CAIE Further Paper 4 (Further Paper 4) 2022 November

1 A basketball club has a large number of players. The heights, \(x \mathrm {~m}\), of a random sample of 10 of these players are measured. A \(90 \%\) confidence interval for the population mean height, \(\mu \mathrm { m }\), of players in this club is calculated. It is assumed that heights are normally distributed. The confidence interval is \(1.78 \leqslant \mu \leqslant 2.02\). Find the values of \(\sum x\) and \(\sum x ^ { 2 }\) for this sample.

2 In the colleges in three regions of a particular country, students are given individual targets to achieve. Their performance is measured against their individual target and graded as 'above target', 'on target' or 'below target'. For a random sample of students from each of the three regions, the observed frequencies are summarised in the following table. Test, at the 10\% significance level, whether performance is independent of region.

3 A scientist is investigating the masses of birds of a certain species in country \(X\) and country \(Y\). She takes a random sample of 50 birds of this species from country \(X\) and a random sample of 80 birds of this species from country \(Y\). She records their masses in \(\mathrm { kg } , x\) and \(y\), respectively. Her results are summarised as follows. $$\sum x = 75.5 \quad \sum x ^ { 2 } = 115.2 \quad \sum y = 116.8 \quad \sum y ^ { 2 } = 172.6$$ The population mean masses of these birds in countries \(X\) and \(Y\) are \(\mu _ { x } \mathrm {~kg}\) and \(\mu _ { y } \mathrm {~kg}\) respectively.
Test, at the \(5 \%\) significance level, the null hypothesis \(\mu _ { \mathrm { x } } = \mu _ { \mathrm { y } }\) against the alternative hypothesis \(\mu _ { \mathrm { x } } > \mu _ { \mathrm { y } }\). State your conclusion in the context of the question.

4 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} k & 0 \leqslant x < 1
k x & 1 \leqslant x \leqslant 2
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac { 2 } { 5 }\).
2. Find the interquartile range of \(X\).
3. Find \(\operatorname { Var } ( X )\).

5 A 6 -sided dice, \(A\), with faces numbered \(1,2,3,4,5,6\) is biased so that the probability of throwing a 6 is \(\frac { 1 } { 4 }\). The random variable \(X\) is the number of 6s obtained when dice \(A\) is thrown twice.

1. Find the probability generating function of \(X\).
   A second dice, \(B\), with faces numbered \(1,2,3,4,5,6\) is unbiased. The random variable \(Y\) is the number of 6s obtained when dice \(B\) is thrown twice. The random variable \(Z\) is the total number of 6s obtained when both dice are thrown twice.
2. Find the probability generating function of \(Z\), expressing your answer as a polynomial.
3. Find \(\operatorname { Var } ( Z )\).
4. Use the probability generating function of \(Z\) to find the most probable value of \(Z\).

6 The manager of a technology company \(A\) claims that his employees earn more per year than the employees at technology company \(B\). The amounts earned per year, in hundreds of dollars, by a random sample of 12 employees from company \(A\) and an independent random sample of 12 employees from company \(B\) are shown below.

1. Carry out a Wilcoxon rank-sum test at the \(5 \%\) significance level to test whether the manager's claim is supported by the data.
2. Explain whether a paired sample \(t\)-test would be appropriate to test the manager's claim if earnings are normally distributed.
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