Name the distribution of \(L\), the length of the longer part of string, and sketch the probability density function for \(L\).
Find the probability that one part of the string is more than twice as long as the other.
\item A supplier of widgets claims that only \(10 \%\) of his widgets have faults.
In a consignment of 50 widgets, 9 are faulty. Test, at the \(5 \%\) significance level, whether this suggests that the supplier's claim is false.
Find how many faulty widgets would be needed to provide evidence against the claim at the \(1 \%\) significance level.
\item In a survey of 22 families, the number of children, \(X\), in each family was given by the following table, where \(f\) denotes the frequency:
\end{enumerate}
\(X\)
0
1
2
3
4
5
\(f\)
3
8
5
3
2
1
Find the mean and variance of \(X\).
Explain why these results suggest that \(X\) may follow a Poisson distribution.
State another feature of the data that suggests a Poisson distribution.
It is sometimes suggested that the number of children in a family follows a Poisson distribution with mean 2.4. Assuming that this is correct,
find the probability that a family has less than two children.
Use this result to find the probability that, in a random sample of 22 families, exactly 11 of the families have less than two children.
\section*{STATISTICS 2 (A) TEST PAPER 7 Page 2}