CAIE Further Paper 4 (Further Paper 4) 2022 June

1 The times taken by members of a large quiz club to complete a challenge have a normal distribution with mean \(\mu\) minutes. The times, \(x\) minutes, are recorded for a random sample of 8 members of the club. The results are summarised as follows, where \(\bar { x }\) is the sample mean. $$\bar { x } = 33.8 \quad \sum ( x - \bar { x } ) ^ { 2 } = 94.5$$ Find a 95\% confidence interval for \(\mu\).

2 A scientist is investigating the size of shells at various beach locations. She selects four beach locations and takes a random sample of shells from each of these beaches. She classifies each shell as large or small. Her results are summarised in the following table. Test, at the 10\% significance level, whether the size of shell is independent of the beach location.

3 George throws two coins, \(A\) and \(B\), at the same time. Coin \(A\) is biased so that the probability of obtaining a head is \(a\). Coin \(B\) is biased so that the probability of obtaining a head is \(b\), where \(\mathrm { b } < \mathrm { a }\). The probability generating function of \(X\), the number of heads obtained by George, is \(\mathrm { G } _ { \mathrm { X } } ( \mathrm { t } )\). The coefficients of \(t\) and \(t ^ { 2 }\) in \(\mathrm { G } _ { \mathrm { X } } ( \mathrm { t } )\) are \(\frac { 5 } { 12 }\) and \(\frac { 1 } { 12 }\) respectively.

1. Find the value of \(a\).
   The random variable \(Y\) is the sum of two independent observations of \(X\).
2. Find the probability generating function of \(Y\), giving your answer as a polynomial in \(t\).
3. Find \(\operatorname { Var } ( Y )\).

4 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} \frac { 3 } { 8 } \left( 1 + \frac { 1 } { x ^ { 2 } } \right) & 1 \leqslant x \leqslant 3
0 & \text { otherwise } \end{cases}$$

1. Find \(\mathrm { E } ( \sqrt { X } )\).
   The random variable \(Y\) is given by \(Y = X ^ { 2 }\).
2. Find the probability density function of \(Y\).
3. Find the 40th percentile of \(Y\).

5 A manager claims that the lengths of the rubber tubes that his company produces have a median of 5.50 cm . The lengths, in cm , of a random sample of 11 tubes produced by this company are as follows. It is required to test at the \(10 \%\) significance level the null hypothesis that the population median length is 5.50 cm against the alternative hypothesis that the population median length is not equal to 5.50 cm . Show that both a sign test and a Wilcoxon signed-rank test give the same conclusion and state this conclusion.

6 A company has two machines, \(A\) and \(B\), which independently fill small bottles with a liquid. The volumes of liquid per bottle, in suitable units, filled by machines \(A\) and \(B\) are denoted by \(x\) and \(y\) respectively. A scientist at the company takes a random sample of 40 bottles filled by machine \(A\) and a random sample of 50 bottles filled by machine \(B\). The results are summarised as follows. $$\sum x = 1120 \quad \sum x ^ { 2 } = 31400 \quad \sum y = 1370 \quad \sum y ^ { 2 } = 37600$$ The population means of the volumes of liquid in the bottles filled by machines \(A\) and \(B\) are denoted by \(\mu _ { A }\) and \(\mu _ { B }\).

1. Test at the \(2 \%\) significance level whether there is any difference between \(\mu _ { A }\) and \(\mu _ { B }\).
2. Find the set of values of \(\alpha\) for which there would be evidence at the \(\alpha \%\) significance level that \(\mu _ { \mathrm { A } } - \mu _ { \mathrm { B } }\) is greater than 0.25.
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