Questions — Edexcel (9670 questions)

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Edexcel FS1 2024 June Q5
10 marks Easy -1.2
  1. Some of the components produced by a factory are defective. The management requires that no more than \(3 \%\) of the components produced are defective.
    Niluki monitors the production process and takes a random sample of \(n\) components.
    1. Write down the hypotheses Niluki should use in a test to assess whether or not the proportion of defective components is greater than 0.03
    Niluki defines the random variable \(D _ { n }\) to represent the number of defective components in a sample of size \(n\). She considers two tests \(\mathbf { A }\) and \(\mathbf { B }\) In test \(\mathbf { A }\), Niluki uses \(n = 100\) and if \(D _ { 100 } \geqslant 5\) she rejects \(H _ { 0 }\)
  2. Find the size of test \(\mathbf { A }\) In test B, Niluki uses \(n = 80\) and
    • if \(D _ { 80 } \geqslant 5\) she rejects \(\mathrm { H } _ { 0 }\)
    • if \(D _ { 80 } \leqslant 3\) she does not reject \(\mathrm { H } _ { 0 }\)
    • if \(D _ { 80 } = 4\) she takes a second random sample of size 80 and if \(D _ { 80 } \geqslant 1\) in this second sample then she rejects \(\mathrm { H } _ { 0 }\) otherwise she does not reject \(\mathrm { H } _ { 0 }\)
    • Find the size of test \(\mathbf { B }\)
    Given that the actual proportion of defective components is 0.06
    1. find the power of test \(\mathbf { A }\)
    2. find the expected number of components sampled using test \(\mathbf { B }\) Given also that, when the actual proportion of defective components is 0.06 , the power of test \(\mathbf { B }\) is 0.713
  4. suggest, giving your reasons, which test Niluki should use.
Edexcel FS1 2024 June Q6
16 marks Challenging +1.2
  1. The random variable \(X\) has probability generating function \(\mathrm { G } _ { X } ( t )\) where
$$\mathrm { G } _ { X } ( t ) = \frac { 1 } { \sqrt { 4 - 3 t } }$$
  1. Use calculus to find \(\operatorname { Var } ( X )\) Show your working clearly.
  2. Find the exact value of \(\mathrm { P } ( X \leqslant 2 )\) The independent random variables \(X _ { 1 }\) and \(X _ { 2 }\) each have the same distribution as \(X\) The random variable \(Y = X _ { 1 } + X _ { 2 } + 1\)
  3. By finding the probability generating function of \(Y\), state the name of the distribution of \(Y\)
  4. Hence, or otherwise, find \(\mathrm { P } \left( X _ { 1 } + X _ { 2 } > 5 \right)\)
Edexcel FS1 2024 June Q7
18 marks Challenging +1.2
  1. The probability of winning a prize when playing a single game of Pento is \(\frac { 1 } { 5 }\)
When more than one game is played the games are independent.
Sam plays 20 games.
  1. Find the probability that Sam wins 4 or more prizes. Tessa plays a series of games.
  2. Find the probability that Tessa wins her 4th prize on her 20th game. Rama invites Sam and Tessa to play some new games of Pento. They must pay Rama \(\pounds 1\) for each game they play but Rama will pay them \(\pounds 2\) for the first time they win a prize, \(\pounds 4\) for the second time and \(\pounds ( 2 w )\) when they win their \(w\) th prize ( \(w > 2\) ) Sam decides to play \(n\) games of Pento with Rama.
  3. Show that Sam's expected profit is \(\pounds \frac { 1 } { 25 } \left( n ^ { 2 } - 16 n \right)\) Given that Sam chose \(n = 15\)
  4. find the probability that Sam does not make a loss. Tessa agrees to play Pento with Rama. She will play games until she wins \(r\) prizes and then she will stop.
  5. Find, in terms of \(r\), Tessa's expected profit.
Edexcel FS1 Specimen Q1
5 marks Standard +0.3
  1. Bacteria are randomly distributed in a river at a rate of 5 per litre of water. A new factory opens and a scientist claims it is polluting the river with bacteria. He takes a sample of 0.5 litres of water from the river near the factory and finds that it contains 7 bacteria. Stating your hypotheses clearly test, at the \(5 \%\) level of significance, whether there is evidence that the level of pollution has increased.
\section*{Q uestion 1 continued}
Edexcel FS1 Specimen Q2
12 marks Standard +0.8
  1. A call centre routes incoming telephone calls to agents who have specialist knowledge to deal with the call. The probability of a caller, chosen at random, being connected to the wrong agent is p.
The probability of at least 1 call in 5 consecutive calls being connected to the wrong agent is 0.049 The call centre receives 1000 calls each day.
  1. Find the mean and variance of the number of wrongly connected calls a day.
  2. Use a Poisson approximation to find, to 3 decimal places, the probability that more than 6 calls each day are connected to the wrong agent.
  3. Explain why the approximation used in part (b) is valid. The probability that more than 6 calls each day are connected to the wrong agent using the binomial distribution is 0.8711 to 4 decimal places.
  4. Comment on the accuracy of your answer in part (b).
Edexcel FS1 Specimen Q3
14 marks Standard +0.8
  1. Bags of \(\pounds 1\) coins are paid into a bank. Each bag contains 20 coins.
The bank manager believes that \(5 \%\) of the \(\pounds 1\) coins paid into the bank are fakes. He decides to use the distribution \(X \sim B ( 20,0.05 )\) to model the random variable \(X\), the number of fake \(\pounds 1\) coins in each bag. The bank manager checks a random sample of 150 bags of \(\pounds 1\) coins and records the number of fake coins found in each bag. His results are summarised in Table 1. He then calculates some of the expected frequencies, correct to 1 decimal place. \begin{table}[h]
Number of fake coins in each bag01234 or more
Observed frequency436226136
Expected frequency53.856.68.9
\captionsetup{labelformat=empty} \caption{Table 1}
\end{table}
  1. Carry out a hypothesis test, at the \(5 \%\) significance level, to see if the data supports the bank manager's statistical model. State your hypotheses clearly. The assistant manager thinks that a binomial distribution is a good model but suggests that the proportion of fake coins is higher than \(5 \%\). She calculates the actual proportion of fake coins in the sample and uses this value to carry out a new hypothesis test on the data. Her expected frequencies are shown in Table 2. \begin{table}[h]
    Number of fake coins in each bag01234 or more
    Observed frequency436226136
    Expected frequency44.555.733.212.54.1
    \captionsetup{labelformat=empty} \caption{Table 2}
    \end{table}
  2. Explain why there are 2 degrees of freedom in this case.
  3. Given that she obtains a \(\chi ^ { 2 }\) test statistic of 2.67 , test the assistant manager's hypothesis that the binomial distribution is a good model for the number of fake coins in each bag. Use a \(5 \%\) level of significance and state your hypotheses clearly.
Edexcel FS1 Specimen Q4
4 marks Standard +0.3
  1. A random sample of 100 observations is taken from a Poisson distribution with mean 2.3
Estimate the probability that the mean of the sample is greater than 2.5
Edexcel FS1 Specimen Q5
8 marks Standard +0.3
  1. The probability of Richard winning a prize in a game at the fair is 0.15
Richard plays a number of games.
  1. Find the probability of Richard winning his second prize on his 8th game,
  2. State two assumptions that have to be made, for the model used in part (a) to be valid. M ary plays the same game, but has a different probability of winning a prize. She plays until she has won r prizes. The random variable \(G\) represents the total number of games M ary plays.
  3. Given that the mean and standard deviation of G are 18 and 6 respectively, determine whether Richard or Mary has the greater probability of winning a prize in a game.
Edexcel FS1 Specimen Q6
14 marks Standard +0.8
  1. The probability generating function of the discrete random variable \(X\) is given by
$$G _ { x } ( t ) = k \left( 3 + t + 2 t ^ { 2 } \right) ^ { 2 }$$
  1. Show that \(\mathrm { k } = \frac { 1 } { 36 }\)
  2. Find \(\mathrm { P } ( \mathrm { X } = 3 )\)
  3. Show that \(\operatorname { Var } ( \mathrm { X } ) = \frac { 29 } { 18 }\)
  4. Find the probability generating function of \(2 \mathrm { X } + 1\)
    \section*{Q uestion 6 continued} \section*{Q uestion 6 continued} \section*{Q uestion 6 continued}
Edexcel FS1 Specimen Q7
18 marks Challenging +1.2
  1. Sam and Tessa are testing a spinner to see if the probability, p , of it landing on red is less than \(\frac { 1 } { 5 }\). They both use a \(10 \%\) significance level.
Sam decides to spin the spinner 20 times and record the number of times it lands on red.
  1. Find the critical region for Sam's test.
  2. Write down the size of Sam's test. Tessa decides to spin the spinner until it lands on red and she records the number of spins.
  3. Find the critical region for Tessa's test.
  4. Find the size of Tessa's test.
    1. Show that the power function for Sam's test is given by $$( 1 - p ) ^ { 19 } ( 1 + 19 p )$$
    2. Find the power function for Tessa's test.
  6. With reference to parts (b), (d) and (e), state, giving your reasons, whether you would recommend Sam's test or Tessa's test when \(\mathrm { p } = 0.15\)
Edexcel FS2 2019 June Q1
8 marks Moderate -0.3
1 A machine is set to fill pots with yoghurt such that the mean weight of yoghurt in a pot is 505 grams. To check that the machine is working properly, a random sample of 8 pots is selected. The weight of yoghurt, in grams, in each pot is as follows $$\begin{array} { l l l l l l l l } 508 & 510 & 500 & 500 & 498 & 503 & 508 & 505 \end{array}$$ Given that the weights of the yoghurt delivered by the machine follow a normal distribution with standard deviation 5.4 grams,
  1. find a \(95 \%\) confidence interval for the mean weight, \(\mu\) grams, of yoghurt in a pot. Give your answers to 2 decimal places.
  2. Comment on whether or not the machine is working properly, giving a reason for your answer.
  3. State the probability that a \(95 \%\) confidence interval for \(\mu\) will not contain \(\mu\) grams.
  4. Without carrying out any further calculations, explain the changes, if any, that would need to be made in calculating the confidence interval in part (a) if the standard deviation was unknown. Give a reason for your answer.
    You may assume that the weights of the yoghurt delivered by the machine still follow a normal distribution.
Edexcel FS2 2019 June Q2
10 marks Standard +0.3
2 A large field of wheat is split into 8 plots of equal area. Each plot is treated with a different amount of fertiliser, \(f\) grams \(/ \mathrm { m } ^ { 2 }\). The yield of wheat, \(w\) tonnes, from each plot is recorded. The results are summarised below. $$\sum f = 28 \quad \sum w = 303 \quad \sum w ^ { 2 } = 13447 \quad \mathrm {~S} _ { f f } = 42 \quad \mathrm {~S} _ { f w } = 269.5$$
  1. Calculate the product moment correlation coefficient between \(f\) and \(w\)
  2. Interpret the value of your product moment correlation coefficient.
  3. Find the equation of the regression line of \(w\) on \(f\) in the form \(w = a + b f\)
  4. Using your equation, estimate the decrease in yield when the amount of fertiliser decreases by 0.5 grams \(/ \mathrm { m } ^ { 2 }\) The residuals of the data recorded are calculated and plotted on the graph below.
    \includegraphics[max width=\textwidth, alt={}, center]{67df73d4-6ce4-45f7-8a69-aa94292ea814-04_1232_1294_1169_301}
  5. With reference to this graph, comment on the suitability of the model you found in part (c).
  6. Suggest how you might be able to refine your model.
Edexcel FS2 2019 June Q3
8 marks Standard +0.8
3 Yin grows two varieties of potato, plant \(A\) and plant \(B\). A random sample of each variety of potato is taken and the yield, \(x \mathrm {~kg}\), produced by each plant is measured. The following statistics are obtained from the data.
Number of plants\(\sum x\)\(\sum x ^ { 2 }\)
\(A\)25194.71637.37
\(B\)26227.52031.19
  1. Stating your hypotheses clearly, test, at the \(10 \%\) significance level, whether or not the variances of the yields of the two varieties of potato are the same.
  2. State an assumption you have made in order to carry out the test in part (a).
Edexcel FS2 2019 June Q4
8 marks Standard +0.3
4 The continuous random variable \(X\) has cumulative distribution function given by $$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x \leqslant 0 \\ k \left( x ^ { 3 } - \frac { 3 } { 8 } x ^ { 4 } \right) & 0 < x \leqslant 2 \\ 1 & x > 2 \end{array} \right.$$ where \(k\) is a constant.
  1. Show that \(k = \frac { 1 } { 2 }\)
  2. Showing your working clearly, use calculus to find
    1. \(\mathrm { E } ( X )\)
    2. the mode of \(X\)
  3. Describe, giving a reason, the skewness of the distribution of \(X\)
Edexcel FS2 2019 June Q5
7 marks Standard +0.3
5 Alexa believes that students are equally likely to achieve the same percentage score on each of two tests, paper I and paper II. She randomly selects 8 students and gives them each paper I and paper II. The percentage scores for each paper are recorded. The following paired data are collected.
Student\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)
Paper I (\%)7070848064656590
Paper II (\%)6476727468645876
Test, at the \(1 \%\) significance level, whether or not there is evidence to support Alexa's belief. State your hypotheses clearly and show your working.
Edexcel FS2 2019 June Q6
9 marks Standard +0.3
6 A company manufactures bolts. The diameter of the bolts follows a normal distribution with a mean diameter of 5 mm . Stan believes that the mean diameter of the bolts is less than 5 mm . He takes a random sample of 10 bolts and measures their diameters. He calculates some statistics but spills ink on his work before completing them. The only information he has left is as follows
\includegraphics[max width=\textwidth, alt={}, center]{67df73d4-6ce4-45f7-8a69-aa94292ea814-16_394_1150_527_456} Stating your hypotheses clearly, test, at the \(5 \%\) level of significance, whether or not Stan's belief is supported.
Edexcel FS2 2019 June Q7
14 marks Challenging +1.2
7 A manufacturer makes two versions of a toy. One version is made out of wood and the other is made out of plastic. The weights, \(W \mathrm {~kg}\), of the wooden toys are normally distributed with mean 2.5 kg and standard deviation 0.7 kg . The weights, \(X \mathrm {~kg}\), of the plastic toys are normally distributed with mean 1.27 kg and standard deviation 0.4 kg . The random variables \(W\) and \(X\) are independent.
  1. Find the probability that the weight of a randomly chosen wooden toy is more than double the weight of a randomly chosen plastic toy. The manufacturer packs \(n\) of these wooden toys and \(2 n\) of these plastic toys into the same container. The maximum weight the container can hold is 252 kg . The probability of the contents of this container being overweight is 0.2119 to 4 decimal places.
  2. Calculate the value of \(n\).
Edexcel FS2 2019 June Q8
11 marks Challenging +1.8
8 Nine athletes, \(A , B , C , D , E , F , G , H\) and \(I\), competed in both the 100 m sprint and the long jump. After the two events the positions of each athlete were recorded and Spearman's rank correlation coefficient was calculated and found to be 0.85
  1. Stating your hypotheses clearly, test whether or not there is evidence to suggest that the higher an athlete's position is in the 100 m sprint, the higher their position is in the long jump. Use a \(5 \%\) level of significance. The piece of paper the positions were recorded on was mislaid. Although some of the athletes agreed their positions, there was some disagreement between athletes \(B , C\) and \(D\) over their long jump results. The table shows the results that are agreed to be correct.
    Athlete\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)\(I\)
    Position in 100 m sprint467928315
    Position in long jump549312
    Given that there were no tied ranks,
  2. find the correct positions of athletes \(B , C\) and \(D\) in the long jump. You must show your working clearly and give reasons for your answers.
  3. Without recalculating the coefficient, explain how Spearman's rank correlation coefficient would change if athlete \(H\) was disqualified from both the 100 m sprint and the long jump.
Edexcel FS2 2020 June Q1
6 marks Standard +0.3
1 Gina receives a large number of packages from two companies, \(A\) and \(B\). She believes that the variance of the weights of packages from company \(A\) is greater than the variance of the weights of packages from company \(B\). Gina takes a random sample of 7 packages from company \(A\) and an independent random sample of 10 packages from company \(B\). Her results are summarised below $$\bar { a } = 300 \quad \mathrm {~S} _ { a a } = 145496 \quad \bar { b } = 233.4 \quad \mathrm {~S} _ { b b } = 56364.4$$ [You may assume that the weights of packages from the two companies are normally distributed.]
Test Gina's belief. Use a \(5 \%\) level of significance and state your hypotheses clearly.
Edexcel FS2 2020 June Q2
6 marks Standard +0.3
2 Jemima makes jam to sell in a local shop. The jam is sold in jars and the weight of jam in a jar is normally distributed. Jemima takes a random sample of 8 of her jars of jam and weighs the contents of each jar, \(x\) grams. Her results are summarised as follows $$\sum x = 3552 \quad \sum x ^ { 2 } = 1577314$$
  1. Calculate a 95\% confidence interval for the mean weight of jam in a jar. The labels on the jars state that the average contents weigh 440 grams.
  2. State, giving a reason, whether or not Jemima should be concerned about the labels on her jars of jam.
Edexcel FS2 2020 June Q3
6 marks Standard +0.3
3 Below are 3 sketches from some students of the residuals from their linear regressions of \(y\) on \(x\).
\includegraphics[max width=\textwidth, alt={}, center]{54bf68ab-7934-432a-890f-20093082ab07-06_252_704_342_660}
\includegraphics[max width=\textwidth, alt={}, center]{54bf68ab-7934-432a-890f-20093082ab07-06_266_718_625_660}
\includegraphics[max width=\textwidth, alt={}, center]{54bf68ab-7934-432a-890f-20093082ab07-06_248_599_936_660} \section*{III} III For each sketch you should state, giving your reason,
  1. whether or not the sketch is feasible
    and if it is feasible
  2. whether or not the sketch suggests a linear or a non-linear relationship between \(y\) and \(x\).
Edexcel FS2 2020 June Q4
7 marks Challenging +1.2
4 A biased coin has a probability \(p\) of landing on heads, where \(0 < p < 1\) Simon spins the coin \(n\) times and the random variable \(X\) represents the number of heads. Taruni spins the coin \(m\) times, \(m \neq n\), and the random variable \(Y\) represents the number of heads. Simon and Taruni want to combine their results to find unbiased estimators of \(p\).
Simon proposes the estimator \(S = \frac { X + Y } { m + n }\) and Taruni proposes \(T = \frac { 1 } { 2 } \left[ \frac { X } { n } + \frac { Y } { m } \right]\)
  1. Show that both \(S\) and \(T\) are unbiased estimators of \(p\).
  2. Prove that, for all values of \(m\) and \(n , S\) is the better estimator.
Edexcel FS2 2020 June Q5
10 marks Standard +0.8
5 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{54bf68ab-7934-432a-890f-20093082ab07-12_446_1105_242_479} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The random variable \(X\) has probability density function \(\mathrm { f } ( x )\) and Figure 1 shows a sketch of \(\mathrm { f } ( x )\) where $$f ( x ) = \left\{ \begin{array} { c c } k ( 1 - \cos x ) & 0 \leqslant x \leqslant 2 \pi \\ 0 & \text { otherwise } \end{array} \right.$$
  1. Show that \(k = \frac { 1 } { 2 \pi }\) The random variable \(Y \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\) and \(\mathrm { E } ( Y ) = \mathrm { E } ( X )\)
    The probability density function of \(Y\) is \(g ( y )\), where $$g ( y ) = \frac { 1 } { \sigma \sqrt { 2 \pi } } e ^ { - \frac { 1 } { 2 } \left( \frac { y - \mu } { \sigma } \right) ^ { 2 } } \quad - \infty < y < \infty$$ Given that \(\mathrm { g } ( \mu ) = \mathrm { f } ( \mu )\)
  2. find the exact value of \(\sigma\)
  3. Calculate the error in using \(\mathrm { P } \left( \frac { \pi } { 2 } < Y < \frac { 3 \pi } { 2 } \right)\) as an approximation to \(\mathrm { P } \left( \frac { \pi } { 2 } < X < \frac { 3 \pi } { 2 } \right)\)
Edexcel FS2 2020 June Q6
12 marks Standard +0.8
6 A new employee, Kim, joins an existing employee, Jiang, to work in the quality control department of a company producing steel rods.
Each day a random sample of rods is taken, their lengths measured and a \(95 \%\) confidence interval for the mean length of the rods, in metres, is calculated. It is assumed that the lengths of the rods produced are normally distributed. Kim took a random sample of 25 rods and used the \(t\) distribution to obtain a \(95 \%\) confidence interval of \(( 1.193,1.367 )\) for the mean length of the rods. Jiang commented that this interval was a little wider than usual and explained that they usually assume that the standard deviation does not change and can be taken as 0.175 metres.
  1. Test, at the \(10 \%\) level of significance, whether or not Kim's sample suggests that the standard deviation is different from 0.175 metres. State your hypotheses clearly. Using Kim's sample and the normal distribution with a standard deviation of 0.175 metres, (b) find a 95\% confidence interval for the mean length of the rods.
Edexcel FS2 2020 June Q7
17 marks Challenging +1.2
7 Fence panels come in two sizes, large and small. The lengths of the large panels are normally distributed with mean 198 cm and standard deviation 5 cm . The lengths of the small panels are normally distributed with mean 74 cm and standard deviation 3 cm .
  1. Find the probability that the total length of a random sample of 3 large panels is greater than the total length of a random sample of 8 small panels. One large panel and one small panel are selected at random.
  2. Find the probability that the length of the large panel is more than \(\frac { 8 } { 3 }\) times the length of the small panel. Rosa needs 1000 cm of fencing. The large panels cost \(\pounds 80\) each and the small panels cost \(\pounds 30\) each. Rosa's plan is to buy 5 large panels and measure the total length. If the total length is less than 1000 cm she will then buy one small panel as well.
  3. Calculate whether or not the expected cost of Rosa's plan is cheaper than simply buying 14 small panels.