| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2012 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Normal Distribution |
| Type | Find p then binomial probability |
| Difficulty | Standard +0.3 This is a straightforward two-part normal distribution question requiring (i) finding σ from a given probability using inverse normal tables, then (ii) applying binomial distribution. Both parts use standard techniques with no novel problem-solving required, making it slightly easier than average. |
| 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 |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(z = -1.036 = \frac{73 - 75}{\sigma}\) | B1 | \(\pm\) correct z value, accept \(\pm 1.037\) |
| M1 | Equation with 73, 75, \(\sigma\) and a z value | |
| \(\sigma = 1.93\) | A1 [3] | Rounding to correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(P(>77) = 0.15\); \(P(<3) = P(0,1,2)\) | M1 | Prob rounding to 0.15 and 0.85 |
| \(= (0.85)^8 + {_8}C_1(0.15)(0.85)^7 + {_8}C_2(0.15)^2(0.85)^6\) | M1 | \(_8C_x p^x(1-p)^{8-x}\) seen, any \(p\), \(0 |
| \(= 0.895\) | A1 [3] | Correct answer |
## Question 3:
### Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $z = -1.036 = \frac{73 - 75}{\sigma}$ | B1 | $\pm$ correct z value, accept $\pm 1.037$ |
| | M1 | Equation with 73, 75, $\sigma$ and a z value |
| $\sigma = 1.93$ | A1 [3] | Rounding to correct answer |
### Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $P(>77) = 0.15$; $P(<3) = P(0,1,2)$ | M1 | Prob rounding to 0.15 and 0.85 |
| $= (0.85)^8 + {_8}C_1(0.15)(0.85)^7 + {_8}C_2(0.15)^2(0.85)^6$ | M1 | $_8C_x p^x(1-p)^{8-x}$ seen, any $p$, $0<p<1$ |
| $= 0.895$ | A1 [3] | Correct answer |
---3 Lengths of rolls of parcel tape have a normal distribution with mean 75 m , and 15\% of the rolls have lengths less than 73 m .\\
(i) Find the standard deviation of the lengths.
Alison buys 8 rolls of parcel tape.\\
(ii) Find the probability that fewer than 3 of these rolls have lengths more than 77 m .
\hfill \mbox{\textit{CAIE S1 2012 Q3 [6]}}