CAIE Further Paper 4 (Further Paper 4) 2024 June

1 The times taken by members of a large cycling club to complete a cross-country circuit have a normal distribution with mean \(\mu\) minutes. The times taken, \(x\) minutes, are recorded for a random sample of 14 members of the club. The results are summarised as follows, where \(\bar { x }\) is the sample mean. $$\bar { x } = 42.8 \quad \sum ( x - \bar { x } ) ^ { 2 } = 941.5$$ Find a 95\% confidence interval for \(\mu\).

2 A large number of students are taking a Physics course. They are assessed by a practical examination and a written examination. The marks out of 100 obtained by a random sample of 15 students in each of the examinations are as follows. Use a sign test, at the \(10 \%\) significance level, to test whether, on average, the practical examination marks are higher than the written examination marks.

3 A factory produces metal discs. The manager claims that the diameters of these discs have a median of 22.0 mm . The diameters, in mm , of a random sample of 12 discs produced by this factory are as follows. $$\begin{array} { l l l l l l l l l l l l } 22.4 & 20.9 & 22.8 & 21.5 & 23.2 & 22.9 & 23.9 & 21.7 & 19.8 & 23.6 & 22.6 & 23.0 \end{array}$$

1. Carry out a Wilcoxon signed-rank test, at the \(10 \%\) significance level, to test whether there is any evidence against the manager's claim.
2. State an assumption that is necessary for this test to be valid.

4 The random variable \(Y\) is the sum of two independent observations of the random variable \(X\). The probability generating function \(\mathrm { G } _ { Y } ( \mathrm { t } )\) of \(Y\) is given by $$G _ { Y } ( t ) = \frac { t ^ { 2 } } { ( 4 - 3 t ) ^ { 4 } }$$

1. Find \(\mathrm { E } ( \mathrm { Y } )\).
2. Write down an expression for the probability generating function of \(X\).
3. Find \(\mathrm { P } ( X = 4 )\).

5 Two companies, \(P\) and \(Q\), produce a certain type of paint brush. An independent examiner rates the quality of the brushes produced as poor, satisfactory or good. He takes a random sample of brushes from each company. The examiner's ratings are summarised in the table.

1. Test, at the \(5 \%\) significance level, whether quality of brushes is independent of company.
2. Compare the quality of the brushes produced by the two companies.

6 Jade is a swimming instructor at a sports college. She claims that, as a result of an intensive training course, the mean time taken by students to swim 50 metres has reduced by more than 1 second. She chooses a random sample of 10 students. The times taken, in seconds, before and after the training course are recorded in the table.

1. Test, at the 10\% significance level, whether Jade's claim is justified.
2. State an assumption that is necessary for this test to be valid.

7 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \left\{ \begin{array} { c c } \frac { x } { 4 } \left( 4 - x ^ { 2 } \right) & 0 \leqslant x \leqslant 2
0 & \text { otherwise } \end{array} \right.$$

1. Find \(\operatorname { Var } ( \sqrt { X } )\).
   The continuous random variable \(Y\) is defined by \(Y = X ^ { 2 }\).
2. Find the probability density function of \(Y\).
3. Find the exact value of the median of \(Y\).
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