of these occasions on which the consumer would reach the incorrect conclusion?
4. Now suppose that the proportion of blue sweets is \(7 \%\). Find the probability of a Type II error. Interpret your answer in context.

5. In March 2020, the coronavirus pandemic caused major disruption to the lives of individuals across the world. A newspaper published the following graph from the \href{http://gov.uk}{gov.uk} website, along with an article which included the following excerpt.
"The daily number of vaccines administered continues to fall. In order to get control of the virus, we need the number of people receiving a second dose of the vaccine to keep rocketing. The fear is it will start to drop off soon, which will leave many people still unprotected." \begin{figure}[h] \end{figure}

1. By referring to the graph, explain how the quote could be misleading.
   The daily numbers of second dose vaccines, in thousands, over the period April 1st 2021 to May 31st 2021 are shown in the table below.

   Daily number of 2nd dose vaccines (1000s)	Midpoint \(x\)	Frequency \(f\)	Percentage
   \(0 \leqslant v < 100\)	50	2	3.3
   \(100 \leqslant v < 200\)	150	8	13.1
   \(200 \leqslant v < 300\)	250	10	16.4
   \(300 \leqslant v < 400\)	350	13	21.3
   \(400 \leqslant v < 500\)	450	26	\(42 \cdot 6\)
   \(500 \leqslant v < 600\)	550	2	3.3
   Total		61	100

2. 1. Calculate estimates of the mean and standard deviation for the daily number of second dose vaccines given over this period. You may use \(\sum x ^ { 2 } f = 8272500\). [4]
   2. Comment on the skewness of these data.
      A second graph in the article shows the number of people receiving a third dose of the vaccine. This graph has a repeated pattern of rising then falling. An extract is shown below. \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Number of people (millions) who received 3rd dose vaccinations daily in the UK, by report date} \includegraphics[alt={},max width=\textwidth]{d9ef2033-bf8b-4aec-bc88-34dbc8b9c208-14_556_1115_571_466}
      \end{figure}
3. Give a possible reason for the pattern observed in this graph.
   Another extract shows the number of people who received the third dose of the vaccine between 27th March 2022 and 25th April 2022. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Number of people (1000s) who received 3rd dose vaccinations daily in the UK, by report date} \includegraphics[alt={},max width=\textwidth]{d9ef2033-bf8b-4aec-bc88-34dbc8b9c208-15_537_1246_571_406}
   \end{figure}
4. State, with a reason, whether or not you think the data for April 15th to April 18th are incorrect.

1. A ship \(S\) is moving with constant velocity \(( 4 \mathbf { i } - 7 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\), where \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors due east and due north respectively.
   Find the speed and direction of \(S\), giving the direction as a three-figure bearing, correct to the nearest degree.
   
2. The diagram below shows a forklift truck being used to raise two boxes, \(P\) and \(Q\), vertically. Box \(Q\) rests on horizontal forks and box \(P\) rests on top of box \(Q\). Box \(P\) has mass 25 kg and box \(Q\) has mass 55 kg .
   \includegraphics[max width=\textwidth, alt={}, center]{d9ef2033-bf8b-4aec-bc88-34dbc8b9c208-17_504_814_504_589}
   1. When the boxes are moving upwards with uniform acceleration, the reaction of the horizontal forks on box \(Q\) is 820 N . Calculate the magnitude of the acceleration.
   2. Calculate the reaction of box \(Q\) on box \(P\) when they are moving vertically upwards with constant speed.
      
   3. A particle, of mass 4 kg , moves in a straight line under the action of a single force \(F \mathrm {~N}\), whose magnitude at time \(t\) seconds is given by
   $$F = 12 \sqrt { t } - 32 \text { for } t \geqslant 0 .$$
3. Find the acceleration of the particle when \(t = 9\).
4. Given that the particle has velocity \(- 1 \mathrm {~ms} ^ { - 1 }\) when \(t = 4\), find an expression for the velocity of the particle at \(t \mathrm {~s}\).
5. Determine whether the speed of the particle is increasing or decreasing when \(t = 9\). [2]

9. The diagram below shows an object \(A\), of mass \(2 m \mathrm {~kg}\), lying on a horizontal table. It is connected to another object \(B\), of mass \(m \mathrm {~kg}\), by a light inextensible string, which passes over a smooth pulley \(P\), fixed at the edge of the table. Initially, object \(A\) is held at rest so that object \(B\) hangs freely with the string taut.
\includegraphics[max width=\textwidth, alt={}, center]{d9ef2033-bf8b-4aec-bc88-34dbc8b9c208-20_589_871_593_605} Object \(A\) is then released.

1. When object \(B\) has moved downwards a vertical distance of 0.4 m , its speed is \(1.2 \mathrm {~ms} ^ { - 1 }\). Use a formula for motion in a straight line with constant acceleration to show that the magnitude of the acceleration of \(B\) is \(1.8 \mathrm {~ms} ^ { - 2 }\).
2. During the motion, object \(A\) experiences a constant resistive force of 22 N . Find the value of \(m\) and hence determine the tension in the string.
3. What assumption did the word 'inextensible' in the description of the string enable you to make in your solution?
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1. A car, starting from rest at a point \(A\), travels along a straight horizontal road towards a point \(B\). The distance between points \(A\) and \(B\) is 1.9 km . Initially, the car accelerates uniformly for 12 seconds until it reaches a speed of \(26 \mathrm {~ms} ^ { - 1 }\). The car continues at \(26 \mathrm {~ms} ^ { - 1 }\) for 1 minute, before decelerating at a constant rate of \(0.75 \mathrm {~ms} ^ { - 2 }\) until it passes the point \(B\).
   1. Show that the car travels 156 m while it is accelerating.
   2. 1. Work out the distance travelled by the car while travelling at a constant speed. [1]
      2. Hence calculate the length of time for which the car is decelerating until it passes the point \(B\).
   3. Sketch a displacement-time graph for the motion of the car between \(A\) and \(B\).
   Additional page, if required. Write the question number(s) in the left-hand margin. Additional page, if required. Write the question number(s) in the left-hand margin. \section*{PLEASE DO NOT WRITE ON THIS PAGE}