Edexcel S2 (Statistics 2) 2021 June

1. Spany sells seeds and claims that \(5 \%\) of its pansy seeds do not germinate. A packet of pansy seeds contains 20 seeds. Each seed germinates independently of the other seeds.
   1. Find the probability that in a packet of Spany's pansy seeds
      1. more than 2 but fewer than 5 seeds do not germinate,
      2. more than 18 seeds germinate.
   Jem buys 5 packets of Spany’s pansy seeds.
2. Calculate the probability that all of these packets contain more than 18 seeds that germinate. Jem believes that Spany's claim is incorrect. She believes that the percentage of pansy seeds that do not germinate is greater than 5\%
3. Write down the hypotheses for a suitable test to examine Jem's belief. Jem planted all of the 100 seeds she bought from Spany and found that 8 did not germinate.
4. Using a suitable approximation, carry out the test using a \(5 \%\) level of significance.

1. Luis makes and sells rugs. He knows that faults occur randomly in his rugs at a rate of 3 every \(4 \mathrm {~m} ^ { 2 }\)
   1. Find the probability of there being exactly 5 faults in one of his rugs that is \(4 \mathrm {~m} ^ { 2 }\) in size.
   2. Find the probability that there are more than 5 faults in one of his rugs that is \(6 \mathrm {~m} ^ { 2 }\) in size.
   Luis makes a rug that is \(4 \mathrm {~m} ^ { 2 }\) in size and finds it has exactly 5 faults in it.
2. Write down the probability that the next rug that Luis makes, which is \(4 \mathrm {~m} ^ { 2 }\) in size, will have exactly 5 faults. Give a reason for your answer. A small rug has dimensions 80 cm by 150 cm . Faults still occur randomly at a rate of 3 every \(4 \mathrm {~m} ^ { 2 }\)
   Luis makes a profit of \(\pounds 80\) on each small rug he sells that contains no faults but a profit of \(\pounds 60\) on any small rug he sells that contains faults. Luis sells \(n\) small rugs and expects to make a profit of at least \(\pounds 4000\)
3. Calculate the minimum value of \(n\) Luis wishes to increase the productivity of his business and employs Rhiannon. Faults also occur randomly in Rhiannon's rugs and independently to faults made by Luis. Luis randomly selects 10 small rugs made by Rhiannon and finds 13 faults.
4. Test, at the \(5 \%\) level of significance, whether or not there is evidence to support the suggestion that the rate at which faults occur is higher for Rhiannon than for Luis. State your hypotheses clearly.

1. The continuous random variable \(Y\) has the following probability density function

$$f ( y ) = \begin{cases} \frac { 6 } { 25 } ( y - 1 ) & 1 \leqslant y < 2
\frac { 3 } { 50 } \left( 4 y ^ { 2 } - y ^ { 3 } \right) & 2 \leqslant y < 4
0 & \text { otherwise } \end{cases}$$

1. Sketch f(y)
2. Find the mode of \(Y\)
3. Use algebraic integration to calculate \(\mathrm { E } \left( Y ^ { 2 } \right)\) Given that \(\mathrm { E } ( Y ) = 2.696\)
4. find \(\operatorname { Var } ( Y )\)
5. Find the value of \(y\) for which \(\mathrm { P } ( Y \geqslant y ) = 0.9\) Give your answer to 3 significant figures.

1. A bag contains a large number of balls, each with one of the numbers 1,2 or 5 written on it in the ratio \(2 : 3 : 4\) respectively.

A random sample of 3 balls is taken from the bag.
The random variable \(B\) represents the range of the numbers written on the balls in the sample.

1. Find \(\mathrm { P } ( B = 4 )\)
2. Find the sampling distribution of \(B\).

1. A game uses two turntables, one red and one yellow. Each turntable has a point marked on the circumference that is lined up with an arrow at the start of the game. Jim spins both turntables and measures the distance, in metres, each point is from the arrow, around the circumference in an anticlockwise direction when the turntables stop spinning.

The continuous random variable \(Y\) represents the distance, in metres, the point is from the arrow for the yellow turntable. The cumulative distribution function of \(Y\) is given by \(\mathrm { F } ( y )\) where $$F ( y ) = \left\{ \begin{array} { c r } 0 & y < 0
1 - \left( \alpha + \beta y ^ { 2 } \right) & 0 \leqslant y \leqslant 5
1 & y > 5 \end{array} \right.$$

1. Explain why (i) \(\alpha = 1\) $$\text { (ii) } \beta = - \frac { 1 } { 25 }$$
2. Find the probability density function of \(Y\) The continuous random variable \(R\) represents the distance, in metres, the point is from the arrow for the red turntable. The distribution of \(R\) is modelled by a continuous uniform distribution over the interval \([ d , 3 d ]\) Given that \(\mathrm { P } \left( R > \frac { 11 } { 5 } \right) = \mathrm { P } \left( Y > \frac { 5 } { 3 } \right)\)
3. find the value of \(d\) In the game each turntable is spun 3 times. The distance between the point and the arrow is determined for each spin. To win a prize, at least 5 of the distances the point is from the arrow when a turntable is spun must be less than \(\frac { 11 } { 5 } \mathrm {~m}\)
   Jo plays the game once.
4. Calculate the probability of Jo winning a prize.

1. The random variable \(Y \sim \mathrm {~B} ( 225 , p )\)

Using a normal approximation, the probability that \(Y\) is at least 188 is 0.1056 to 4 decimal places.

1. Show that \(p\) satisfies \(145 p ^ { 2 } - 241 p + 100 = 0\) when the normal probability tables are used.
2. Hence find the value of \(p\), justifying your answer.