S2 June 2014 (R) Q2

EdexcelOld spec7 marksBinomial Distribution Probability

2. A bag contains a large number of counters. Each counter has a single digit number on it and the mean of all the numbers in the bag is the unknown parameter \(\mu\). The number 2 is on 40% of the counters and the number 5 is on 25% of the counters. All the remaining counters have numbers greater than 5 on them.

A random sample of 10 counters is taken from the bag.

(a) State whether or not each of the following is a statistic

(i) \(S\) = the sum of the numbers on the counters in the sample,

(ii) \(D\) = the difference between the highest number in the sample and \(\mu\),

(iii) \(F\) = the number of counters in the sample with a number 5 on them. (3)

The random variable \(T\) represents the number of counters in a random sample of 10 with the number 2 on them.

(b) Specify the sampling distribution of \(T\). (2)

The counters are selected one by one.

Scheme	Marks
(i) \(S\) is a statistic, (ii) \(D\) is not a statistic, (iii) \(F\) is a statistic	B1, B1, B1
	(3)

Scheme	Marks
\(T \sim \mathrm{B}(10, 0.4)\)	M1A1
	(2)

Mark scheme (c)

Scheme	Marks
\(\mathrm{P}(2^{\prime}\ 2^{\prime}\ 2)\) or \(\mathrm{P}(5\ 5\ 2,\ 5\ {\gt}5\ 2,\ {\gt}5\ {\gt}5\ 2)\) \(= 0.6^2 \times 0.4\) or \(= (0.25)^2(0.4) + 2\times(0.25)(0.35)(0.4) + (0.35)^2(0.4)\)	M1
\(= 0.144\)	A1
	(2)
	(7 marks)

Notes

M1 for identifying the correct possibilities \(2^{\prime}\ 2^{\prime}\ 2\) or 5 5 2 and 5 >5 2 and >5 5 2 and >5 >5 2 or a correct probability statement. The possibilities must be in the correct order. Condone \(2\times\) (5 >5 2) or \(2\times\) (>5 5 2). Implied a correct answer.

A1 for 0.144 or exact equivalent e.g. \(\dfrac{18}{125}\)