3 The random variable \(X\) is the length of time in minutes that Jannon takes to mend a bicycle puncture. \(X\) has a normal distribution with mean \(\mu\) and variance \(\sigma ^ { 2 }\). It is given that \(\mathrm { P } ( X > 30.0 ) = 0.1480\) and \(\mathrm { P } ( X > 20.9 ) = 0.6228\). Find \(\mu\) and \(\sigma\).

$(+/-) 1.045, (+/-) 0.313$ | B1, B1 | 1 correct $z$-value, the other correct $z$-value | 

$20.9 - \mu = -0.313\sigma$ and $30 - \mu = 1.045\sigma$ | M1 | Valid attempt to solve 2 equations relating to $\mu$, $\sigma$, 30, 20.9. No $\sqrt{\sigma}, \sigma^2$ |

$\sigma = 6.70$ and $\mu = 23.0$ | A1, A1 | correct answer; correct answer | **Total: [5]**

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3 The random variable $X$ is the length of time in minutes that Jannon takes to mend a bicycle puncture. $X$ has a normal distribution with mean $\mu$ and variance $\sigma ^ { 2 }$. It is given that $\mathrm { P } ( X > 30.0 ) = 0.1480$ and $\mathrm { P } ( X > 20.9 ) = 0.6228$. Find $\mu$ and $\sigma$.

\hfill \mbox{\textit{CAIE S1 2010 Q3 [5]}}