Questions — Edexcel S2 (494 questions)

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Edexcel S2 2015 June Q2
2. A company produces chocolate chip biscuits. The number of chocolate chips per biscuit has a Poisson distribution with mean 8
  1. Find the probability that one of these biscuits, selected at random, does not contain 8 chocolate chips. A small packet contains 4 of these biscuits, selected at random.
  2. Find the probability that each biscuit in the packet contains at least 8 chocolate chips. A large packet contains 9 of these biscuits, selected at random.
  3. Use a suitable approximation to find the probability that there are more than 75 chocolate chips in the packet. A shop sells packets of biscuits, randomly, at a rate of 1.5 packets per hour. Following an advertising campaign, 11 packets are sold in 4 hours.
  4. Test, at the \(5 \%\) level of significance, whether or not there is evidence that the rate of sales of packets of biscuits has increased. State your hypotheses clearly.
Edexcel S2 2015 June Q3
  1. A piece of spaghetti has length \(2 c\), where \(c\) is a positive constant. It is cut into two pieces at a random point. The continuous random variable \(X\) represents the length of the longer piece and is uniformly distributed over the interval \([ c , 2 c ]\).
    1. Sketch the graph of the probability density function of \(X\)
    2. Use integration to prove that \(\operatorname { Var } ( X ) = \frac { c ^ { 2 } } { 12 }\)
    3. Find the probability that the longer piece is more than twice the length of the shorter piece.
Edexcel S2 2015 June Q4
  1. A single observation \(x\) is to be taken from a Poisson distribution with parameter \(\lambda\) This observation is to be used to test, at a \(5 \%\) level of significance,
$$\mathrm { H } _ { 0 } : \lambda = k \quad \mathrm { H } _ { 1 } : \lambda \neq k$$ where \(k\) is a positive integer.
Given that the critical region for this test is \(( X = 0 ) \cup ( X \geqslant 9 )\)
  1. find the value of \(k\), justifying your answer.
  2. Find the actual significance level of this test.
Edexcel S2 2015 June Q5
5. A bag contains a large number of counters with \(35 \%\) of the counters having a value of 6 and \(65 \%\) of the counters having a value of 9 A random sample of size 2 is taken from the bag and the value of each counter is recorded as \(X _ { 1 }\) and \(X _ { 2 }\) respectively. The statistic \(Y\) is calculated using the formula $$Y = \frac { 2 X _ { 1 } + X _ { 2 } } { 3 }$$
  1. List all the possible values of \(Y\).
  2. Find the sampling distribution of \(Y\).
  3. Find \(\mathrm { E } ( Y )\).
Edexcel S2 2015 June Q6
6. Past information at a computer shop shows that \(40 \%\) of customers buy insurance when they purchase a product. In a random sample of 30 customers, \(X\) buy insurance.
  1. Write down a suitable model for the distribution of \(X\).
  2. State an assumption that has been made for the model in part (a) to be suitable. The probability that fewer than \(r\) customers buy insurance is less than 0.05
  3. Find the largest possible value of \(r\). A second random sample, of 100 customers, is taken.
    The probability that at least \(t\) of these customers buy insurance is 0.938 , correct to 3 decimal places.
  4. Using a suitable approximation, find the value of \(t\). The shop now offers an extended warranty on all products. Following this, a random sample of 25 customers is taken and 6 of them buy insurance.
  5. Test, at the \(10 \%\) level of significance, whether or not there is evidence that the proportion of customers who buy insurance has decreased. State your hypotheses clearly.
Edexcel S2 2015 June Q7
  1. A random variable \(X\) has probability density function
$$f ( x ) = \begin{cases} \frac { 2 x } { 15 } & 0 \leqslant x \leqslant k
\frac { 1 } { 5 } ( 5 - x ) & k < x \leqslant 5
0 & \text { otherwise } \end{cases}$$
  1. Showing your working clearly, find the value of \(k\).
  2. Write down the mode of \(X\).
  3. Find \(\mathrm { P } \left( \left. X \leqslant \frac { k } { 2 } \right\rvert \, X \leqslant k \right)\)
    郬
Edexcel S2 2016 June Q1
  1. During a typical day, a school website receives visits randomly at a rate of 9 per hour.
The probability that the school website receives fewer than \(v\) visits in a randomly selected one hour period is less than 0.75
  1. Find the largest possible value of \(v\)
  2. Find the probability that in a randomly selected one hour period, the school website receives at least 4 but at most 11 visits.
  3. Find the probability that in a randomly selected 10 minute period, the school website receives more than 1 visit.
  4. Using a suitable approximation, find the probability that in a randomly selected 8 hour period the school website receives more than 80 visits.
Edexcel S2 2016 June Q2
2. The random variable \(X \sim \mathrm {~B} ( 10 , p )\)
    1. Write down an expression for \(\mathrm { P } ( X = 3 )\) in terms of \(p\)
    2. Find the value of \(p\) such that \(\mathrm { P } ( X = 3 )\) is 16 times the value of \(\mathrm { P } ( X = 7 )\) The random variable \(Y \sim \operatorname { Po } ( \lambda )\)
  2. Find the value of \(\lambda\) such that \(\mathrm { P } ( Y = 3 )\) is 5 times the value of \(\mathrm { P } ( Y = 5 )\) The random variable \(W \sim \mathrm {~B} ( n , 0.4 )\)
  3. Find the value of \(n\) and the value of \(\alpha\) such that \(W\) can be approximated by the normal distribution, \(\mathrm { N } ( 32 , \alpha )\)
Edexcel S2 2016 June Q3
3. A single observation \(x\) is to be taken from \(X \sim \mathrm {~B} ( 12 , p )\) This observation is used to test \(\mathrm { H } _ { 0 } : p = 0.45\) against \(\mathrm { H } _ { 1 } : p > 0.45\)
  1. Using a \(5 \%\) level of significance, find the critical region for this test.
  2. State the actual significance level of this test. The value of the observation is found to be 9
  3. State the conclusion that can be made based on this observation.
  4. State whether or not this conclusion would change if the same test was carried out at the
    1. 10\% level of significance,
    2. \(1 \%\) level of significance.
Edexcel S2 2016 June Q4
  1. The waiting times, in minutes, between flight take-offs at an airport are modelled by the continuous random variable \(X\) with probability density function
$$f ( x ) = \begin{cases} \frac { 1 } { 5 } & 2 \leqslant x \leqslant 7
0 & \text { otherwise } \end{cases}$$
  1. Write down the name of this distribution. A randomly selected flight takes off at 9am
  2. Find the probability that the next flight takes off before 9.05 am
  3. Find the probability that at least 1 of the next 5 flights has a waiting time of more than 6 minutes.
  4. Find the cumulative distribution function of \(X\), for all \(x\)
  5. Sketch the cumulative distribution function of \(X\) for \(2 \leqslant x \leqslant 7\) On foggy days, an extra 2 minutes is added to each waiting time.
  6. Find the mean and variance of the waiting times between flight take-offs on foggy days.
Edexcel S2 2016 June Q5
5. A bag contains a large number of coins. It contains only \(1 \mathrm { p } , 5 \mathrm { p }\) and 10 p coins. The fraction of 1 p coins in the bag is \(q\), the fraction of 5 p coins in the bag is \(r\) and the fraction of 10p coins in the bag is \(s\). Two coins are selected at random from the bag and the coin with the highest value is recorded. Let \(M\) represent the value of the highest coin. The sampling distribution of \(M\) is given below
\(m\)1510
\(\mathrm { P } ( M = m )\)\(\frac { 1 } { 25 }\)\(\frac { 13 } { 80 }\)\(\frac { 319 } { 400 }\)
  1. List all the possible samples of two coins which may be selected.
  2. Find the value of \(q\), the value of \(r\) and the value of \(s\)
Edexcel S2 2016 June Q6
6. A continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} a x - b x ^ { 2 } & 0 \leqslant x \leqslant 2
0 & \text { otherwise } \end{cases}$$ Given that the mode is 1
  1. show that \(a = 2 b\)
  2. Find the value of \(a\) and the value of \(b\)
  3. Calculate F(1.5)
  4. State whether the upper quartile of \(X\) is greater than 1.5, equal to 1.5, or less than 1.5 Give a reason for your answer.
Edexcel S2 2016 June Q7
7. Last year \(4 \%\) of cars tested in a large chain of garages failed an emissions test. A random sample of \(n\) of these cars is taken. The number of cars that fail the test is represented by \(X\) Given that the standard deviation of \(X\) is 1.44
    1. find the value of \(n\)
    2. find \(\mathrm { E } ( X )\) A random sample of 20 of the cars tested is taken.
  2. Find the probability that all of these cars passed the emissions test. Given that at least 1 of these cars failed the emissions test,
  3. find the probability that exactly 3 of these cars failed the emissions test. A car mechanic claims that more than \(4 \%\) of the cars tested at the garage chain this year are failing the emissions test. A random sample of 125 of these cars is taken and 10 of these cars fail the emissions test.
  4. Using a suitable approximation, test whether or not there is evidence to support the mechanic's claim. Use a \(5 \%\) level of significance and state your hypotheses clearly.
Edexcel S2 2017 June Q1
  1. At a particular junction on a train line, signal failures are known to occur randomly at a rate of 1 every 4 days.
    1. Find the probability that there are no signal failures on a randomly selected day.
    2. Find the probability that there is at least 1 signal failure on each of the next 3 days.
    3. Find the probability that in a randomly selected 7 -day week, there are exactly 5 days with no signal failures.
    Repair works are carried out on the line. After these repair works, the number, \(f\), of signal failures in a 32-day period is recorded. A test is carried out, at the \(5 \%\) level of significance, to determine whether or not there has been a decrease in the rate of signal failures following the repair works.
  2. State the hypotheses for this test.
  3. Find the largest value of \(f\) for which the null hypothesis should be rejected.
Edexcel S2 2017 June Q2
2. Crispy-crisps produces packets of crisps. During a promotion, a prize is placed in \(25 \%\) of the packets. No more than 1 prize is placed in any packet. A box contains 6 packets of crisps.
    1. Write down a suitable distribution to model the number of prizes found in a box.
    2. Write down one assumption required for the model.
  2. Find the probability that in 2 randomly selected boxes, only 1 box contains exactly 1 prize.
  3. Find the probability that a randomly selected box contains at least 2 prizes. Neha buys 80 boxes of crisps.
  4. Using a normal approximation, find the probability that no more than 30 of the boxes contain at least 2 prizes.
Edexcel S2 2017 June Q3
3. The random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} a x + b & 1 \leqslant x < 4
\frac { 3 } { 2 } - \frac { 1 } { 4 } x & 4 \leqslant x \leqslant 6
0 & \text { otherwise } \end{cases}$$ as shown in Figure 1, where \(a\) and \(b\) are constants. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1a1534ea-4c62-4945-850a-9460ea87fd64-08_634_1132_694_397} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure}
  1. Show that the median of \(X\) is 4
  2. Find the value of \(a\) and the value of \(b\)
  3. Specify fully the cumulative distribution function of \(X\)
Edexcel S2 2017 June Q4
4. In a large population, past records show that 1 in 200 adults has a particular allergy. In a random sample of 700 adults selected from the population, estimate
    1. the mean number of adults with the allergy,
    2. the standard deviation of the number of adults with the allergy. Give your answer to 3 decimal places. A doctor claims that the past records are out of date and the proportion of adults with the allergy is higher than the records indicate. A random sample of 500 adults is taken from the population and 5 are found to have the allergy. A test of the doctor's claim is to be carried out at the 5\% level of significance.
    1. State the hypotheses for this test.
    2. Using a suitable approximation, carry out the test.
      (6) It is also claimed that \(30 \%\) of those with the allergy take medication for it daily. To test this claim, a random sample of \(n\) people with the allergy is taken. The random variable \(Y\) represents the number of people in the sample who take medication for the allergy daily. A two-tailed test, at the \(1 \%\) level of significance, is carried out to see if the proportion differs from 30\% The critical region for the test is \(Y = 0\) or \(Y \geqslant w\)
  3. Find the smallest possible value of \(n\) and the corresponding value of \(w\)
Edexcel S2 2017 June Q5
5. A call centre records the length of time, \(T\) minutes, its customers wait before being connected to an agent. The random variable \(T\) has a cumulative distribution function given by $$\mathrm { F } ( t ) = \left\{ \begin{array} { l r } 0 & t < 0
0.3 t - 0.004 t ^ { 3 } & 0 \leqslant t \leqslant 5
1 & t > 5 \end{array} \right.$$
  1. Find the proportion of customers waiting more than 4 minutes to be connected to an agent.
  2. Given that a customer waits more than 2 minutes to be connected to an agent, find the probability that the customer waits more than 4 minutes.
  3. Show that the upper quartile lies between 2.7 and 2.8 minutes.
  4. Find the mean length of time a customer waits to be connected to an agent.
Edexcel S2 2017 June Q6
6. At a men's tennis tournament there are 3 , 4 or 5 sets in a match. Over many years, data collected show that 50\% of matches last for exactly 3 sets, 30\% of matches last for exactly 4 sets and 20\% of matches last for exactly 5 sets. A random sample of 3 tennis matches is taken. The number of sets in each match is recorded as \(S _ { 1 } , S _ { 2 }\) and \(S _ { 3 }\) respectively. The random variable \(M\) represents the maximum value of \(S _ { 1 } , S _ { 2 }\) and \(S _ { 3 }\)
  1. List all the samples where \(M \neq 5\)
  2. Find the sampling distribution of \(M\)
  3. Write down the mode of \(S _ { 1 }\) and the mode of \(M\)
Edexcel S2 2017 June Q7
7. The continuous random variable \(X\) is uniformly distributed over the interval \([ a , b ]\)
  1. Find an expression, in terms of \(a\) and \(b\), for \(\mathrm { E } ( 3 - 2 X )\)
  2. Find \(\mathrm { P } \left( X > \frac { 1 } { 3 } b + \frac { 2 } { 3 } a \right)\) Given that \(\mathrm { E } ( X ) = 0\)
  3. find an expression, in terms of \(b\) only, for \(\mathrm { E } \left( 3 X ^ { 2 } \right)\) Given also that the range of \(X\) is 18
  4. find \(\operatorname { Var } ( X )\)
    \includegraphics[max width=\textwidth, alt={}]{1a1534ea-4c62-4945-850a-9460ea87fd64-24_2630_1828_121_121}
Edexcel S2 2018 June Q1
  1. A salesman sells insurance to people. Each day he chooses a number of people to contact. The probability that the salesman sells insurance to a person he contacts is 0.05
On Monday he chooses to contact 10 people.
  1. Find the probability that on Monday the salesman sells insurance to
    1. exactly 1 person,
    2. at least 3 people.
  2. Find the number of people he should contact each day in order to sell insurance, on average, to 3 people per day.
  3. Calculate the least number of people he must choose to contact on Friday, so that the probability of selling insurance to at least 1 person on Friday exceeds 0.99
Edexcel S2 2018 June Q2
2. John weaves cloth by hand. Emma believes that faults are randomly distributed in John's cloth at a rate of more than 4 per 50 metres of cloth. To check her belief, Emma takes a random sample of 100 metres of the cloth and finds that it contains 14 faults.
  1. Stating your hypotheses clearly, test, at the \(5 \%\) level of significance, Emma's belief. Armani also weaves cloth by hand. He knows that faults are randomly distributed in his cloth at a rate of 4 per 50 metres of cloth. Emma decides to buy a large amount of Armani's cloth to sell in pieces of length \(l\) metres. She chooses \(l\) so that the probability of no faults in a piece is exactly 0.9
  2. Show that \(l = 1.3\) to 2 significant figures. Emma sells 5000 of these pieces of cloth of length 1.3 metres. She makes a profit of \(\pounds 2.50\) on each piece of cloth that does not contain any faults but a loss of \(\pounds 0.50\) on any piece that contains at least one fault.
  3. Find Emma's expected profit.
Edexcel S2 2018 June Q3
3. A machine pours oil into bottles. It is electronically controlled to cut off the flow of oil randomly between 100 ml and \(k \mathrm { ml }\), where \(k > 100\). It is equally likely to cut off the flow at any point in this range. The random variable \(X\) is the volume of oil poured into a bottle. Given that \(\mathrm { P } ( 102 \leqslant X \leqslant k ) = \frac { 2 } { 3 }\)
  1. show that \(k = 106\)
  2. Find the probability that the volume of oil poured into a bottle is
    1. less than 105 ml ,
    2. exactly 105 ml .
  3. Write down the value of \(\mathrm { E } ( X )\)
  4. Find the 15th percentile of this distribution.
  5. Determine the value of \(x\) such that \(3 \mathrm { P } ( X \leqslant x - 1.5 ) = \mathrm { P } ( X \geqslant x + 1.5 )\)
Edexcel S2 2018 June Q4
4. The volume of milk, \(M\) litres, in cartons produced by a dairy, has distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\), where \(\mu\) and \(\sigma\) are unknown. A random sample of 12 cartons is taken and the volume of milk in each carton is measured ( \(M _ { 1 } , M _ { 2 } , \ldots , M _ { 12 }\) ). A statistic \(X\) is based on this sample.
  1. Explain what is meant by "a random sample" in this case.
  2. State the population in this case.
  3. Write down the distribution of \(\frac { M _ { 12 } - \mu } { \sigma }\)
  4. Explain what you understand by the sampling distribution of \(X\).
  5. State, giving a reason, which of the following is not a statistic based on this sample.
    (I) \(3 M _ { 1 } + \frac { 2 M _ { 11 } } { 6 }\)
    (II) \(\sum _ { i = 1 } ^ { 12 } \left( \frac { M _ { i } - \mu } { \sigma } \right) ^ { 2 }\)
    (III) \(\sum _ { i = 1 } ^ { 12 } \left( 2 M _ { i } - 3 \right)\)
Edexcel S2 2018 June Q5
5. Cars stop at a service station randomly at a rate of 3 every 5 minutes.
  1. Calculate the probability that in a randomly selected 10 minute period,
    1. exactly 7 cars will stop at the service station,
    2. more than 7 cars will stop at the service station. Using a normal approximation, the probability that more than 40 cars will stop at the service station during a randomly selected \(n\) minute period is 0.2266 correct to 4 significant figures.
  2. Find the value of \(n\).