Edexcel S2 (Statistics 2) 2024 October

1. During an annual beach-clean, the people doing the clean are asked to conduct a litter survey.
   At a particular beach-clean, litter was found at a rate of 4 items per square metre.
   1. Find the probability that, in a randomly selected area of 2 square metres on this beach, exactly 5 items of litter were found.
   Of the litter found on the beach, 30\% of the items were face masks.
2. Find the probability that, in a randomly selected area of 5 square metres on this beach, more than 4 face masks were found.
3. Using a suitable approximation, find the probability that, in a randomly selected area of 20 square metres on this beach, less than 60 items of litter were found that were not face masks.

1. A multiple-choice test consists of 25 questions, each having 5 responses, only one of which is correct.

Each correct answer gains 4 marks but each incorrect answer loses 1 mark.
Sam answers all 25 questions by choosing at random one response for each question.
Let \(X\) be the number of correct answers that Sam achieves.

1. State the distribution of \(X\) Let \(M\) be the number of marks that Sam achieves.
2. 1. State the distribution of \(M\) in terms of \(X\)
   2. Hence, show clearly that the number of marks that Sam is expected to achieve is zero. In order to pass the test at least 30 marks are required.
3. Find the probability that Sam will pass the test. Past records show that when the test is done properly, the probability that a student answers the first question correctly is 0.5 A random sample of 50 students that did the test properly was taken.
   Given that the probability that more than \(n\) but at most 30 students answered the first question correctly was 0.9328 to 4 decimal places,
4. find the value of \(n\)

1. During Monday afternoons, customers are known to enter a certain shop at a mean rate of 7 customers every 10 minutes.
   1. Suggest a suitable distribution to model the number of customers that enter this shop in a 10-minute interval on Monday afternoons.
   2. State two assumptions necessary for this distribution to be a suitable model of this situation.
   A new shop manager wants to find out if the rate of customers has changed since they took over.
2. Write down suitable null and alternative hypotheses that the shop manager should use. The shop manager decides to monitor the number of customers entering the shop in a random 10-minute interval next Monday afternoon.
3. Using a \(3 \%\) level of significance, find the critical region to test whether the rate of customers has changed.
4. Find the actual significance level of this test based on your critical region from part (d) During the random 10-minute interval that Monday afternoon, 12 customers entered the shop.
5. Comment on this finding, using the critical region in part (d)

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

Given that

• \(\mathrm { P } ( X > 27 ) = \frac { 3 } { 4 }\)
• \(\operatorname { Var } ( X ) = 300\)
  1. find the value of \(a\) and the value of \(b\)

Given also that $$4 \times \mathrm { P } ( X < k - 10 ) = \mathrm { P } ( X > k + 20 )$$

find the value of \(k\) (ii) A piece of wire of length 42 cm is cut into 2 pieces at a random point. Each of the two pieces of the wire is bent to form the outline of a square.
Find the probability that the side length of the larger square minus the side length of the smaller square will be greater than 2 cm .

1. The continuous random variable \(X\) has a probability density function given by

$$f ( x ) = \begin{cases} \frac { 1 } { 4 } ( 3 - x ) & 1 \leqslant x \leqslant 2 \\ \frac { 1 } { 4 } & 2 < x \leqslant 3 \\ \frac { 1 } { 4 } ( x - 2 ) & 3 < x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$ The cumulative distribution function of \(X\) is \(\mathrm { F } ( x )\)

1. Show that \(\mathrm { F } ( x ) = \frac { 1 } { 4 } \left( 3 x - \frac { x ^ { 2 } } { 2 } \right) - \frac { 5 } { 8 }\) for \(1 \leqslant x \leqslant 2\)
2. Find \(\mathrm { F } ( x )\) for all values of \(x\)
3. Find \(\mathrm { P } ( 1.2 < X < 3.1 )\)

1. Two boxes, A and B , each contain a large number of coins.

In box A

• there are only 1 p coins and 2 p coins
• the ratio of 1 p coins to 2 p coins is \(1 : 3\)

In box B

• there are only 2 p coins and 5 p coins
• the ratio of 2 p coins to 5 p coins is \(1 : 4\)

One coin is randomly selected from box A and two coins are randomly selected from box B The random variable \(T\) represents the total of the values of the three coins selected.

1. Find the sampling distribution of \(T\) The random variable \(M\) represents the median of the values of the three coins selected.
2. Find the sampling distribution of \(M\)

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

$$f ( x ) = \begin{cases} a x & 0 \leqslant x \leqslant 4 \\ b x + c & 4 < x \leqslant 8 \\ 0 & \text { otherwise } \end{cases}$$ where \(a\), \(b\) and \(c\) are constants. \begin{figure}[h] \end{figure} Figure 1 shows the graph of the probability density function \(\mathrm { f } ( x )\) The graph consists of two straight line segments of equal length joined at the point where \(x = 4\)

1. Show that \(a = \frac { 1 } { 16 }\)
2. Hence find
   1. the value of \(b\)
   2. the value of \(c\)
3. Using algebraic integration, show that \(\operatorname { Var } ( X ) = \frac { 8 } { 3 }\)
4. Find, to 2 decimal places, the lower quartile and the upper quartile of \(X\) A statistician claims that $$\mathrm { P } ( - \sigma < X - \mu < \sigma ) > 0.5$$ where \(\mu\) and \(\sigma\) are the mean and standard deviation of \(X\)
5. Show that the statistician's claim is correct.