| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2015 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Approximating Binomial to Normal Distribution |
| Type | Normal distribution probability then binomial/normal approximation on sample |
| Difficulty | Standard +0.3 This question involves standard normal distribution calculations followed by straightforward binomial probability. Part (i) requires finding P(X>29) then calculating binomial probabilities for n=8, k<2. Part (ii) involves solving 1-(1-p)^n > 0.98 using logarithms. All steps are routine applications of formulas with no novel problem-solving required, making it slightly easier than average. |
| Spec | 2.04c Calculate binomial probabilities2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(P(\text{large}) = 1 - \Phi\!\left(\frac{29 - 21.7}{6.5}\right)\) | M1 | Standardising, no cc, no sq rt |
| M1 | Correct area \(1 - \Phi\) | |
| \(= 1 - \Phi(1.123) = 1 - 0.8692 = 0.1308\) | A1 | Rounding to 0.13 |
| \(P(0,1) = (0.8692)^8 + {}^8C_1(0.1308)(0.8692)^7\) | M1 | Any bin term with \({}^8C_x p^x(1-p)^{8-x}\), \(0 < p < 1\) |
| M1 | Summing bin \(P(0) + P(1)\) only with \(n = 8\), oe | |
| \(= 0.718\) | A1 [6] | Correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(1 - (0.8692)^n > 0.98\) | M1 | eq/ineq involving their \((0.8692)^n\) or \((0.1308)^n\), 0.02 or 0.98 oe with or without a 1 |
| \((0.8692)^n < 0.02\) | M1 | Solving attempt (could be trial and error) — may be implied by their answer |
| Least number \(= 28\) | A1 [3] | Correct answer |
## Question 5:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $P(\text{large}) = 1 - \Phi\!\left(\frac{29 - 21.7}{6.5}\right)$ | M1 | Standardising, no cc, no sq rt |
| | M1 | Correct area $1 - \Phi$ |
| $= 1 - \Phi(1.123) = 1 - 0.8692 = 0.1308$ | A1 | Rounding to 0.13 |
| $P(0,1) = (0.8692)^8 + {}^8C_1(0.1308)(0.8692)^7$ | M1 | Any bin term with ${}^8C_x p^x(1-p)^{8-x}$, $0 < p < 1$ |
| | M1 | Summing bin $P(0) + P(1)$ only with $n = 8$, oe |
| $= 0.718$ | A1 **[6]** | Correct answer |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $1 - (0.8692)^n > 0.98$ | M1 | eq/ineq involving their $(0.8692)^n$ or $(0.1308)^n$, 0.02 or 0.98 oe with or without a 1 |
| $(0.8692)^n < 0.02$ | M1 | Solving attempt (could be trial and error) — may be implied by their answer |
| Least number $= 28$ | A1 **[3]** | Correct answer |
---5 The heights of books in a library, in cm, have a normal distribution with mean 21.7 and standard deviation 6.5. A book with a height of more than 29 cm is classified as 'large'.\\
(i) Find the probability that, of 8 books chosen at random, fewer than 2 books are classified as large.\\
(ii) $n$ books are chosen at random. The probability of there being at least 1 large book is more than 0.98 . Find the least possible value of $n$.
\hfill \mbox{\textit{CAIE S1 2015 Q5 [9]}}