CAIE Further Paper 4 (Further Paper 4) 2021 June

1 Farmer \(A\) grows apples of a certain variety. Each tree produces 14.8 kg of apples, on average, per year. Farmer \(B\) grows apples of the same variety and claims that his apple trees produce a higher mass of apples per year than Farmer \(A\) 's trees. The masses of apples from Farmer \(B\) 's trees may be assumed to be normally distributed. A random sample of 10 trees from Farmer \(B\) is chosen. The masses, \(x \mathrm {~kg}\), of apples produced in a year are summarised as follows. $$\sum x = 152.0 \quad \sum x ^ { 2 } = 2313.0$$ Test, at the \(5 \%\) significance level, whether Farmer \(B\) 's claim is justified.

2 A company is developing a new flavour of chocolate by varying the quantities of the ingredients. A random selection of 9 flavours of chocolate are judged by two tasters who each give marks out of 100 to each flavour of chocolate. Carry out a Wilcoxon matched-pairs signed-rank test at the \(10 \%\) significance level to investigate whether, on average, there is a difference between marks awarded by the two tasters.

3 The heights, \(x \mathrm {~m}\), of a random sample of 50 adult males from country \(A\) were recorded. The heights, \(y \mathrm {~m}\), of a random sample of 40 adult males from country \(B\) were also recorded. The results are summarised as follows. $$\Sigma x = 89.0 \quad \Sigma x ^ { 2 } = 159.4 \quad \Sigma y = 67.2 \quad \Sigma y ^ { 2 } = 113.1$$ Find a 95\% confidence interval for the difference between the mean heights of adult males from country \(A\) and adult males from country \(B\).
\(4 X\) is a discrete random variable which takes the values \(0,2,4 , \ldots\). The probability generating function of \(X\) is given by $$G _ { X } ( t ) = \frac { 1 } { 3 - 2 t ^ { 2 } }$$

1. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
2. Find \(\mathrm { P } ( X = 4 )\).

5 Chai packs china mugs into cardboard boxes. Chai's manager suspects that breakages occur at random times and that the number of breakages may follow a Poisson distribution. He takes a small sample of observations and finds that the number of breakages in a one-hour period has a mean of 2.4 and a standard deviation of 1.5.

1. Explain how this information tends to support the manager's suspicion.
   The manager now takes a larger sample and claims that the numbers of breakages in a one-hour period follow a Poisson distribution. The numbers of breakages in a random sample of 180 one-hour periods are summarised in the following table.

   Number of breakages	0	1	2	3	4	5	6	7 or more
   Frequency	21	33	46	31	23	16	10	0

   The mean number of breakages calculated from this sample is 2.5.
2. Use the data from this larger sample to carry out a goodness of fit test, at the \(10 \%\) significance level, to test the claim.

6 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} \frac { 1 } { 8 } & 0 \leqslant x < 1
\frac { 1 } { 28 } ( 8 - x ) & 1 \leqslant x \leqslant 8
0 & \text { otherwise } \end{cases}$$

1. Find the cumulative distribution function of \(X\).
2. Find the value of the constant \(a\) such that \(\mathrm { P } ( \mathrm { X } \leqslant \mathrm { a } ) = \frac { 5 } { 7 }\).
   The random variable \(Y\) is given by \(Y = \sqrt [ 3 ] { X }\).
3. Find the probability density function of \(Y\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.