Questions S2 (1690 questions)

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CAIE S2 2002 November Q6
10 marks Moderate -0.3
The average speed of a bus, \(x\) km h\(^{-1}\), on a certain journey is a continuous random variable \(X\) with probability density function given by $$\text{f}(x) = \begin{cases} \frac{k}{x^2} & 20 \leq x \leq 28, \\ 0 & \text{otherwise}. \end{cases}$$
  1. Show that \(k = 70\). [3]
  2. Find E\((X)\). [3]
  3. Find P\((X < \text{E}(X))\). [2]
  4. Hence determine whether the mean is greater or less than the median. [2]
CAIE S2 2002 November Q7
10 marks Standard +0.3
Bottles of wine are stacked in racks of 12. The weights of these bottles are normally distributed with mean 1.3 kg and standard deviation 0.06 kg. The weights of the empty racks are normally distributed with mean 2 kg and standard deviation 0.3 kg.
  1. Find the probability that the total weight of a full rack of 12 bottles of wine is between 17 kg and 18 kg. [5]
  2. Two bottles of wine are chosen at random. Find the probability that they differ in weight by more than 0.05 kg. [5]
CAIE S2 2011 November Q1
4 marks Moderate -0.8
Test scores, \(X\), have mean 54 and variance 144. The scores are scaled using the formula \(Y = a + bX\), where \(a\) and \(b\) are constants and \(b > 0\). The scaled scores, \(Y\), have mean 50 and variance 100. Find the values of \(a\) and \(b\). [4]
CAIE S2 2011 November Q2
5 marks Standard +0.3
35% of a random sample of \(n\) students walk to college. This result is used to construct an approximate 98% confidence interval for the population proportion of students who walk to college. Given that the width of this confidence interval is 0.157, correct to 3 significant figures, find \(n\). [5]
CAIE S2 2011 November Q3
7 marks Easy -1.2
Jack has to choose a random sample of 8 people from the 750 members of a sports club.
  1. Explain fully how he can use random numbers to choose the sample. [3]
Jack asks each person in the sample how much they spent last week in the club café. The results, in dollars, were as follows. 15 \quad 25 \quad 30 \quad 8 \quad 12 \quad 18 \quad 27 \quad 25
  1. Find unbiased estimates of the population mean and variance. [3]
  2. Explain briefly what is meant by 'population' in this question. [1]
CAIE S2 2011 November Q4
7 marks Moderate -0.3
The random variable \(X\) has probability density function given by $$f(x) = \begin{cases} ke^{-x} & 0 \leqslant x \leqslant 1, \\ 0 & \text{otherwise}. \end{cases}$$
  1. Show that \(k = \frac{e}{e-1}\). [3]
  2. Find E(\(X\)) in terms of \(e\). [4]
CAIE S2 2011 November Q5
8 marks Standard +0.3
Records show that the distance driven by a bus driver in a week is normally distributed with mean 1150 km and standard deviation 105 km. New driving regulations are introduced and in the next 20 weeks he drives a total of 21 800 km.
  1. Stating any assumption(s), test, at the 1% significance level, whether his mean weekly driving distance has decreased. [6]
  2. A similar test at the 1% significance level was carried out using the data from another 20 weeks. State the probability of a Type I error and describe what is meant by a Type I error in this context. [2]
CAIE S2 2011 November Q6
8 marks Standard +0.3
Ranjit goes to mathematics lectures and physics lectures. The length, in minutes, of a mathematics lecture is modelled by the variable \(X\) with distribution N(36, 3.5²). The length, in minutes, of a physics lecture is modelled by the independent variable \(Y\) with distribution N(55, 5.2²).
  1. Find the probability that the total length of two mathematics lectures and one physics lecture is less than 140 minutes. [4]
  2. Ranjit calculates how long he will need to spend revising the content of each lecture as follows. Each minute of a mathematics lecture requires 1 minute of revision and each minute of a physics lecture requires 1½ minutes of revision. Find the probability that the total revision time required for one mathematics lecture and one physics lecture is more than 100 minutes. [4]
CAIE S2 2011 November Q7
11 marks Standard +0.8
The numbers of men and women who visit a clinic each hour are independent Poisson variables with means 2.4 and 2.8 respectively.
  1. Find the probability that, in a half-hour period,
    1. 2 or more men and 1 or more women will visit the clinic, [4]
    2. a total of 3 or more people will visit the clinic. [3]
  2. Find the probability that, in a 10-hour period, a total of more than 60 people will visit the clinic. [4]
CAIE S2 2020 Specimen Q1
4 marks Moderate -0.8
Leaves from a certain type of tree have lengths that are distributed with standard deviation 3 cm. A random sample of 6 of these leaves is taken and the mean length of this sample is found to be 8 cm.
  1. Calculate a 95\% confidence interval for the population mean length. [3]
  2. Write down the probability that the whole 95\% confidence interval will lie below the population mean. [1]
CAIE S2 2020 Specimen Q2
3 marks Easy -1.8
Describe briefly how to use a random number generator to obtain a sample of 10 students from a group of 50 students. [3]
CAIE S2 2020 Specimen Q3
5 marks Moderate -0.3
The number of calls received at a small call centre has a Poisson distribution with mean 2 calls per 5 minute period.
  1. Find the probability exactly 4 calls in a 10 minute period. [2]
  2. Find the probability at least 3 calls in a 3 minute period. [3]
CAIE S2 2020 Specimen Q3
4 marks Standard +0.3
The number of calls received at a large call centre has a Poisson distribution with mean 4 calls per 5 minute period.
  1. [(c)] Use an approximation to find the probability that the number of calls received in a 5 minute period is between 4 and 9 inclusive. [4]
CAIE S2 2020 Specimen Q4
10 marks Standard +0.3
The lifetimes, in hours, of light bulbs have an exponential distribution with parameter \(\frac{1}{500}\). Each bulb is tested and rejected if the lifetime is less than 500 hours.
  1. Find the probability that a bulb of this type has a lifetime of more than 500 hours. [4]
  2. Find the probability that the lifetime is at least three times the expected lifetime. [6]
CAIE S2 2020 Specimen Q5
7 marks Standard +0.3
The diagram shows the graph of the probability density function of a random variable \(X\), where $$f(x) = \begin{cases} \frac{1}{6}(3x - x^2) & 0 \leq x \leq 3, \\ 0 & \text{otherwise}. \end{cases}$$ \includegraphics{figure_1}
  1. State the values of E(\(X\)) and Var(\(X\)). [4]
  2. State the values of P(\(0.5 < X < 1\)). [1]
  3. Given that P(\(1 < X < 2\)) = \(\frac{13}{27}\), find P(\(X > 2\)). [2]
CAIE S2 2020 Specimen Q6
9 marks Standard +0.3
At a certain hospital it was found that the probability that a patient did not arrive for an appointment was 0.1. The hospital's business model assumed that this probability will be reduced. They wish to test whether this probability is now less than 0.1. A random sample of 50 appointments is selected and the number of patients that did not arrive is noted. This figure is used as a test statistic at the 5\% significance level.
  1. Explain why this test is a one-tailed test and state suitable null and alternative hypotheses. [2]
  2. Use a binomial distribution to find the critical region and find the probability of a Type I error. [5]
  3. In fact 3 patients out of the 50 did not arrive. State the conclusion of the test, explaining your answer. [2]
CAIE S2 2020 Specimen Q7
7 marks Standard +0.3
The mean weight of bags of carrots is \(\mu\) kilograms. An inspector wishes to test whether \(\mu = 20\). He weighs a random sample of 6 bags and the results are summarised as follows: $$\Sigma x = 430 \quad \Sigma x^2 = 40$$ Carry out the test at the 5\% significance level. [7]
Edexcel S2 2016 January Q1
5 marks Easy -1.2
The manager of a clothing shop wishes to investigate how satisfied customers are with the quality of service they receive. A database of the shop's customers is used as a sampling frame for this investigation.
  1. Identify one potential problem with this sampling frame. [1]
Customers are asked to complete a survey about the quality of service they receive. Past information shows that 35\% of customers complete the survey. A random sample of 20 customers is taken.
  1. Write down a suitable distribution to model the number of customers in this sample that complete the survey. [2]
  2. Find the probability that more than half of the customers in the sample complete the survey. [2]
Edexcel S2 2016 January Q2
10 marks Moderate -0.3
The continuous random variable \(X\) is uniformly distributed over the interval \([a, b]\) Given that \(\mathrm{P}(3 < X < 5) = \frac{1}{8}\) and \(\mathrm{E}(X) = 4\)
  1. find the value of \(a\) and the value of \(b\) [3]
  2. find the value of the constant, \(c\), such that \(\mathrm{E}(cX - 2) = 0\) [2]
  3. find the exact value of \(\mathrm{E}(X^2)\) [3]
  4. find \(\mathrm{P}(2X - b > a)\) [2]
Edexcel S2 2016 January Q3
11 marks Moderate -0.3
Left-handed people make up 10\% of a population. A random sample of 60 people is taken from this population. The discrete random variable \(Y\) represents the number of left-handed people in the sample.
    1. Write down an expression for the exact value of \(\mathrm{P}(Y \leq 1)\)
    2. Evaluate your expression, giving your answer to 3 significant figures. [3]
  2. Using a Poisson approximation, estimate \(\mathrm{P}(Y \leq 1)\) [2]
  3. Using a normal approximation, estimate \(\mathrm{P}(Y \leq 1)\) [5]
  4. Give a reason why the Poisson approximation is a more suitable estimate of \(\mathrm{P}(Y \leq 1)\) [1]
Edexcel S2 2016 January Q4
12 marks Standard +0.3
A continuous random variable \(X\) has cumulative distribution function $$\mathrm{F}(x) = \begin{cases} 0 & x < 0 \\ \frac{1}{4}x & 0 \leq x \leq 1 \\ \frac{1}{20}x^4 + \frac{1}{5} & 1 < x \leq d \\ 1 & x > d \end{cases}$$
  1. Show that \(d = 2\) [2]
  2. Find \(\mathrm{P}(X < 1.5)\) [2]
  3. Write down the value of the lower quartile of \(X\) [1]
  4. Find the median of \(X\) [3]
  5. Find, to 3 significant figures, the value of \(k\) such that \(\mathrm{P}(X > 1.9) = \mathrm{P}(X < k)\) [4]
Edexcel S2 2016 January Q5
10 marks Standard +0.3
The number of eruptions of a volcano in a 10 year period is modelled by a Poisson distribution with mean 1
  1. Find the probability that this volcano erupts at least once in each of 2 randomly selected 10 year periods. [2]
  2. Find the probability that this volcano does not erupt in a randomly selected 20 year period. [2]
The probability that this volcano erupts exactly 4 times in a randomly selected \(w\) year period is 0.0443 to 3 significant figures.
  1. Use the tables to find the value of \(w\) [3]
A scientist claims that the mean number of eruptions of this volcano in a 10 year period is more than 1 She selects a 100 year period at random in order to test her claim.
  1. State the null hypothesis for this test. [1]
  2. Determine the critical region for the test at the 5\% level of significance. [2]
Edexcel S2 2016 January Q6
15 marks Standard +0.3
A continuous random variable \(X\) has probability density function $$\mathrm{f}(x) = \begin{cases} ax^2 + bx & 1 \leq x \leq 7 \\ 0 & \text{otherwise} \end{cases}$$ where \(a\) and \(b\) are constants.
  1. Show that \(114a + 24b = 1\) [4]
Given that \(a = \frac{1}{90}\)
  1. use algebraic integration to find \(\mathrm{E}(X)\) [4]
  2. find the cumulative distribution function of \(X\), specifying it for all values of \(x\) [3]
  3. find \(\mathrm{P}(X > \mathrm{E}(X))\) [2]
  4. use your answer to part (d) to describe the skewness of the distribution. [2]
Edexcel S2 2016 January Q7
12 marks Standard +0.3
A fisherman is known to catch fish at a mean rate of 4 per hour. The number of fish caught by the fisherman in an hour follows a Poisson distribution. The fisherman takes 5 fishing trips each lasting 1 hour.
  1. Find the probability that this fisherman catches at least 6 fish on exactly 3 of these trips. [6]
The fisherman buys some new equipment and wants to test whether or not there is a change in the mean number of fish caught per hour. Given that the fisherman caught 14 fish in a 2 hour period using the new equipment,
  1. carry out the test at the 5\% level of significance. State your hypotheses clearly. [6]
Edexcel S2 Q1
6 marks Easy -1.8
The small village of Tornep has a preservation society which is campaigning for a new by-pass to be built. The society needs to measure
  1. the strength of opinion amongst the residents of Tornep for the scheme and
  2. the flow of traffic through the village on weekdays. The society wants to know whether to use a census or a sample survey for each of these measures.
    1. In each case suggest which they should use and specify a suitable sampling frame. [4] For the measurement of traffic flow through Tornep,
    2. suggest a suitable statistic and a possible statistical model for this statistic. [2]