CAIE Further Paper 4 (Further Paper 4) 2023 June

1 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} \frac { 1 } { 6 } \left( x ^ { - \frac { 1 } { 3 } } - x ^ { - \frac { 2 } { 3 } } \right) & 1 \leqslant x \leqslant 27
0 & \text { otherwise } \end{cases}$$

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

2 Shane is studying the lengths of the tails of male red kangaroos. He takes a random sample of 14 male red kangaroos and measures the length of the tail, \(x \mathrm {~m}\), for each kangaroo. He then calculates a \(90 \%\) confidence interval for the population mean tail length, \(\mu \mathrm { m }\), of male red kangaroos. He assumes that the tail lengths are normally distributed and finds that \(1.11 \leqslant \mu \leqslant 1.14\). Find the values of \(\sum x\) and \(\sum x ^ { 2 }\) for this sample.

3 A large number of students took two test papers in mathematics. The teacher believes that the marks obtained in Paper 1 will be higher than the marks obtained in Paper 2. She chooses a random sample of 9 students and compares their marks. The marks are shown in the table.

1. Carry out a Wilcoxon matched-pairs signed-rank test, at the \(5 \%\) significance level, to test whether the data supports the teacher’s belief.
2. State an assumption that you have made in carrying out the test in part (a).

4 An inspector is checking the lengths of metal rods produced by two machines, \(X\) and \(Y\). These rods should be of the same length, but the inspector suspects that those made by machine \(X\) are shorter, on average, than those made by machine \(Y\). The inspector chooses a random sample of 80 rods made by machine \(X\) and a random sample of 60 rods made by machine \(Y\). The lengths of these rods are \(x \mathrm {~cm}\) and \(y \mathrm {~cm}\) respectively. Her results are summarised as follows. $$\sum x = 164.0 \quad \sum x ^ { 2 } = 338.1 \quad \sum y = 124.8 \quad \sum y ^ { 2 } = 261.1$$

1. Test at the \(10 \%\) significance level whether the data supports the inspector's suspicion.
2. Give a reason why it is not necessary to make any assumption about the distributions of the lengths of the rods.

5 The random variable \(X\) has probability generating function \(\mathrm { G } _ { X } ( \mathrm { t } )\) given by $$\mathrm { G } _ { \mathrm { X } } ( \mathrm { t } ) = \mathrm { k } \left( 1 + 3 \mathrm { t } + 4 \mathrm { t } ^ { 2 } \right)$$ where \(k\) is a constant.

1. Show that \(\mathrm { E } ( X ) = \frac { 11 } { 8 }\).
   The random variable \(Y\) has probability generating function \(\mathrm { G } _ { \gamma } ( \mathrm { t } )\) given by $$G _ { \gamma } ( t ) = \frac { 1 } { 3 } t ^ { 2 } ( 1 + 2 t )$$ The random variables \(X\) and \(Y\) are independent and \(\mathrm { Z } = \mathrm { X } + \mathrm { Y }\).
2. Find the probability generating function of \(Z\), expressing your answer as a polynomial in \(t\).
3. Use your answer to part (b) to find the value of \(\operatorname { Var } ( Z )\).
4. Write down the most probable value of \(Z\).

6 A scientist is investigating whether the ability to remember depends on age. A random sample of 150 students in different age groups is chosen. Each student is shown a set of 20 objects for thirty seconds and then asked to list as many as they can remember. The students are graded \(A\) or \(B\) according to how many objects they remembered correctly: grade \(A\) for 16 or more correct and grade \(B\) for fewer than 16 correct. The results are shown in the table.

1. Carry out a \(\chi ^ { 2 }\)-test at the \(2.5 \%\) significance level to test whether grade is independent of age of student.
   The scientist decides instead to use three grades: grade \(A\) for 16 or more correct, grade \(B\) for 10 to 15 correct and grade \(C\) for fewer than 10 correct. The results are shown in the following table.

   \multirow{2}{*}{}	Age of students
   	11-12 years	13-14 years	15-16 years
   Grade \(A\)	25	16	19
   Grade \(B\)	12	27	11
   Grade \(C\)	16	18	6

   With this second set of data, the test statistic is calculated as 10.91.
2. Complete the \(\chi ^ { 2 }\)-test at the \(2.5 \%\) significance level for this second set of data.
3. State, with a reason, whether you would prefer to use the result from part (a) or part (b) to investigate whether the ability to remember depends on age.
   If you use the following page to complete the answer to any question, the question number must be clearly shown.