CAIE Further Paper 4 (Further Paper 4) 2020 Specimen

1

1. State briefly the circumstances under which a non-parametric test of significance should be used rather than a parametric test. The level of pollution in a river was measured at 12 randomly chosen locations. The results, in suitable units, are shown below, where higher values represent greater pollution.

   5.62

   5.73

   6.55

   6.81

   6.10

   5.75

   5.87

   6.47

   5.86

   6.26

   6.99

   5.91

2. Use a Wilcoxon signed-rank test to test whether the average pollution level in the river is more than 6.00. Use a \(5\%\) significance level.
   [0pt] [6]

2 Each of 200 identically biased dice is thrown repeatedly until an even number is obtained. The number of throws needed is recorded and the results are summarised in the following table. Carry out a goodness of fit test, at the \(5\%\) significance level, to test whether \(\operatorname{Geo}(0.6)\) is a satisfactory model for the data.

3 Employees at a particular company have been working seven hours each day, from 9 am to 4 pm. To try to reduce absence, the company decides to introduce 'flexi-time' and allow employees to work their seven hours each day at any time between 7 am and 9 pm. For a random sample of 10 employees, the numbers of hours of absence in the year before and the year after the introduction of flexi-time are given in the following table. Test, at the \(10\%\) significance level, whether the population mean number of hours of absence has decreased following the introduction of flexi-time, stating any assumption that you make.

4 The number, \(x\), of a certain type of sea shell was counted at 60 randomly chosen sites, each one metre square, along the coastline in country \(A\). The number, \(y\), of the same type of sea shell was counted at 50 randomly chosen sites, each one metre square, along the coastline in country \(B\). The results are summarised as follows, where \(\bar{x}\) and \(\bar{y}\) denote the sample means of \(x\) and \(y\) respectively. $$\bar{x} = 29.2 \quad \Sigma(x - \bar{x})^{2} = 4341.6 \quad \bar{y} = 24.4 \quad \Sigma(y - \bar{y})^{2} = 3732.0$$ Find a \(95\%\) confidence interval for the difference between the mean number of sea shells, per square metre, on the coastlines in country \(A\) and in country \(B\).

5 The continuous random variable \(X\) has probability density function f given by $$f(x) = \begin{cases} 0 & x < 0 \\ \frac{6}{5} x & 0 \leqslant x \leqslant 1 \\ \frac{6}{5} x^{-4} & x > 1 \end{cases}$$

1. Find \(\mathrm{P}(X > 1)\).
2. Find the median value of \(X\).
3. Given that \(\mathrm{E}(X) = 1\), find the variance of \(X\).
4. Find \(\mathrm{E}(\sqrt{X})\).

6 Aisha has a bag containing 3 red balls and 3 white balls. She selects a ball at random, notes its colour and returns it to the bag; the same process is repeated twice more. The number of red balls selected by Aisha is denoted by \(X\).

1. Find the probability generating function \(\mathrm{G}_{X}(t)\) of \(X\).

Basant also has a bag containing 3 red balls and 3 white balls. He selects three balls at random, without replacement, from his bag. The number of red balls selected by Basant is denoted by \(Y\).

1. Find the probability generating function \(\mathrm{G}_{Y}(t)\) of \(Y\).

The random variable \(Z\) is the total number of red balls selected by Aisha and Basant.

1. Find the probability generating function of \(Z\), expressing your answer as a polynomial.
2. Use the probability generating function of \(Z\) to find \(\mathrm{E}(Z)\) and \(\operatorname{Var}(Z)\).