Edexcel S2 (Statistics 2) 2014 June

1. (a) State the conditions under which the Poisson distribution may be used as an approximation to the binomial distribution.

A farmer supplies a bakery with eggs. The manager of the bakery claims that the proportion of eggs having a double yolk is 0.009 The farmer claims that the proportion of his eggs having a double yolk is more than 0.009
(b) State suitable hypotheses for testing these claims. In a batch of 500 eggs the baker records 9 eggs with a double yolk.
(c) Using a suitable approximation, test at the \(5 \%\) level of significance whether or not this supports the farmer's claim.

2. The amount of flour used by a factory in a week is \(Y\) thousand kg where \(Y\) has probability density function $$\mathrm { f } ( y ) = \left\{ \begin{array} { c c } k \left( 4 - y ^ { 2 } \right) & 0 \leqslant y \leqslant 2
0 & \text { otherwise } \end{array} \right.$$

1. Show that the value of \(k\) is \(\frac { 3 } { 16 }\) Use algebraic integration to find
2. the mean number of kilograms of flour used by the factory in a week,
3. the standard deviation of the number of kilograms of flour used by the factory in a week,
4. the probability that more than 1500 kg of flour will be used by the factory next week.

1. The continuous random variable \(T\) is uniformly distributed on the interval \([ \alpha , \beta ]\) where \(\beta > \alpha\)

Given that \(\mathrm { E } ( T ) = 2\) and \(\operatorname { Var } ( T ) = \frac { 16 } { 3 }\), find

1. the value of \(\alpha\) and the value of \(\beta\),
2. \(\mathrm { P } ( T < 3.4 )\)

4. Pieces of ribbon are cut to length \(L \mathrm {~cm}\) where \(L \sim \mathrm {~N} \left( \mu , 0.5 ^ { 2 } \right)\)

1. Given that \(30 \%\) of the pieces of ribbon have length more than 100 cm , find the value of \(\mu\) to the nearest 0.1 cm . John selects 12 pieces of ribbon at random.
2. Find the probability that fewer than 3 of these pieces of ribbon have length more than 100 cm . Aditi selects 400 pieces of ribbon at random.
3. Using a suitable approximation, find the probability that more than 127 of these pieces of ribbon will have length more than 100 cm .

5. A company claims that \(35 \%\) of its peas germinate. In order to test this claim Ann decides to plant 15 of these peas and record the number which germinate.

1. 1. State suitable hypotheses for a two-tailed test of this claim.
   2. Using a \(5 \%\) level of significance, find an appropriate critical region for this test. The probability in each of the tails should be as close to \(2.5 \%\) as possible.
2. Ann found that 8 of the 15 peas germinated. State whether or not the company's claim is supported. Give a reason for your answer.
3. State the actual significance level of this test.

6. A continuous random variable \(X\) has cumulative distribution function \(\mathrm { F } ( x )\) given by $$F ( x ) = \left\{ \begin{array} { l c } 0 & x < 0
\frac { x ^ { 2 } } { 20 } ( 9 - 2 x ) & 0 \leqslant x \leqslant 2
1 & x > 2 \end{array} \right.$$

1. Verify that the median of \(X\) lies between 1.23 and 1.24
2. Specify fully the probability density function \(\mathrm { f } ( x )\).
3. Find the mode of \(X\).
4. Describe the skewness of this distribution. Justify your answer.

7. Flaws occur at random in a particular type of material at a mean rate of 2 per 50 m .

1. Find the probability that in a randomly chosen 50 m length of this material there will be exactly 5 flaws. This material is sold in rolls of length 200 m . Susie buys 4 rolls of this material.
2. Find the probability that only one of these rolls will have fewer than 7 flaws. A piece of this material of length \(x \mathrm {~m}\) is produced. Using a normal approximation, the probability that this piece of material contains fewer than 26 flaws is 0.5398
3. Find the value of \(x\).