Edexcel S2 (Statistics 2) 2024 January

1. The manager of a supermarket is investigating the number of complaints per day received from customers.

A random sample of 180 days is taken and the results are shown in the table below.

1. Calculate the mean and the variance of these data.
2. Explain why the results in part (a) suggest that a Poisson distribution may be a suitable model for the number of complaints per day. The manager uses a Poisson distribution with mean 3 to model the number of complaints per day.
3. For a randomly selected day find, using the manager's model, the probability that there are
   1. at least 3 complaints,
   2. more than 4 complaints but less than 8 complaints. A week consists of 7 consecutive days.
4. Using the manager's model and a suitable approximation, show that the probability that there are less than 19 complaints in a randomly selected week is 0.29 to 2 decimal places.
   Show your working clearly.
   (Solutions relying on calculator technology are not acceptable.) A period of 13 weeks is selected at random.
5. Find the probability that in this period there are exactly 5 weeks that have less than 19 complaints.
   Show your working clearly.

1. The length of pregnancy for a randomly selected pregnant sheep is \(D\) days where

$$D \sim \mathrm {~N} \left( 112.4 , \sigma ^ { 2 } \right)$$ Given that 5\% of pregnant sheep have a length of pregnancy of less than 108 days,

1. find the value of \(\sigma\) Qiang selects 25 pregnant sheep at random from a large flock.
2. Find the probability that more than 3 of these pregnant sheep have a length of pregnancy of less than 108 days. Charlie takes 200 random samples of 25 pregnant sheep.
3. Use a Poisson approximation to estimate the probability that at least 2 of the samples have more than 3 pregnant sheep with a length of pregnancy of less than 108 days.

1. Rowan believes that \(35 \%\) of type \(A\) vacuum tubes shatter when exposed to alternating high and low temperatures.

Rowan takes a random sample of 15 of these type \(A\) vacuum tubes and uses a two-tailed test, at the \(5 \%\) level of significance, to test his belief.

1. Give two assumptions, in context, that Rowan needs to make for a binomial distribution to be a suitable model for the number of these type \(A\) vacuum tubes that shatter when exposed to alternating high and low temperatures.
2. Using a binomial distribution, find the critical region for the test. You should state the probability of rejection in each tail, which should be as close as possible to 0.025
3. Find the actual level of significance of the test based on your critical region from part (b) Rowan records that in the latest batch of 15 type \(A\) vacuum tubes exposed to alternating high and low temperatures, 4 of them shattered.
4. With reference to part (b), comment on Rowan’s belief. Give a reason for your answer. Rowan changes to type \(B\) vacuum tubes. He takes a random sample of 40 type \(B\) vacuum tubes and finds that 8 of them shatter when exposed to alternating high and low temperatures.
5. Test, at the \(5 \%\) level of significance, whether or not there is evidence that the proportion of type \(B\) vacuum tubes that shatter when exposed to alternating high and low temperatures is lower than \(35 \%\)
   You should state your hypotheses clearly.

1. The continuous random variable \(G\) has probability density function \(\mathrm { f } ( \mathrm { g } )\) given by

$$f ( g ) = \begin{cases} \frac { 1 } { 15 } ( g + 3 ) & - 1 < g \leqslant 2
\frac { 3 } { 20 } & 2 < g \leqslant 4
0 & \text { otherwise } \end{cases}$$

1. Sketch the graph of \(\mathrm { f } ( \mathrm { g } )\)
2. Find \(\mathrm { P } ( ( 1 \leqslant 2 G \leqslant 6 ) \mid G \leqslant 2 )\) The continuous random variable \(H\) is such that \(\mathrm { E } ( H ) = 12\) and \(\operatorname { Var } ( H ) = 2.4\)
3. Find \(\mathrm { E } \left( 2 H ^ { 2 } + 3 G + 3 \right)\) Show your working clearly.
   (Solutions relying on calculator technology are not acceptable.)

1. The random variable \(W\) has a continuous uniform distribution over the interval \([ - 6 , a ]\) where \(a\) is a constant.

Given that \(\operatorname { Var } ( W ) = 27\)

1. show that \(a = 12\) Given that \(\mathrm { P } ( W > b ) = \frac { 3 } { 5 }\)
2. 1. find the value of \(b\)
   2. find \(\mathrm { P } \left( - 12 < W < \frac { b } { 2 } \right)\) A piece of wood \(A B\) has length 160 cm . The wood is cut at random into 2 pieces. Each of the pieces is then cut in half. The four pieces are used to form the sides of a rectangle.
3. Calculate the probability that the area of the rectangle is greater than \(975 \mathrm {~cm} ^ { 2 }\)

1. A bag contains a large number of counters with an odd number or an even number written on each.

Odd and even numbered counters occur in the ratio \(4 : 1\)
In a game a player takes a random sample of 4 counters from the bag.
The player scores
5 points for each counter taken that has an even number written on it
2 points for each counter taken that has an odd number written on it
The random variable \(X\) represents the total score, in points, from the 4 counters.

1. Find the sampling distribution of \(X\) A random sample of \(n\) sets of 4 counters is taken. The random variable \(Y\) represents the number of these \(n\) sets that have a total score of exactly 14
2. Calculate the minimum value of \(n\) such that \(\mathrm { P } ( Y \geqslant 1 ) > 0.95\)

1. A continuous random variable \(X\) has cumulative distribution function \(\mathrm { F } ( x )\) given by

$$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x < 1
k \left( a x + b x ^ { 3 } - x ^ { 4 } - 4 \right) & 1 \leqslant x \leqslant 2
1 & x > 2 \end{array} \right.$$ where \(a\), \(b\) and \(k\) are non-zero constants.
Given that the mode of \(X\) is 1.5

1. show that \(b = 3\)
2. Hence show that \(a = 2\)
3. Show that the median of \(X\) lies between 1.4 and 1.5