Questions Further Paper 4 (133 questions)

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CAIE Further Paper 4 2023 November Q4
9 marks Standard +0.3
  1. Given that \(\mathrm { P } ( X \leqslant 2 ) = \frac { 1 } { 3 }\), show that \(m = \frac { 1 } { 6 }\) and find the values of \(k\) and \(c\).
  2. Find the exact numerical value of the interquartile range of \(X\).
CAIE Further Paper 4 2021 June Q1
6 marks Standard +0.3
Farmer A grows apples of a certain variety. Each tree produces 14.8 kg of apples, on average, per year. Farmer B grows apples of the same variety and claims that his apple trees produce a higher mass of apples per year than Farmer A's trees. The masses of apples from Farmer B's trees may be assumed to be normally distributed. A random sample of 10 trees from Farmer B is chosen. The masses, \(x\) kg, of apples produced in a year are summarised as follows. $$\sum x = 152.0 \qquad \sum x^2 = 2313.0$$ Test, at the 5% significance level, whether Farmer B's claim is justified. [6]
CAIE Further Paper 4 2021 June Q2
7 marks Standard +0.3
A company is developing a new flavour of chocolate by varying the quantities of the ingredients. A random selection of 9 flavours of chocolate are judged by two tasters who each give marks out of 100 to each flavour of chocolate.
ChocolateABCDEFGHI
Taster 1728675929879876062
Taster 2847274958587827568
Carry out a Wilcoxon matched-pairs signed-rank test at the 10% significance level to investigate whether, on average, there is a difference between marks awarded by the two tasters. [7]
CAIE Further Paper 4 2021 June Q3
8 marks Standard +0.8
The heights, \(x\) m, of a random sample of 50 adult males from country A were recorded. The heights, \(y\) m, of a random sample of 40 adult males from country B were also recorded. The results are summarised as follows. $$\sum x = 89.0 \qquad \sum x^2 = 159.4 \qquad \sum y = 67.2 \qquad \sum y^2 = 113.1$$ Find a 95% confidence interval for the difference between the mean heights of adult males from country A and adult males from country B. [8]
CAIE Further Paper 4 2021 June Q4
5 marks Standard +0.8
\(X\) is a discrete random variable which takes the values 0, 2, 4, ... . The probability generating function of \(X\) is given by $$G_X(t) = \frac{1}{3 - 2t^2}.$$
  1. Find E\((X)\) and Var\((X)\). [5]
CAIE Further Paper 4 2021 June Q4
3 marks Standard +0.8
  1. Find P\((X = 4)\). [3]
CAIE Further Paper 4 2021 June Q5
10 marks Standard +0.3
Chai packs china mugs into cardboard boxes. Chai's manager suspects that breakages occur at random times and that the number of breakages may follow a Poisson distribution. He takes a small sample of observations and finds that the number of breakages in a one-hour period has a mean of 2.4 and a standard deviation of 1.5.
  1. Explain how this information tends to support the manager's suspicion. [2]
The manager now takes a larger sample and claims that the numbers of breakages in a one-hour period follow a Poisson distribution. The numbers of breakages in a random sample of 180 one-hour periods are summarised in the following table.
Number of breakages01234567 or more
Frequency213346312316100
The mean number of breakages calculated from this sample is 2.5.
  1. Use the data from this larger sample to carry out a goodness of fit test, at the 10% significance level, to test the claim. [8]
CAIE Further Paper 4 2021 June Q6
14 marks Standard +0.8
The continuous random variable \(X\) has probability density function f given by $$f(x) = \begin{cases} \frac{1}{8} & 0 \leq x < 1, \\ \frac{1}{28}(8 - x) & 1 \leq x \leq 8, \\ 0 & \text{otherwise}. \end{cases}$$
  1. Find the cumulative distribution function of \(X\). [3]
  1. Find the value of the constant \(a\) such that P\((X \leq a) = \frac{5}{7}\). [3]
The random variable \(Y\) is given by \(Y = \sqrt[3]{X}\).
  1. Find the probability density function of \(Y\). [5]