Standard +0.3 This is a standard S2 question requiring students to set up two equations using inverse normal tables (finding z-scores for given probabilities) and solve simultaneously for μ and σ. While it involves multiple steps, the method is routine and commonly practiced in S2 courses, making it slightly easier than average overall but typical for this module.
1 The random variable \(Y\) is normally distributed with mean \(\mu\) and variance \(\sigma ^ { 2 }\). It is found that \(\mathrm { P } ( Y > 150.0 ) = 0.0228\) and \(\mathrm { P } ( Y > 143.0 ) = 0.9332\). Find the values of \(\mu\) and \(\sigma\).
Standardise with \(\sigma\), \(\mu\) at least once, ignore cc, \(\sqrt{}\) errors, equate to \(z\)
\(\frac{143-\mu}{\sigma} = -1.5\)
A1
Both LHS and signs of RHS correct
Both \(z\)-values correct to 3 SF
B1
Correct method for solution
M1
\(\mu \in [145.95, 146.05]\) www
A1
\(\mu = 146\), \(\sigma = 2\)
A1
\(\sigma \in [1.995, 2.005]\) or \(\sigma^2 = 4\) www
Total: 6
\(z\) not used e.g. equated to 0.0228 and 0.9332 or 0.5092 and 0.8246: max M0M1. One \(z\), one not: M1A0B0. \(\sqrt{\sigma}\) or \(\sigma^2\): can get M1A0B1M1A1A0
## Question 1:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{150-\mu}{\sigma} = 2.00$ | M1 | Standardise with $\sigma$, $\mu$ at least once, ignore cc, $\sqrt{}$ errors, equate to $z$ |
| $\frac{143-\mu}{\sigma} = -1.5$ | A1 | Both LHS and signs of RHS correct |
| Both $z$-values correct to 3 SF | B1 | |
| Correct method for solution | M1 | |
| $\mu \in [145.95, 146.05]$ www | A1 | |
| $\mu = 146$, $\sigma = 2$ | A1 | $\sigma \in [1.995, 2.005]$ or $\sigma^2 = 4$ www |
| **Total: 6** | | $z$ not used e.g. equated to 0.0228 and 0.9332 or 0.5092 and 0.8246: max M0M1. One $z$, one not: M1A0B0. $\sqrt{\sigma}$ or $\sigma^2$: can get M1A0B1M1A1A0 |
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1 The random variable $Y$ is normally distributed with mean $\mu$ and variance $\sigma ^ { 2 }$. It is found that $\mathrm { P } ( Y > 150.0 ) = 0.0228$ and $\mathrm { P } ( Y > 143.0 ) = 0.9332$. Find the values of $\mu$ and $\sigma$.
\hfill \mbox{\textit{OCR S2 2015 Q1 [6]}}