| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2016 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Approximating Binomial to Normal Distribution |
| Type | Exact binomial then normal approximation (same context, different n) |
| Difficulty | Standard +0.3 This is a straightforward application of binomial probability (part i) and normal approximation to binomial (part ii). Part (i) requires calculating P(X≥6) for X~B(7, 0.8), which is routine. Part (ii) is a standard textbook exercise in applying continuity correction for X~B(100, 0.2) approximated by N(20, 16). Both parts require only direct application of learned techniques with no problem-solving insight needed, making it slightly easier than average. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities2.04d Normal approximation to binomial |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\text{Bin}(7, 0.8)\) | M1 | \(^7C_n\, p^n(1-p)^{7-n}\) seen |
| \(P(6,7) = {^7C_6}(0.8)^6(0.2)^1 + (0.8)^7\) | M1 | Correct unsimplified expression for \(P(6,7)\) |
| \(= 0.577\) | A1 [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| mean \(= 100 \times 0.2 = 20\) | B1 | Correct unsimplified mean and var |
| Var \(= 100 \times 0.2 \times 0.8 = 16\) | ||
| \(P(\text{at most } 30) = P\!\left(z < \dfrac{30.5 - 20}{\sqrt{16}}\right)\) | M1 | Standardising must have sq rt, their \(\mu\), variance |
| M1 | cc either 29.5 or 30.5 | |
| M1 | Correct area \(\Phi\), from final process | |
| \(= P(z < 2.625) = 0.996\) | A1 [5] |
## Question 3:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\text{Bin}(7, 0.8)$ | M1 | $^7C_n\, p^n(1-p)^{7-n}$ seen |
| $P(6,7) = {^7C_6}(0.8)^6(0.2)^1 + (0.8)^7$ | M1 | Correct unsimplified expression for $P(6,7)$ |
| $= 0.577$ | A1 [3] | |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| mean $= 100 \times 0.2 = 20$ | B1 | Correct unsimplified mean and var |
| Var $= 100 \times 0.2 \times 0.8 = 16$ | | |
| $P(\text{at most } 30) = P\!\left(z < \dfrac{30.5 - 20}{\sqrt{16}}\right)$ | M1 | Standardising must have sq rt, their $\mu$, variance |
| | M1 | cc either 29.5 or 30.5 |
| | M1 | Correct area $\Phi$, from final process |
| $= P(z < 2.625) = 0.996$ | A1 [5] | |
---3 On any day at noon, the probabilities that Kersley is asleep or studying are 0.2 and 0.6 respectively.\\
(i) Find the probability that, in any 7-day period, Kersley is either asleep or studying at noon on at least 6 days.\\
(ii) Use an approximation to find the probability that, in any period of 100 days, Kersley is asleep at noon on at most 30 days.
\hfill \mbox{\textit{CAIE S1 2016 Q3 [8]}}