| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2012 |
| Session | November |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Normal Distribution |
| Type | Rounded or discrete from continuous |
| Difficulty | Standard +0.3 This is a straightforward application of normal distribution techniques with standard table lookups. Part (i) requires understanding rounding (finding P(83.5 < X < 84.5)), part (ii) combines normal probability with binomial calculation, and part (iii) is a reverse normal table lookup. All are routine procedures slightly above average difficulty due to the multi-step nature and discrete-from-continuous conversion. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \([\frac{1}{2} \times 12(7^2 - 3^2)]\) | M1 | For using KE = \(\frac{1}{2}m(v_B^2 - v_A^2)\) |
| Increase is 240 J | A1 | 2 marks total |
| (ii) | M1 | For using \(mgh\) = KE gain |
| \(12g \times AB\sin10° = 240\) | A1ft | |
| Distance is 11.5 m | A1 | 3 marks total |
| SR for candidates who avoid 'hence' (max 2/3): For using Newton's Second Law and \(v^2 = u^2 + 2as\): \([12g\sin10° = 12a\) then \(7^2 = 3^2 + 2(g\sin10° \times AB)]\) | M1 | 11.5 m |
| (iii) | ||
| \(F \times 11.5\cos10° = 240\) or \(F\cos10° - 12g\sin10° = 0\) | A1ft | |
| Magnitude is 21.2 N | A1 | 3 marks total |
**(i)** $[\frac{1}{2} \times 12(7^2 - 3^2)]$ | M1 | For using KE = $\frac{1}{2}m(v_B^2 - v_A^2)$
Increase is 240 J | A1 | 2 marks total
**(ii)** | M1 | For using $mgh$ = KE gain
$12g \times AB\sin10° = 240$ | A1ft |
Distance is 11.5 m | A1 | 3 marks total
| | | SR for candidates who avoid 'hence' (max 2/3): For using Newton's Second Law and $v^2 = u^2 + 2as$: $[12g\sin10° = 12a$ then $7^2 = 3^2 + 2(g\sin10° \times AB)]$ | M1 | 11.5 m | A1
**(iii)** | | | For using $F(AB)\cos10° =$ PE gain or for using Newton's 2nd law with $a = 0$
$F \times 11.5\cos10° = 240$ or $F\cos10° - 12g\sin10° = 0$ | A1ft |
Magnitude is 21.2 N | A1 | 3 marks total
---5 The random variable $X$ is such that $X \sim \mathrm {~N} ( 82,126 )$.\\
(i) A value of $X$ is chosen at random and rounded to the nearest whole number. Find the probability that this whole number is 84 .\\
(ii) Five independent observations of $X$ are taken. Find the probability that at most one of them is greater than 87.\\
(iii) Find the value of $k$ such that $\mathrm { P } ( 87 < X < k ) = 0.3$.
\hfill \mbox{\textit{CAIE S1 2012 Q5 [12]}}