| Exam Board | SPS |
|---|---|
| Module | SPS SM Statistics (SPS SM Statistics) |
| Year | 2025 |
| Session | January |
| Marks | 8 |
| Topic | Normal Distribution |
| Type | Standard two probabilities given |
| Difficulty | Standard +0.3 This is a standard normal distribution problem requiring students to set up two equations using z-scores from given probabilities, then solve simultaneously for μ and σ. Part (ii) adds a simple conceptual check about distribution shape. While it involves multiple steps, the technique is routine and commonly practiced in A-level statistics. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation |
3. The continuous random variable $T$ has mean $\mu$ and standard deviation $\sigma$. It is known that $\mathrm { P } ( T < 140 ) = 0.01$ and $\mathrm { P } ( T < 300 ) = 0.8$.\\
(i) Assuming that $T$ is normally distributed, calculate the values of $\mu$ and $\sigma$.
In fact, $T$ represents the time, in minutes, taken by a randomly chosen runner in a public marathon, in which about $10 \%$ of runners took longer than 400 minutes.\\
(ii) State with a reason whether the mean of $T$ would be higher than, equal to, or lower than the value calculated in part (i).\\[0pt]
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\hfill \mbox{\textit{SPS SPS SM Statistics 2025 Q3 [8]}}