2007 H2 Mathematics Paper 2 Question 8

The Normal Distribution

Answers

(i)

{0.395}

(ii)

{0.160}

(iii)

{0.392}

The event in part (ii) is a subset of the event in part (iii).
That is, if the selling price of the chicken exceeds $7 and the total selling price of a turkey exceeds $55, then the total selling price of a turkey and chicken will exceed $62.
However, the converse is not true. If the total selling price of a turkey and chicken exceeds $62, then the selling price of the chicken may not exceed $7 (eg we could have a $5 chicken and a $60 turkey).

Full solutions

(i)

Let

{C}

and

{T}

denote the selling prices of a randomly chosen chicken and turkey respectively

\begin{align*} C &\sim \textrm{N}(3 \times 2.2, 3^2 \times 0.5^2 ) \\ C &\sim \textrm{N}( 6.6000, 2.25 ) \\ \end{align*}

\begin{align*} & \textrm{P}(C > 7) \\ &= 0.39486 \\ &= 0.395 \textrm{ (3 sf)} \; \blacksquare \end{align*}

(ii)

\begin{align*} T &\sim \textrm{N}(5 \times 10.5, 5^2 \times 2.1^2 ) \\ T &\sim \textrm{N}( 52.5, 110.25 ) \\ \end{align*}

\begin{align*} & \textrm{P}(T > 55) \\ &= 0.40590 \end{align*}

\begin{align*} & \textrm{Probability required} \\ & = 0.39486 \times 0.40590 \\ &= 0.160 \textrm{ (3 sf)} \; \blacksquare \end{align*}

(iii)

\begin{align*} C+T &\sim \textrm{N}(5 \times 10.5, 5^2 \times 2.1^2 ) \\ C + T &\sim \textrm{N}( 59.1, 112.5 ) \\ \end{align*}

\begin{align*} & \textrm{P}(C + T > 62) \\ &= 0.406 \textrm{ (3 sf)} \; \blacksquare \end{align*}

(iv)

{\blacksquare}