6 The volume of shampoo, \(V\) millilitres, delivered by a machine into bottles may be modelled by a normal random variable with mean \(\mu\) and standard deviation \(\sigma\).
- Given that \(\mu = 412\) and \(\sigma = 8\), determine:
- \(\mathrm { P } ( V < 400 )\);
- \(\mathrm { P } ( V > 420 )\);
- \(\mathrm { P } ( V = 410 )\).
- A new quality control specification requires that the values of \(\mu\) and \(\sigma\) are changed so that
$$\mathrm { P } ( V < 400 ) = 0.05 \quad \text { and } \quad \mathrm { P } ( V > 420 ) = 0.01$$
- Show, with the aid of a suitable sketch, or otherwise, that
$$400 - \mu = - 1.6449 \sigma \quad \text { and } \quad 420 - \mu = 2.3263 \sigma$$
- Hence calculate values for \(\mu\) and \(\sigma\).