Edexcel S2 (Statistics 2) 2022 October

1. Bhavna produces rolls of cloth. She knows that faults occur randomly in her cloth at a mean rate of 1.5 every 15 metres.
   1. Find the probability that in 15 metres of her cloth there are
      1. less than 3 faults,
      2. at least 6 faults.
   Each roll contains 100 metres of Bhavna's cloth.
   She selects 15 rolls at random.
2. Find the probability that exactly 10 of these rolls each have fewer than 13 faults. Bhavna decides to sell her cloth in pieces.
   Each piece of her cloth is 4 metres long.
   The cost to make each piece is \(\pounds 5.00\)
   She sells each piece of her cloth that contains no faults for \(\pounds 7.40\)
   She sells each piece of her cloth that contains faults for \(\pounds 2.00\)
3. Find the expected profit that Bhavna will make on each piece of her cloth that she sells.

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

$$f ( x ) = \left\{ \begin{array} { c c } \frac { 1 } { 4 } & - \frac { 1 } { 2 } \leqslant x < \frac { 1 } { 2 }
2 x - \frac { 3 } { 4 } & \frac { 1 } { 2 } \leqslant x \leqslant k
0 & \text { otherwise } \end{array} \right.$$ where \(k\) is a positive constant.

1. Sketch the graph of \(\mathrm { f } ( x )\)
2. By forming and solving an equation in \(k\), show that \(k = 1.25\)
3. Use calculus to find \(\mathrm { E } ( X )\)
4. Calculate the interquartile range of \(X\)

1. A company produces packets of sunflower seeds. Each packet contains 40 seeds. The company claims that, on average, only 35\% of its sunflower seeds do not germinate.

A packet is selected at random.

1. Using a \(5 \%\) level of significance, find an appropriate critical region for a two-tailed test that the proportion of sunflower seeds that do not germinate is 0.35 You should state your hypotheses clearly and state the probability, which should be as close as possible to \(2.5 \%\), for each tail of your critical region.
2. Write down the actual significance level of this test. Past records suggest that \(2.8 \%\) of the company's sunflower seeds grow to a height of more than 3 metres.
   A random sample of 250 of the company's sunflower seeds is taken and 11 of them grow to a height of more than 3 metres.
3. Using a suitable approximation test, at the \(5 \%\) level of significance, whether or not there is evidence that the proportion of sunflower seeds that grow to a height of more than 3 metres is now greater than \(2.8 \%\)
   State your hypotheses clearly.

1. The probability that a person completes a particular task in less than 15 minutes is 0.4 Jeffrey selects 20 people at random and asks them to complete the task. The random variable, \(X\), represents the number of people who complete the task in less than 15 minutes.
   1. Find \(\mathrm { P } ( 5 \leqslant X < 8 )\)
   Mia takes a random sample of 140 people.
   Using a normal approximation, the probability that fewer than \(n\) of these 140 people complete the task in less than 15 minutes is 0.0239 to 4 decimal places.
2. Find the value of \(n\) Show your working clearly.

1. The continuous random variable \(X\) has cumulative distribution function given by

$$\mathrm { F } ( x ) = \left\{ \begin{array} { c r } 0 & x < 3
\frac { 1 } { 6 } ( x - 3 ) ^ { 2 } & 3 \leqslant x < 4
\frac { x } { 3 } - \frac { 7 } { 6 } & 4 \leqslant x < c
1 - \frac { 1 } { 6 } ( d - x ) ^ { 2 } & c \leqslant x < 7
1 & x \geqslant 7 \end{array} \right.$$ where \(c\) and \(d\) are constants.

1. Show that \(c = 6\)
2. Find \(\mathrm { P } ( X > 3.5 )\)
3. Find \(\mathrm { P } ( X > 4.5 \mid 3.5 < X < 5.5 )\)

1. A bag contains a large number of counters with one of the numbers 5 , 10 or 20 written on each of them in the ratio \(5 : 2 : a\)

A jar contains a large number of counters with one of the numbers 5 or 10 written on each of them in the ratio \(1 : 3\) One counter is selected at random from the bag and then two counters are selected at random from the jar.
The random variable \(R\) represents the range of the numbers on the 3 counters.
Given that \(\mathrm { P } ( R = 15 ) = \frac { 63 } { 256 }\)

1. by forming and solving an equation in \(a\), show that \(a = 9\)
2. find the sampling distribution of \(R\)

1. (i) The continuous random variable \(X\) is uniformly distributed over the interval \([ a , b ]\)

Given that \(\mathrm { P } ( 5 < X < 13 ) = \frac { 1 } { 5 }\) and \(\mathrm { E } ( X ) = 9\), find \(\mathrm { P } ( 3 X > a + b )\)
(ii) The continuous random variable \(Y\) is uniformly distributed over the interval \([ 1 , c ]\) Given that \(\operatorname { Var } ( Y ) = 0.48\), find the exact value of \(\mathrm { E } \left( Y ^ { 2 } \right)\)
(iii) A wire of length 20 cm is cut into 2 pieces at a random point. The longest piece of wire is then cut into 2 pieces, equal in length, giving 3 pieces of wire altogether. Find the probability that the length of the shortest piece of wire is less than 6 cm .