Moderate -0.3 This is a standard inverse normal distribution problem requiring students to set up two equations using z-scores from given probabilities (P(X > 12) = 0.04 and P(X > 9) = 0.32), then solve simultaneously for μ and σ. While it requires understanding of standardization and table lookup/inverse, it's a routine textbook exercise with a well-established method, making it slightly easier than average.
2 Lengths of a certain type of white radish are normally distributed with mean \(\mu \mathrm { cm }\) and standard deviation \(\sigma \mathrm { cm } .4 \%\) of these radishes are longer than 12 cm and \(32 \%\) are longer than 9 cm . Find \(\mu\) and \(\sigma\).
An eqn with a z-value, \(\mu\) and \(\sigma\) no \(\sigma^2\)
\(\sigma = 2.34\)
M1
Sensible attempt to eliminate \(\mu\) or \(\sigma\) by substitution or subtraction, need a value
\(\mu = 7.91\)
A1
5
$1.751 = \frac{12 - \mu}{\sigma}$ | B1 | Rounding to ±1.75 seen
$0.468 = \frac{9 - \mu}{\sigma}$ | B1 | ±0.468 seen
| M1 | An eqn with a z-value, $\mu$ and $\sigma$ no $\sigma^2$
$\sigma = 2.34$ | M1 | Sensible attempt to eliminate $\mu$ or $\sigma$ by substitution or subtraction, need a value
$\mu = 7.91$ | A1 | 5 | correct answers
2 Lengths of a certain type of white radish are normally distributed with mean $\mu \mathrm { cm }$ and standard deviation $\sigma \mathrm { cm } .4 \%$ of these radishes are longer than 12 cm and $32 \%$ are longer than 9 cm . Find $\mu$ and $\sigma$.
\hfill \mbox{\textit{CAIE S1 2014 Q2 [5]}}