CAIE Further Paper 4 (Further Paper 4) 2024 June

1 A college uses two assessments, \(X\) and \(Y\), when interviewing applicants for research posts at the college. These assessments have been used for a large number of applicants this year. The scores for a random sample of 9 applicants who took assessment \(X\) are as follows. $$\begin{array} { l l l l l l l l l } 21.4 & 24.6 & 25.3 & 22.7 & 20.8 & 21.5 & 22.9 & 21.3 & 22.3 \end{array}$$ The scores for a random sample of 10 applicants who took assessment \(Y\) are as follows. $$\begin{array} { l l l l l l l l l l } 20.9 & 23.5 & 24.8 & 21.9 & 23.4 & 24.0 & 23.8 & 24.1 & 25.1 & 25.8 \end{array}$$ The interviewer believes that the population median score from assessment \(X\) is lower than the population median score from assessment \(Y\). Carry out a Wilcoxon rank-sum test, at the \(1 \%\) significance level, to test whether the interviewer's belief is supported by the data.
\includegraphics[max width=\textwidth, alt={}, center]{b5ff998a-fcb6-4a1b-ae86-ec66b0dccc3c-02_2714_37_143_2008}

2 A rowing club has a large number of members.A random sample of 12 of these members is taken and the pulse rate,\(x\) beats per minute(bpm),of each is measured after a 30 -minute training session.A \(98 \%\) confidence interval for the population mean pulse rate,\(\mu \mathrm { bpm }\) ,is calculated from the sample as \(64.22 < \mu < 68.66\) .

1. Find the values of \(\sum x\) and \(\sum x ^ { 2 }\) .
2. State an assumption that is necessary for the confidence interval to be valid.
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3 There are three bus companies in a city. The council is investigating whether the buses reliably arrive at their destination on time. The results from random samples of buses from each company are summarised in the following table. Test, at the \(5 \%\) significance level, whether the reliability of buses is independent of bus company.

4 The random variable \(X\) has probability generating function \(\mathrm { G } _ { X } ( t )\) given by $$\mathrm { G } _ { X } ( t ) = \operatorname { ct } ( 1 + t ) ^ { 5 }$$ where \(c\) is a constant.

1. Find the value of \(c\).
2. Find the value of \(\mathrm { E } ( X )\).
   \includegraphics[max width=\textwidth, alt={}, center]{b5ff998a-fcb6-4a1b-ae86-ec66b0dccc3c-06_2718_33_141_2014} The random variable \(Y\) is the sum of two independent values of \(X\).
3. Write down the probability generating function of \(Y\) and hence find \(\operatorname { Var } ( Y )\).
4. Find \(\mathrm { P } ( Y = 5 )\).

5 The continuous random variable \(X\) has cumulative distribution function F given by $$F ( x ) = \begin{cases} 0 & x < 2 ,
\frac { ( x - 2 ) ^ { 2 } } { 12 } & 2 \leqslant x < 4 ,
1 - \frac { ( 8 - x ) ^ { 2 } } { 24 } & 4 \leqslant x \leqslant 8 ,
1 & x > 8 . \end{cases}$$ (c) Find the exact value of the interquartile range of \(X\).

6 Seva is investigating the lengths of the tails of adult wallabies in two regions of Australia, \(X\) and \(Y\). He chooses a random sample of 50 adult wallabies from region \(X\) and records the lengths, \(x \mathrm {~cm}\), of their tails. He also chooses a random sample of 40 adult wallabies from region \(Y\) and records the lengths, \(y \mathrm {~cm}\), of their tails. His results are summarised as follows. $$\sum x = 1080 \quad \sum x ^ { 2 } = 23480 \quad \sum y = 940 \quad \sum y ^ { 2 } = 22220$$ It cannot be assumed that the population variances of the two distributions are the same.

1. Find a \(90 \%\) confidence interval for the difference between the population mean lengths of the tails of adult wallabies in regions \(X\) and \(Y\).
   \includegraphics[max width=\textwidth, alt={}, center]{b5ff998a-fcb6-4a1b-ae86-ec66b0dccc3c-10_2718_38_141_2010} The population mean lengths of the tails of adult wallabies in regions \(X\) and \(Y\) are \(\mu _ { X } \mathrm {~cm}\) and \(\mu _ { Y } \mathrm {~cm}\) respectively.
2. Test, at the \(10 \%\) significance level, the null hypothesis \(\mu _ { Y } - \mu _ { X } = 1.1\) against the alternative hypothesis \(\mu _ { Y } - \mu _ { X } > 1.1\). State your conclusion in the context of the question.
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