Edexcel S2 (Statistics 2) 2013 June

1. A bag contains a large number of counters. A third of the counters have a number 5 on them and the remainder have a number 1 .

A random sample of 3 counters is selected.

1. List all possible samples.
2. Find the sampling distribution for the range.

2. The continuous random variable \(Y\) has cumulative distribution function $$\mathrm { F } ( y ) = \left\{ \begin{array} { c c } 0 & y < 0
\frac { 1 } { 4 } \left( y ^ { 3 } - 4 y ^ { 2 } + k y \right) & 0 \leqslant y \leqslant 2
1 & y > 2 \end{array} \right.$$ where \(k\) is a constant.

1. Find the value of \(k\).
2. Find the probability density function of \(Y\), specifying it for all values of \(y\).
3. Find \(\mathrm { P } ( Y > 1 )\).

3. The random variable \(X\) has a continuous uniform distribution on \([ a , b ]\) where \(a\) and \(b\) are positive numbers. Given that \(\mathrm { E } ( X ) = 23\) and \(\operatorname { Var } ( X ) = 75\)

1. find the value of \(a\) and the value of \(b\). Given that \(\mathrm { P } ( X > c ) = 0.32\)
2. find \(\mathrm { P } ( 23 < X < c )\).

4. The random variable \(X\) has probability density function \(\mathrm { f } ( x )\) given by $$f ( x ) = \left\{ \begin{array} { c c } k \left( 3 + 2 x - x ^ { 2 } \right) & 0 \leqslant x \leqslant 3
0 & \text { otherwise } \end{array} \right.$$ where \(k\) is a constant.

1. Show that \(k = \frac { 1 } { 9 }\)
2. Find the mode of \(X\).
3. Use algebraic integration to find \(\mathrm { E } ( X )\). By comparing your answers to parts (b) and (c),
4. describe the skewness of \(X\), giving a reason for your answer.

1. In a village shop the customers must join a queue to pay. The number of customers joining the queue in a 10 minute interval is modelled by a Poisson distribution with mean 3

Find the probability that

1. exactly 4 customers join the queue in the next 10 minutes,
2. more than 10 customers join the queue in the next 20 minutes. When a customer reaches the front of the queue the customer pays the assistant. The time each customer takes paying the assistant, \(T\) minutes, has a continuous uniform distribution over the interval \([ 0,5 ]\). The random variable \(T\) is independent of the number of people joining the queue.
3. Find \(\mathrm { P } ( T > 3.5 )\) In a random sample of 5 customers, the random variable \(C\) represents the number of customers who took more than 3.5 minutes paying the assistant.
4. Find \(\mathrm { P } ( C \geqslant 3 )\) Bethan has just reached the front of the queue and starts paying the assistant.
5. Find the probability that in the next 4 minutes Bethan finishes paying the assistant and no other customers join the queue.

6. Frugal bakery claims that their packs of 10 muffins contain on average 80 raisins per pack. A Poisson distribution is used to describe the number of raisins per muffin. A muffin is selected at random to test whether or not the mean number of raisins per muffin has changed.

1. Find the critical region for a two-tailed test using a \(10 \%\) level of significance. The probability of rejection in each tail should be less than 0.05
2. Find the actual significance level of this test. The bakery has a special promotion claiming that their muffins now contain even more raisins. A random sample of 10 muffins is selected and is found to contain a total of 95 raisins.
3. Use a suitable approximation to test the bakery's claim. You should state your hypotheses clearly and use a \(5 \%\) level of significance.

7. As part of a selection procedure for a company, applicants have to answer all 20 questions of a multiple choice test. If an applicant chooses answers at random the probability of choosing a correct answer is 0.2 and the number of correct answers is represented by the random variable \(X\).

1. Suggest a suitable distribution for \(X\).
   (2) Each applicant gains 4 points for each correct answer but loses 1 point for each incorrect answer. The random variable \(S\) represents the final score, in points, for an applicant who chooses answers to this test at random.
2. Show that \(S = 5 X - 20\)
3. Find \(\mathrm { E } ( S )\) and \(\operatorname { Var } ( S )\). An applicant who achieves a score of at least 20 points is invited to take part in the final stage of the selection process.
4. Find \(\mathrm { P } ( S \geqslant 20 )\)
   (4) Cameron is taking the final stage of the selection process which is a multiple choice test consisting of 100 questions. He has been preparing for this test and believes that his chance of answering each question correctly is 0.4
5. Using a suitable approximation, estimate the probability that Cameron answers more than half of the questions correctly.