Edexcel S2 (Statistics 2) 2005 January

1. The random variables \(R , S\) and \(T\) are distributed as follows

$$R \sim \mathrm {~B} ( 15,0.3 ) , \quad S \sim \mathrm { Po } ( 7.5 ) , \quad T \sim \mathrm {~N} \left( 8,2 ^ { 2 } \right) .$$ Find

1. \(\mathrm { P } ( R = 5 )\),
2. \(\mathrm { P } ( S = 5 )\),
3. \(\mathrm { P } ( T = 5 )\).

2. (a) Explain what you understand by (i) a population and (ii) a sampling frame. The population and the sampling frame may not be the same.
(b) Explain why this might be the case.
(c) Give an example, justifying your choices, to illustrate when you might use

1. a census,
2. a sample.

3. A rod of length \(2 l\) was broken into 2 parts. The point at which the rod broke is equally likely to be anywhere along the rod. The length of the shorter piece of rod is represented by the random variable \(X\).

1. Write down the name of the probability density function of \(X\), and specify it fully.
2. Find \(\mathrm { P } \left( X < \frac { 1 } { 3 } l \right)\).
3. Write down the value of \(\mathrm { E } ( X )\). Two identical rods of length \(2 l\) are broken.
4. Find the probability that both of the shorter pieces are of length less than \(\frac { 1 } { 3 } l\).

4. In an experiment, there are 250 trials and each trial results in a success or a failure.

1. Write down two other conditions needed to make this into a binomial experiment. It is claimed that \(10 \%\) of students can tell the difference between two brands of baked beans. In a random sample of 250 students, 40 of them were able to distinguish the difference between the two brands.
2. Using a normal approximation, test at the \(1 \%\) level of significance whether or not the claim is justified. Use a one-tailed test.
3. Comment on the acceptability of the assumptions you needed to carry out the test.

5. From company records, a manager knows that the probability that a defective article is produced by a particular production line is 0.032 . A random sample of 10 articles is selected from the production line.

1. Find the probability that exactly 2 of them are defective. On another occasion, a random sample of 100 articles is taken.
2. Using a suitable approximation, find the probability that fewer than 4 of them are defective. At a later date, a random sample of 1000 is taken.
3. Using a suitable approximation, find the probability that more than 42 are defective.
   (6)

6. Over a long period of time, accidents happened on a stretch of road at random at a rate of 3 per month. Find the probability that

1. in a randomly chosen month, more than 4 accidents occurred,
2. in a three-month period, more than 4 accidents occurred. At a later date, a speed restriction was introduced on this stretch of road. During a randomly chosen month only one accident occurred.
3. Test, at the \(5 \%\) level of significance, whether or not there is evidence to support the claim that this speed restriction reduced the mean number of road accidents occurring per month. The speed restriction was kept on this road. Over a two-year period, 55 accidents occurred.
4. Test, at the \(5 \%\) level of significance, whether or not there is now evidence that this speed restriction reduced the mean number of road accidents occurring per month.

7. The random variable \(X\) has probability density function $$\mathrm { f } ( x ) = \begin{cases} k \left( - x ^ { 2 } + 5 x - 4 \right) , & 1 \leq x \leq 4
0 , & \text { otherwise } \end{cases}$$

1. Show that \(k = \frac { 2 } { 9 }\). Find
2. \(\mathrm { E } ( X )\),
3. the mode of \(X\).
4. the cumulative distribution function \(\mathrm { F } ( x )\) for all \(x\).
5. Evaluate \(\mathrm { P } ( X \leq 2.5 )\),
6. Deduce the value of the median and comment on the shape of the distribution.