A2 October 2020 Q4

EdexcelCurrent spec7 marksConfidence Intervals

4. A biased coin has a probability \(p\) of landing on heads, where \(0 \lt p \lt 1\)
Simon spins the coin \(n\) times and the random variable \(X\) represents the number of heads.
Taruni spins the coin \(m\) times, \(m \neq n\), and the random variable \(Y\) represents the number of heads.

Simon and Taruni want to combine their results to find unbiased estimators of \(p\).

Simon proposes the estimator \(S = \dfrac{X + Y}{m + n}\) and Taruni proposes \(T = \dfrac{1}{2}\left[\dfrac{X}{n} + \dfrac{Y}{m}\right]\)

(a) Show that both \(S\) and \(T\) are unbiased estimators of \(p\). (3)

(b) Prove that, for all values of \(m\) and \(n\), \(S\) is the better estimator. (4)

Mark scheme (a)

Scheme	Marks	AO
\(X \sim \mathrm{B}(n, p)\) so \(\mathrm{E}(X) = np\) and \(Y \sim \mathrm{B}(m, p)\) so \(\mathrm{E}(Y) = mp\)	M1	3.3
\(\mathrm{E}(S) = \dfrac{\mathrm{E}(X + Y)}{n + m} = \dfrac{np + mp}{n + m} = p\) so \(S\) is unbiased	M1	3.4
\(\mathrm{E}(T) = \dfrac{1}{2}\left[\dfrac{\mathrm{E}(X)}{n} + \dfrac{\mathrm{E}(Y)}{m}\right] = \dfrac{1}{2}\left[\dfrac{np}{n} + \dfrac{mp}{m}\right] = \dfrac{1}{2} \times 2p = p\) so \(T\) is unbiased	A1cso	1.1b
	(3)

Notes

1^st M1 for selecting correct models for \(X\) and \(Y\)

2^nd M1 for using these models to show that either \(S\) or \(T\) is unbiased

A1cso for correctly showing that both are unbiased.

Mark scheme (b)

Scheme	Marks	AO
\(\mathrm{Var}(S) = \dfrac{np(1-p) + mp(1-p)}{(n+m)^2} = \dfrac{p(1-p)}{n+m}\)	M1	2.1
\(\mathrm{Var}(T) = \dfrac{1}{4}\left[\dfrac{np(1-p)}{n^2} + \dfrac{mp(1-p)}{m^2}\right] = \dfrac{p(1-p)(m+n)}{4nm}\)	A1	1.1b
\(\mathrm{Var}(S) \lt \mathrm{Var}(T) \Rightarrow \dfrac{p(1-p)}{n+m} \lt \dfrac{p(1-p)(m+n)}{4mn} \Leftrightarrow 4mn \lt (n+m)^2\)	M1	1.1b
\(\Leftrightarrow \quad 0 \lt m^2 + 2mn + n^2 - 4mn \quad \Leftrightarrow \quad 0 \lt (m-n)^2\) So \(S\) always has the smaller variance and is the better estimator	A1cso	2.2a
	(4)
	(7 marks)

Notes

1^st M1 for a correct attempt at \(\mathrm{Var}(S)\) or \(\mathrm{Var}(T)\) (need not be simplified)

1^st A1 for both correct variances

2^nd M1 for a correct inequality in \(m\) and \(n\) and a first step to clear denominators

2^nd A1cso for a correct proof and conclusion