Edexcel S2 (Statistics 2) 2021 October

1. A research project into food purchases found that \(35 \%\) of people who buy eggs do not buy free range eggs.

A random sample of 30 people who bought eggs is taken. The random variable \(F\) denotes the number of people who do not buy free range eggs.

1. Find \(\mathrm { P } ( F \geqslant 12 )\)
2. Find \(\mathrm { P } ( 8 \leqslant F < 15 )\) A farm shop gives 3 loyalty points with every purchase of free range eggs. With every purchase of eggs that are not free range the farm shop gives 1 loyalty point. A random sample of 30 customers who buy eggs from the farm shop is taken.
3. Find the probability that the total number of points given to these customers is less than 70 The manager of the farm shop believes that the proportion of customers who buy eggs but do not buy free range eggs is more than \(35 \%\) In a survey of 200 customers who buy eggs, 86 do not buy free range eggs. Using a suitable test and a normal approximation,
4. determine, at the \(5 \%\) level of significance, whether there is evidence to support the manager's belief. State your hypotheses clearly.

2. (i) The continuous random variable \(X\) is uniformly distributed over the interval \([ a , b ]\) Given that \(\mathrm { P } ( 8 < X < 14 ) = \frac { 1 } { 5 }\) and \(\mathrm { E } ( X ) = 11\)

1. write down \(\mathrm { P } ( X > 14 )\)
2. find \(\mathrm { P } ( 6 X > a + b )\)
   (ii) Susie makes a strip of pasta 45 cm long. She then cuts the strip of pasta, at a randomly chosen point, into two pieces. The random variable \(S\) is the length of the shortest piece of pasta.
3. Write down the distribution of \(S\)
4. Calculate the probability that the shortest piece of pasta is less than 12 cm long. Susie makes 20 strips of pasta, all 45 cm long, and separately cuts each strip of pasta, at a randomly chosen point, into two pieces.
5. Calculate the probability that exactly 6 of the pieces of pasta are less than 12 cm long.

3. A continuous random variable \(X\) has cumulative distribution function $$\mathrm { F } ( x ) = \left\{ \begin{array} { l r } 0 & x < 0
4 a x ^ { 2 } & 0 \leqslant x \leqslant 1
a \left( b x ^ { 3 } - x ^ { 4 } + 1 \right) & 1 < x \leqslant 3
1 & x > 3 \end{array} \right.$$ where \(a\) and \(b\) are positive constants.

1. Show that \(b = 4\)
2. Find the exact value of \(a\)
3. Find \(\mathrm { P } ( X > 2.25 )\)
4. Showing your working clearly,
   1. sketch the probability density function of \(X\)
   2. calculate the mode of \(X\)

1. The number of cars entering a safari park per 10 -minute period can be modelled by a Poisson distribution with mean 6
   1. Find the probability that in a given 10 -minute period exactly 8 cars will enter the safari park.
   2. Find the smallest value of \(n\) such that the probability that at least \(n\) cars enter the safari park in 10 minutes is less than 0.05
   The probability that no cars enter the safari park in \(m\) minutes, where \(m\) is an integer, is less than 0.05
2. Find the smallest value of \(m\) A car enters the safari park.
3. Find the probability that there is less than 5 minutes before the next car enters the safari park. Given that exactly 15 cars entered the safari park in a 30-minute period,
4. find the probability that exactly 1 car entered the safari park in the first 5 minutes of the 30-minute period. Aston claims that the mean number of cars entering the safari park per 10-minute period is more than 6 He selects a 15-minute period at random in order to test whether there is evidence to support his claim.
5. Determine the critical region for the test at the \(5 \%\) level of significance.

1. A bag contains a large number of counters.

40\% of the counters are numbered 1
\(35 \%\) of the counters are numbered 2
\(25 \%\) of the counters are numbered 3 In a game Alif draws two counters at random from the bag. His score is 4 times the number on the first counter minus 2 times the number on the second counter.

1. Show that Alif gets a score of 8 with probability 0.0875
2. Find the sampling distribution of Alif's score.
3. Calculate Alif's expected score.

6. The continuous random variable \(Y\) has probability density function \(\mathrm { f } ( y )\) given by $$f ( y ) = \begin{cases} \frac { 1 } { 14 } ( y + 2 ) & - 1 < y \leqslant 1
\frac { 3 } { 14 } & 1 < y \leqslant 3
\frac { 1 } { 14 } ( 6 - y ) & 3 < y \leqslant 5
0 & \text { otherwise } \end{cases}$$

1. Sketch the probability density function \(\mathrm { f } ( \mathrm { y } )\) Given that \(\mathrm { E } \left( Y ^ { 2 } \right) = \frac { 131 } { 21 }\)
2. find \(\operatorname { Var } ( 2 Y - 3 )\) The cumulative distribution function of \(Y\) is \(\mathrm { F } ( y )\)
3. Show that \(\mathrm { F } ( y ) = \frac { 1 } { 14 } \left( \frac { y ^ { 2 } } { 2 } + 2 y + \frac { 3 } { 2 } \right)\) for \(- 1 < y \leqslant 1\)
4. Find \(\mathrm { F } ( y )\) for all values of \(y\)
5. Find the exact value of the 30th percentile of \(Y\)
6. Find \(\mathrm { P } ( 4 Y \leqslant 5 \mid Y \leqslant 3 )\)