Specimen 2017 H2 Mathematics Paper 2 Question 9

The Normal Distribution

Answers

{81.9\%.}

{C \sim N( 5.0894,0.0056849 )}

{5.21 \textrm{ cm}.}

{0.712.}

{\mu = 1.19,}

{\sigma = 0.0584.}

Let

{B}

denote the r.v. of the radius of a randomly chosen bolt

B \sim N( 0.8,0.01^2 )

\begin{align*} & \textrm{Percentage required} \\ & = \textrm{P}(0.79 < B < 0.82) \times 100\% \\ & = 81.9\% \textrm{ (3 s.f.)} \; \blacksquare \end{align*}

Let

{W}

and

{C}

denote the r.v. of the inside radius and the inside circumference of a randomly chosen washer respectively

\begin{gather*} W \sim N( 0.81,0.012^2 ) \\ C = 2\pi W \sim \textrm{N}( 2\pi(0.81), (2\pi)^2 (0.012)^2 ) \\ C \sim N( 5.0894,0.0056849 ) \; \blacksquare \end{gather*}

Let

{c}

denote the circumference exceeded by

{5\%}

of the washers

\begin{align*} \textrm{P}(C > c) &= 0.05 \\ c &= 5.21 \textrm{ (3 s.f.) cm} \; \blacksquare \end{align*}

W-B \sim \textrm{N} (0.01, 0.000244)

\begin{align*} & \textrm{P}(\textrm{good fit}) \\ & = \textrm{P}(B < W < B+0.04) \\ & = \textrm{P}(0 < W-B < 0.04) \\ & = 0.712 \textrm{( 3 s.f.)} \; \blacksquare \end{align*}

Let

{O}

denote the r.v. of the outside radius of a randomly chosen washer

O \sim \textrm{N}(\mu, \sigma^2)

\begin{align*} \textrm{P}(O > 1.25) &= 0.15 \\ \textrm{P}(Z > \frac{1.25-\mu}{\sigma}) &= 0.15 \\ \frac{1.25-\mu}{\sigma} &= 1.0364 \\ \end{align*}

\begin{align*} \textrm{P}(O < 1.15) &= 0.25 \\ \textrm{P}(Z < \frac{1.15-\mu}{\sigma}) &= 0.25 \\ \frac{1.15-\mu}{\sigma} &= -0.67449 \\ \end{align*}

\begin{align} && \quad \mu + 1.0364\sigma &= 1.25 \\ && \quad \mu -0.67449\sigma &= 1.15 \\ \end{align}

Solving with a GC,

\begin{gather*} \mu = 1.19 \textrm{ (3 s.f.)} \; \blacksquare \\ \sigma = 0.0584 \textrm{ (3 s.f.)} \; \blacksquare \end{gather*}