Edexcel S2 (Statistics 2)

\begin{enumerate} \item A company that makes ropes for mountaineering wants to assess the breaking strain of its ropes.

1. Explain why a sample survey, and not a census, should be used.
2. Suggest an appropriate sampling frame. \item It is thought that a random variable \(X\) has a Poisson distribution whose mean, \(\lambda\), is equal to 8 . Find the critical region to test the hypothesis \(\mathrm { H } _ { 0 } : \lambda = 8\) against the hypothesis \(\mathrm { H } _ { 1 } : \lambda < 8\), working at the \(1 \%\) significance level. \item A child cuts a 30 cm piece of string into two parts, cutting at a random point.

1. Name the distribution of \(L\), the length of the longer part of string, and sketch the probability density function for \(L\).
2. 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.
3. 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.
4. 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

5. Find the mean and variance of \(X\).
6. Explain why these results suggest that \(X\) may follow a Poisson distribution.
7. 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,
8. find the probability that a family has less than two children.
9. 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}

1. When a park is redeveloped, it is claimed that \(70 \%\) of the local population approve of the new design. Assuming this to be true, find the probability that, in a group of 10 residents selected at random,
   1. 6 or more approve,
   2. exactly 7 approve.
   A conservation group, however, carries out a survey of 20 people, and finds that only 9 approve.
2. Use this information to carry out a hypothesis test on the original claim, working at the \(5 \%\) significance level. State your conclusion clearly. If the conservationists are right, and only \(45 \%\) approve of the new park,
3. use a suitable approximation to the binomial distribution to estimate the probability that in a larger survey, of 500 people, less than half will approve.

7. A continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) given by $$\begin{array} { l l } \mathrm { f } ( x ) = \frac { 2 x } { 3 } & 0 \leq x < 1
\mathrm { f } ( x ) = 1 - \frac { x } { 3 } & 1 \leq x \leq 3
\mathrm { f } ( x ) = 0 & \text { otherwise. } \end{array}$$

1. Sketch the graph of \(\mathrm { f } ( x )\) for all \(x\).
2. Find the mean of \(X\).
3. Find the standard deviation of \(X\).
4. Show that the cumulative distribution function of \(X\) is given by $$\mathrm { F } ( x ) = \frac { x ^ { 2 } } { 3 } \quad 0 \leq x < 1$$ and find \(\mathrm { F } ( x )\) for \(1 \leq x \leq 3\).