Edexcel S2 (Statistics 2) 2004 June

1. Explain briefly what you understand by
   1. a sampling frame,
   2. a statistic.
   3. The continuous random variable \(X\) is uniformly distributed over the interval \([ - 1,4 ]\).
   Find
2. \(\mathrm { P } ( X < 2.7 )\),
3. \(\mathrm { E } ( X )\),
4. \(\operatorname { Var } ( X )\).

3. Brad planted 25 seeds in his greenhouse. He has read in a gardening book that the probability of one of these seeds germinating is 0.25 . Ten of Brad's seeds germinated. He claimed that the gardening book had underestimated this probability. Test, at the \(5 \%\) level of significance, Brad's claim. State your hypotheses clearly.

4. (a) State two conditions under which a random variable can be modelled by a binomial distribution. In the production of a certain electronic component it is found that \(10 \%\) are defective.
The component is produced in batches of 20 .
(b) Write down a suitable model for the distribution of defective components in a batch. Find the probability that a batch contains
(c) no defective components,
(d) more than 6 defective components.
(e) Find the mean and the variance of the defective components in a batch. A supplier buys 100 components. The supplier will receive a refund if there are more than 15 defective components.
(f) Using a suitable approximation, find the probability that the supplier will receive a refund.

5. (a) Explain what you understand by a critical region of a test statistic. The number of breakdowns per day in a large fleet of hire cars has a Poisson distribution with mean \(\frac { 1 } { 7 }\).
(b) Find the probability that on a particular day there are fewer than 2 breakdowns.
(c) Find the probability that during a 14-day period there are at most 4 breakdowns. The cars are maintained at a garage. The garage introduced a weekly check to try to decrease the number of cars that break down. In a randomly selected 28-day period after the checks are introduced, only 1 hire car broke down.
(d) Test, at the \(5 \%\) level of significance, whether or not the mean number of breakdowns has decreased. State your hypotheses clearly.

6. Minor defects occur in a particular make of carpet at a mean rate of 0.05 per \(\mathrm { m } ^ { 2 }\).

1. Suggest a suitable model for the distribution of the number of defects in this make of carpet. Give a reason for your answer. A carpet fitter has a contract to fit this carpet in a small hotel. The hotel foyer requires \(30 \mathrm {~m} ^ { 2 }\) of this carpet. Find the probability that the foyer carpet contains
2. exactly 2 defects,
3. more than 5 defects. The carpet fitter orders a total of \(355 \mathrm {~m} ^ { 2 }\) of the carpet for the whole hotel.
4. Using a suitable approximation, find the probability that this total area of carpet contains 22 or more defects.
   (6)

7. A random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { 1 } { 3 } , & 0 \leq x \leq 1
\frac { 8 x ^ { 3 } } { 45 } , & 1 \leq x \leq 2
0 , & \text { otherwise } \end{cases}$$

1. Calculate the mean of \(X\).
2. Specify fully the cumulative distribution function \(\mathrm { F } ( x )\).
3. Find the median of \(X\).
4. Comment on the skewness of the distribution of \(X\). END