CAIE Further Paper 4 (Further Paper 4) 2021 November

1 The number, \(x\), of pine trees was counted in each of 40 randomly chosen regions of equal size in country \(A\). The number, \(y\), of pine trees was counted in each of 60 randomly chosen regions of the same equal size in country \(B\). The results are summarised as follows. $$\sum x = 752 \quad \sum x ^ { 2 } = 14320 \quad \sum y = 1548 \quad \sum y ^ { 2 } = 40200$$ Find a 95\% confidence interval for the difference between the mean number of pine trees in regions of this size in countries \(A\) and \(B\).

2 It is claimed that the heights of a particular age group of boys follow a normal distribution with mean 125 cm and standard deviation 12 cm . Observations for a randomly chosen group of 60 boys in this age group are summarised in the following table. The table also gives the expected frequencies, correct to 2 decimal places, based on the normal distribution with mean 125 cm and standard deviation 12 cm .

1. Show how the expected frequency for \(130 \leqslant x < 140\) is obtained.
2. Carry out a goodness of fit test, at the \(5 \%\) significance level, to determine whether the claim is supported by the data.

3 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} a + \frac { 1 } { 5 } x & 0 \leqslant x < 1
2 a - \frac { 1 } { 5 } x & 1 \leqslant x \leqslant 2
0 & \text { otherwise } \end{cases}$$ where \(a\) is a constant.

1. Find the value of \(a\).
2. Find \(\mathrm { E } \left( X ^ { 2 } \right)\).
3. Find the cumulative distribution function of \(X\).

4 Applicants for a particular college take a written test when they attend for interview. There are two different written tests, \(A\) and \(B\), and each applicant takes one or the other. The interviewer wants to determine whether the medians of the distribution of marks obtained in the two tests are equal. The marks obtained by a random sample of 8 applicants who took test \(A\) and a random sample of 8 applicants who took test \(B\) are as follows.

1. Carry out a Wilcoxon rank-sum test at the \(5 \%\) significance level to determine whether there is a difference in the population median marks obtained in the two tests.
   The interviewer considers using the given information to carry out a paired sample \(t\)-test to determine whether there is a difference in the population means for the two tests.
2. Give two reasons why it is not appropriate to use this test.

5 The random variable \(X\) is such that \(\mathrm { P } ( \mathrm { X } = \mathrm { r } ) = \mathrm { kr } ^ { 2 }\) for \(r = 1,2,3,4\), where \(k\) is a constant.

1. Find the value of \(k\).
2. Find the probability generating function \(\mathrm { G } _ { X } ( \mathrm { t } )\) of \(X\).
   The random variable \(Y\) has probability generating function \(\mathrm { G } _ { Y } ( \mathrm { t } ) = \frac { 1 } { 4 } + \frac { 1 } { 2 } \mathrm { t } + \frac { 1 } { 4 } \mathrm { t } ^ { 2 }\).
   The random variable \(Z\) is the sum of \(X\) and \(Y\).
3. Assuming that \(X\) and \(Y\) are independent, find the probability generating function \(\mathrm { G } _ { \mathrm { Z } } ( \mathrm { t } )\) of \(Z\) as a polynomial in \(t\).
4. Given that \(\mathrm { E } ( \mathrm { Z } ) = \frac { 13 } { 3 }\), use \(\mathrm { G } _ { \mathrm { Z } } ( \mathrm { t } )\) to find \(\operatorname { Var } ( \mathrm { Z } )\).

6 A scientist is investigating the masses of a particular type of fish found in lakes \(A\) and \(B\). He chooses a random sample of 10 fish of this type from lake \(A\) and records their masses, \(x \mathrm {~kg}\), as follows.
0.9
1.8
1.8
1.9
2.1
2.4
2.6
2.2
2.5
3.0 The scientist also chooses a random sample of 12 fish of this type from lake \(B\), but he only has a summary of their masses, \(y \mathrm {~kg}\), as follows. $$\sum y = 24.48 \quad \sum y ^ { 2 } = 53.75$$ Test at the \(10 \%\) significance level whether the mean mass of fish of this type in lake \(A\) is greater than the mean mass of fish of this type in lake \(B\). You should state any assumptions that you need to make for the test to be valid.
[0pt] [10]
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.