Edexcel S2 (Statistics 2) 2015 June

1. A continuous random variable \(X\) has cumulative distribution function

$$\mathrm { F } ( x ) = \left\{ \begin{array} { l r } 0 & x < 2
\frac { 1 } { 20 } \left( x ^ { 2 } - 4 \right) & 2 \leqslant x \leqslant 4
\frac { 1 } { 5 } ( 2 x - 5 ) & 4 < x \leqslant 5
1 & x > 5 \end{array} \right.$$

1. Calculate \(\mathrm { P } ( X > 4 )\)
2. Find the probability density function of \(X\), specifying it for all values of \(x\).
3. Find the value of \(a\) such that \(\mathrm { P } ( 3 < X < a ) = 0.642\)
4. Find the probability density function of \(X\), specifying it for all values of \(x\).

2. A company produces chocolate chip biscuits. The number of chocolate chips per biscuit has a Poisson distribution with mean 8

1. Find the probability that one of these biscuits, selected at random, does not contain 8 chocolate chips. A small packet contains 4 of these biscuits, selected at random.
2. Find the probability that each biscuit in the packet contains at least 8 chocolate chips. A large packet contains 9 of these biscuits, selected at random.
3. Use a suitable approximation to find the probability that there are more than 75 chocolate chips in the packet. A shop sells packets of biscuits, randomly, at a rate of 1.5 packets per hour. Following an advertising campaign, 11 packets are sold in 4 hours.
4. Test, at the \(5 \%\) level of significance, whether or not there is evidence that the rate of sales of packets of biscuits has increased. State your hypotheses clearly.

1. A piece of spaghetti has length \(2 c\), where \(c\) is a positive constant. It is cut into two pieces at a random point. The continuous random variable \(X\) represents the length of the longer piece and is uniformly distributed over the interval \([ c , 2 c ]\).
   1. Sketch the graph of the probability density function of \(X\)
   2. Use integration to prove that \(\operatorname { Var } ( X ) = \frac { c ^ { 2 } } { 12 }\)
   3. Find the probability that the longer piece is more than twice the length of the shorter piece.

1. A single observation \(x\) is to be taken from a Poisson distribution with parameter \(\lambda\) This observation is to be used to test, at a \(5 \%\) level of significance,

$$\mathrm { H } _ { 0 } : \lambda = k \quad \mathrm { H } _ { 1 } : \lambda \neq k$$ where \(k\) is a positive integer.
Given that the critical region for this test is \(( X = 0 ) \cup ( X \geqslant 9 )\)

1. find the value of \(k\), justifying your answer.
2. Find the actual significance level of this test.

5. A bag contains a large number of counters with \(35 \%\) of the counters having a value of 6 and \(65 \%\) of the counters having a value of 9 A random sample of size 2 is taken from the bag and the value of each counter is recorded as \(X _ { 1 }\) and \(X _ { 2 }\) respectively. The statistic \(Y\) is calculated using the formula $$Y = \frac { 2 X _ { 1 } + X _ { 2 } } { 3 }$$

1. List all the possible values of \(Y\).
2. Find the sampling distribution of \(Y\).
3. Find \(\mathrm { E } ( Y )\).

6. Past information at a computer shop shows that \(40 \%\) of customers buy insurance when they purchase a product. In a random sample of 30 customers, \(X\) buy insurance.

1. Write down a suitable model for the distribution of \(X\).
2. State an assumption that has been made for the model in part (a) to be suitable. The probability that fewer than \(r\) customers buy insurance is less than 0.05
3. Find the largest possible value of \(r\). A second random sample, of 100 customers, is taken.
   The probability that at least \(t\) of these customers buy insurance is 0.938 , correct to 3 decimal places.
4. Using a suitable approximation, find the value of \(t\). The shop now offers an extended warranty on all products. Following this, a random sample of 25 customers is taken and 6 of them buy insurance.
5. Test, at the \(10 \%\) level of significance, whether or not there is evidence that the proportion of customers who buy insurance has decreased. State your hypotheses clearly.

1. A random variable \(X\) has probability density function

$$f ( x ) = \begin{cases} \frac { 2 x } { 15 } & 0 \leqslant x \leqslant k
\frac { 1 } { 5 } ( 5 - x ) & k < x \leqslant 5
0 & \text { otherwise } \end{cases}$$

1. Showing your working clearly, find the value of \(k\).
2. Write down the mode of \(X\).
3. Find \(\mathrm { P } \left( \left. X \leqslant \frac { k } { 2 } \right\rvert \, X \leqslant k \right)\)
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