| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2006 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Normal Distribution |
| Type | Find p then binomial probability |
| Difficulty | Standard +0.3 This is a straightforward normal distribution question requiring standard techniques: (i) using inverse normal to find σ from a given probability, (ii) calculating a probability using the found σ, and (iii) applying binomial probability with the complement rule. All steps are routine S1 procedures with no novel problem-solving required, making it slightly easier than average. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(1.645 = \dfrac{50-38}{\sigma}\) | B1 | Using \(z = \pm 1.645\) or \(1.65\) |
| M1 | Equation with \(38, 50, \sigma\) and a recognisable \(z\)-value | |
| \(\sigma = 7.29\) | A1 | 3 Correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(z = \dfrac{30-38}{\text{their } \sigma}\ (= -1.097)\) | M1 | Standardising, no cc |
| \(P(z < 30) = 1 - \Phi(1.097)\) | M1 | Finding correct area i.e. \(< 0.5\) |
| \(= 1 - 0.8637\) | ||
| \(= 0.136\) | A1 | 3 Correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(1-(0.95)^8\) | B1 | \((0.95)^8\) seen |
| \(= 0.370\) | B1 | 2 Correct answer |
## Question 3(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $1.645 = \dfrac{50-38}{\sigma}$ | B1 | Using $z = \pm 1.645$ or $1.65$ |
| | M1 | Equation with $38, 50, \sigma$ and a recognisable $z$-value |
| $\sigma = 7.29$ | A1 | **3** Correct answer |
## Question 3(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $z = \dfrac{30-38}{\text{their } \sigma}\ (= -1.097)$ | M1 | Standardising, no cc |
| $P(z < 30) = 1 - \Phi(1.097)$ | M1 | Finding correct area i.e. $< 0.5$ |
| $= 1 - 0.8637$ | | |
| $= 0.136$ | A1 | **3** Correct answer |
## Question 3(iii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $1-(0.95)^8$ | B1 | $(0.95)^8$ seen |
| $= 0.370$ | B1 | **2** Correct answer |
---3 The lengths of fish of a certain type have a normal distribution with mean 38 cm . It is found that $5 \%$ of the fish are longer than 50 cm .\\
(i) Find the standard deviation.\\
(ii) When fish are chosen for sale, those shorter than 30 cm are rejected. Find the proportion of fish rejected.\\
(iii) 9 fish are chosen at random. Find the probability that at least one of them is longer than 50 cm .
\hfill \mbox{\textit{CAIE S1 2006 Q3 [8]}}