| Exam Board | OCR |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2010 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Normal Distribution |
| Type | Standard two probabilities given |
| Difficulty | Moderate -0.8 Part (i) requires only basic understanding that probabilities sum to 1 and that normal distributions are monotonic, with no calculations needed. Part (ii) is a standard S2 exercise using symmetry of the normal distribution and z-tables, requiring straightforward algebraic manipulation. The question tests fundamental properties rather than problem-solving skills, making it easier than average A-level maths questions. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working: Probabilities don't total 1 | Marks: B1 | Guidance: |
| Answer/Working: P(> 70) must be < P(> 50) | Marks: B1 | Guidance: |
| Answer/Working: P(< 50) = 0.3 ⇒ μ < 50 | Marks: B1 | Guidance: Any relevant valid statement, e.g. "P(< 50) = 0.7 but P(< 50) must be < P(< 70)" |
| Answer/Working: P(< 70) = 0.3 ⇒ μ > 70 | Marks: B1 | Guidance: |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working: μ = 60 by symmetry | Marks: B1 | Guidance: μ = 60 obtained at any point, allow from Φ |
| Answer/Working: \(\frac{10}{\sigma} = \Phi^{-1}(0.7) = 0.5244(4)\) | Marks: M1 | Guidance: One standardisation, equate to \(\Phi^{-1}\), not 0.758 |
| Answer/Working: σ = 10/0.5243 | Marks: B1 | Guidance: \(\Phi^{-1} \in [0.524, 0.5245]\) seen |
| Answer/Working: = 19.084 | Marks: A1, 4 | Guidance: σ in range [19.07, 19.1], e.g. 19.073 |
### (i)
**Answer/Working:** Probabilities don't total 1 | **Marks:** B1 | **Guidance:**
**Answer/Working:** P(> 70) must be < P(> 50) | **Marks:** B1 | **Guidance:**
**Answer/Working:** P(< 50) = 0.3 ⇒ μ < 50 | **Marks:** B1 | **Guidance:** Any relevant valid statement, e.g. "P(< 50) = 0.7 but P(< 50) must be < P(< 70)"
**Answer/Working:** P(< 70) = 0.3 ⇒ μ > 70 | **Marks:** B1 | **Guidance:**
### (ii)
**Answer/Working:** μ = 60 by symmetry | **Marks:** B1 | **Guidance:** μ = 60 obtained at any point, allow from Φ
**Answer/Working:** $\frac{10}{\sigma} = \Phi^{-1}(0.7) = 0.5244(4)$ | **Marks:** M1 | **Guidance:** One standardisation, equate to $\Phi^{-1}$, not 0.758
**Answer/Working:** σ = 10/0.5243 | **Marks:** B1 | **Guidance:** $\Phi^{-1} \in [0.524, 0.5245]$ seen
**Answer/Working:** = 19.084 | **Marks:** A1, 4 | **Guidance:** σ in range [19.07, 19.1], e.g. 19.073
---The continuous random variable $X$ has the distribution N($\mu$, $\sigma^2$).
\begin{enumerate}[label=(\roman*)]
\item Each of the three following sets of probabilities is impossible. Give a reason in each case why the probabilities cannot both be correct. (You should not attempt to find $\mu$ or $\sigma$.)
\begin{enumerate}[label=(\alph*)]
\item P($X > 50$) = 0.7 and P($X < 50$) = 0.2 [1]
\item P($X > 50$) = 0.7 and P($X > 70$) = 0.8 [1]
\item P($X > 50$) = 0.3 and P($X < 70$) = 0.3 [1]
\end{enumerate}
\item Given that P($X > 50$) = 0.7 and P($X < 70$) = 0.7, find the values of $\mu$ and $\sigma$. [4]
\end{enumerate}
\hfill \mbox{\textit{OCR S2 2010 Q6 [7]}}