Questions S2 (1690 questions)

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Edexcel S2 2008 January Q4
7 marks Standard +0.3
  1. The continuous random variable \(Y\) has cumulative distribution function \(\mathrm { F } ( y )\) given by
$$\mathrm { F } ( y ) = \left\{ \begin{array} { c l } 0 & y < 1 \\ k \left( y ^ { 4 } + y ^ { 2 } - 2 \right) & 1 \leqslant y \leqslant 2 \\ 1 & y > 2 \end{array} \right.$$
  1. Show that \(k = \frac { 1 } { 18 }\).
  2. Find \(\mathrm { P } ( Y > 1.5 )\).
  3. Specify fully the probability density function f(y).
Edexcel S2 2008 January Q5
7 marks Moderate -0.3
  1. Dhriti grows tomatoes. Over a period of time, she has found that there is a probability 0.3 of a ripe tomato having a diameter greater than 4 cm . She decides to try a new fertiliser. In a random sample of 40 ripe tomatoes, 18 have a diameter greater than 4 cm . Dhriti claims that the new fertiliser has increased the probability of a ripe tomato being greater than 4 cm in diameter.
Test Dhriti's claim at the 5\% level of significance. State your hypotheses clearly.
Edexcel S2 2008 January Q6
12 marks Standard +0.3
6. The probability that a sunflower plant grows over 1.5 metres high is 0.25 . A random sample of 40 sunflower plants is taken and each sunflower plant is measured and its height recorded.
  1. Find the probability that the number of sunflower plants over 1.5 m high is between 8 and 13 (inclusive) using
    1. a Poisson approximation,
    2. a Normal approximation.
  2. Write down which of the approximations used in part (a) is the most accurate estimate of the probability. You must give a reason for your answer.
Edexcel S2 2008 January Q7
14 marks Standard +0.3
  1. Explain what you understand by
    1. a hypothesis test,
    2. a critical region. During term time, incoming calls to a school are thought to occur at a rate of 0.45 per minute. To test this, the number of calls during a random 20 minute interval, is recorded.
  2. Find the critical region for a two-tailed test of the hypothesis that the number of incoming calls occurs at a rate of 0.45 per 1 minute interval. The probability in each tail should be as close to \(2.5 \%\) as possible.
  3. Write down the actual significance level of the above test. In the school holidays, 1 call occurs in a 10 minute interval.
  4. Test, at the \(5 \%\) level of significance, whether or not there is evidence that the rate of incoming calls is less during the school holidays than in term time.
Edexcel S2 2008 January Q8
13 marks Moderate -0.3
  1. The continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) given by
$$f ( x ) = \left\{ \begin{array} { c c } 2 ( x - 2 ) & 2 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{array} \right.$$
  1. Sketch \(\mathrm { f } ( x )\) for all values of \(x\).
  2. Write down the mode of \(X\). Find
  3. \(\mathrm { E } ( X )\),
  4. the median of \(X\).
  5. Comment on the skewness of this distribution. Give a reason for your answer.
Edexcel S2 2010 January Q1
8 marks Easy -1.2
A manufacturer supplies DVD players to retailers in batches of 20 . It has \(5 \%\) of the players returned because they are faulty.
  1. Write down a suitable model for the distribution of the number of faulty DVD players in a batch. Find the probability that a batch contains
  2. no faulty DVD players,
  3. more than 4 faulty DVD players.
  4. Find the mean and variance of the number of faulty DVD players in a batch.
Edexcel S2 2010 January Q2
10 marks Moderate -0.8
2. A continuous random variable \(X\) has cumulative distribution function $$\mathrm { F } ( x ) = \begin{cases} 0 , & x < - 2 \\ \frac { x + 2 } { 6 } , & - 2 \leqslant x \leqslant 4 \\ 1 , & x > 4 \end{cases}$$
  1. Find \(\mathrm { P } ( X < 0 )\).
  2. Find the probability density function \(\mathrm { f } ( x )\) of \(X\).
  3. Write down the name of the distribution of \(X\).
  4. Find the mean and the variance of \(X\).
  5. Write down the value of \(\mathrm { P } ( X = 1 )\).
Edexcel S2 2010 January Q3
10 marks Moderate -0.3
A robot is programmed to build cars on a production line. The robot breaks down at random at a rate of once every 20 hours.
  1. Find the probability that it will work continuously for 5 hours without a breakdown. Find the probability that, in an 8 hour period,
  2. the robot will break down at least once,
  3. there are exactly 2 breakdowns. In a particular 8 hour period, the robot broke down twice.
  4. Write down the probability that the robot will break down in the following 8 hour period. Give a reason for your answer.
Edexcel S2 2010 January Q4
17 marks Standard +0.3
  1. The continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) given by
$$f ( x ) = \begin{cases} k \left( x ^ { 2 } - 2 x + 2 \right) & 0 < x \leqslant 3 \\ 3 k & 3 < x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = \frac { 1 } { 9 }\)
  2. Find the cumulative distribution function \(\mathrm { F } ( x )\).
  3. Find the mean of \(X\).
  4. Show that the median of \(X\) lies between \(x = 2.6\) and \(x = 2.7\)
Edexcel S2 2010 January Q5
9 marks Standard +0.3
  1. A café serves breakfast every morning. Customers arrive for breakfast at random at a rate of 1 every 6 minutes.
Find the probability that
  1. fewer than 9 customers arrive for breakfast on a Monday morning between 10 am and 11 am. The café serves breakfast every day between 8 am and 12 noon.
  2. Using a suitable approximation, estimate the probability that more than 50 customers arrive for breakfast next Tuesday.
Edexcel S2 2010 January Q6
10 marks Standard +0.3
6.
  1. Define the critical region of a test statistic. A discrete random variable \(X\) has a Binomial distribution \(\mathrm { B } ( 30 , p )\). A single observation is used to test \(\mathrm { H } _ { 0 } : p = 0.3\) against \(\mathrm { H } _ { 1 } : p \neq 0.3\)
  2. Using a \(1 \%\) level of significance find the critical region of this test. You should state the probability of rejection in each tail which should be as close as possible to 0.005
  3. Write down the actual significance level of the test. The value of the observation was found to be 15 .
  4. Comment on this finding in light of your critical region.
Edexcel S2 2010 January Q7
11 marks Moderate -0.5
A bag contains a large number of coins. It contains only \(1 p\) and \(2 p\) coins in the ratio \(1 : 3\)
  1. Find the mean \(\mu\) and the variance \(\sigma ^ { 2 }\) of the values of this population of coins. A random sample of size 3 is taken from the bag.
  2. List all the possible samples.
  3. Find the sampling distribution of the mean value of the samples.
Edexcel S2 2012 January Q1
8 marks Moderate -0.8
  1. The time in minutes that Elaine takes to checkout at her local supermarket follows a continuous uniform distribution defined over the interval [3,9].
Find
  1. Elaine's expected checkout time,
  2. the variance of the time taken to checkout at the supermarket,
  3. the probability that Elaine will take more than 7 minutes to checkout. Given that Elaine has already spent 4 minutes at the checkout,
  4. find the probability that she will take a total of less than 6 minutes to checkout.
Edexcel S2 2012 January Q2
7 marks Moderate -0.3
2. David claims that the weather forecasts produced by local radio are no better than those achieved by tossing a fair coin and predicting rain if a head is obtained or no rain if a tail is obtained. He records the weather for 30 randomly selected days. The local radio forecast is correct on 21 of these days. Test David's claim at the \(5 \%\) level of significance. State your hypotheses clearly.
Edexcel S2 2012 January Q3
9 marks Moderate -0.8
3. The probability of a telesales representative making a sale on a customer call is 0.15 Find the probability that
  1. no sales are made in 10 calls,
  2. more than 3 sales are made in 20 calls. Representatives are required to achieve a mean of at least 5 sales each day.
  3. Find the least number of calls each day a representative should make to achieve this requirement.
  4. Calculate the least number of calls that need to be made by a representative for the probability of at least 1 sale to exceed 0.95
Edexcel S2 2012 January Q4
16 marks Moderate -0.3
4. A website receives hits at a rate of 300 per hour.
  1. State a distribution that is suitable to model the number of hits obtained during a 1 minute interval.
  2. State two reasons for your answer to part (a). Find the probability of
  3. 10 hits in a given minute,
  4. at least 15 hits in 2 minutes. The website will go down if there are more than 70 hits in 10 minutes.
  5. Using a suitable approximation, find the probability that the website will go down in a particular 10 minute interval.
Edexcel S2 2012 January Q5
7 marks Moderate -0.3
The probability of an electrical component being defective is 0.075 The component is supplied in boxes of 120
  1. Using a suitable approximation, estimate the probability that there are more than 3 defective components in a box. A retailer buys 2 boxes of components.
  2. Estimate the probability that there are at least 4 defective components in each box.
Edexcel S2 2012 January Q6
18 marks Standard +0.3
6. A random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { 1 } { 2 } & 0 \leqslant x < 1 \\ x - \frac { 1 } { 2 } & 1 \leqslant x \leqslant k \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a positive constant.
  1. Sketch the graph of \(\mathrm { f } ( x )\).
  2. Show that \(k = \frac { 1 } { 2 } ( 1 + \sqrt { 5 } )\).
  3. Define fully the cumulative distribution function \(\mathrm { F } ( x )\).
  4. Find \(\mathrm { P } ( 0.5 < X < 1.5 )\).
  5. Write down the median of \(X\) and the mode of \(X\).
  6. Describe the skewness of the distribution of \(X\). Give a reason for your answer.
Edexcel S2 2012 January Q7
10 marks Standard +0.3
7. (a) Explain briefly what you understand by
  1. a critical region of a test statistic,
  2. the level of significance of a hypothesis test.
    (b) An estate agent has been selling houses at a rate of 8 per month. She believes that the rate of sales will decrease in the next month.
  3. Using a \(5 \%\) level of significance, find the critical region for a one tailed test of the hypothesis that the rate of sales will decrease from 8 per month.
  4. Write down the actual significance level of the test in part (b)(i). The estate agent is surprised to find that she actually sold 13 houses in the next month. She now claims that this is evidence of an increase in the rate of sales per month.
    (c) Test the estate agent's claim at the \(5 \%\) level of significance. State your hypotheses clearly.
Edexcel S2 2013 January Q1
5 marks Easy -1.2
  1. Write down the conditions under which the Poisson distribution can be used as an approximation to the binomial distribution. The probability of any one letter being delivered to the wrong house is 0.01 On a randomly selected day Peter delivers 1000 letters.
  2. Using a Poisson approximation, find the probability that Peter delivers at least 4 letters to the wrong house. Give your answer to 4 decimal places.
Edexcel S2 2013 January Q2
11 marks Moderate -0.3
2. In a village, power cuts occur randomly at a rate of 3 per year.
  1. Find the probability that in any given year there will be
    1. exactly 7 power cuts,
    2. at least 4 power cuts.
  2. Use a suitable approximation to find the probability that in the next 10 years the number of power cuts will be less than 20
Edexcel S2 2013 January Q3
10 marks Standard +0.3
A random variable \(X\) has the distribution \(\mathrm { B } ( 12 , p )\).
  1. Given that \(p = 0.25\) find
    1. \(\mathrm { P } ( X < 5 )\)
    2. \(\mathrm { P } ( X \geqslant 7 )\)
  2. Given that \(\mathrm { P } ( X = 0 ) = 0.05\), find the value of \(p\) to 3 decimal places.
  3. Given that the variance of \(X\) is 1.92 , find the possible values of \(p\).
Edexcel S2 2013 January Q4
14 marks Standard +0.3
4. The continuous random variable \(X\) is uniformly distributed over the interval \([ - 4,6 ]\).
  1. Write down the mean of \(X\).
  2. Find \(\mathrm { P } ( X \leqslant 2.4 )\)
  3. Find \(\mathrm { P } ( - 3 < X - 5 < 3 )\) The continuous random variable \(Y\) is uniformly distributed over the interval \([ a , 4 a ]\).
  4. Use integration to show that \(\mathrm { E } \left( Y ^ { 2 } \right) = 7 a ^ { 2 }\)
  5. Find \(\operatorname { Var } ( Y )\).
  6. Given that \(\mathrm { P } \left( X < \frac { 8 } { 3 } \right) = \mathrm { P } \left( Y < \frac { 8 } { 3 } \right)\), find the value of \(a\).
Edexcel S2 2013 January Q5
10 marks Moderate -0.8
5. The continuous random variable \(T\) is used to model the number of days, \(t\), a mosquito survives after hatching. The probability that the mosquito survives for more than \(t\) days is $$\frac { 225 } { ( t + 15 ) ^ { 2 } } , \quad t \geqslant 0$$
  1. Show that the cumulative distribution function of \(T\) is given by $$\mathrm { F } ( t ) = \begin{cases} 1 - \frac { 225 } { ( t + 15 ) ^ { 2 } } & t \geqslant 0 \\ 0 & \text { otherwise } \end{cases}$$
  2. Find the probability that a randomly selected mosquito will die within 3 days of hatching.
  3. Given that a mosquito survives for 3 days, find the probability that it will survive for at least 5 more days. A large number of mosquitoes hatch on the same day.
  4. Find the number of days after which only \(10 \%\) of these mosquitoes are expected to survive.
Edexcel S2 2013 January Q6
10 marks Standard +0.3
6.
  1. Explain what you understand by a hypothesis.
  2. Explain what you understand by a critical region. Mrs George claims that 45\% of voters would vote for her.
    In an opinion poll of 20 randomly selected voters it was found that 5 would vote for her.
  3. Test at the \(5 \%\) level of significance whether or not the opinion poll provides evidence to support Mrs George's claim. In a second opinion poll of \(n\) randomly selected people it was found that no one would vote for Mrs George.
  4. Using a \(1 \%\) level of significance, find the smallest value of \(n\) for which the hypothesis \(\mathrm { H } _ { 0 } : p = 0.45\) will be rejected in favour of \(\mathrm { H } _ { 1 } : p < 0.45\)