| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2012 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear combinations of normal random variables |
| Type | Expectation and variance with context application |
| Difficulty | Moderate -0.8 This is a straightforward application of linear transformations of random variables (E[aX+b] and SD[aX+b]) followed by combining independent variables. The context is simple, the formulas are standard, and no problem-solving insight is required—just direct application of memorized rules for expectation and variance. |
| Spec | 5.04a Linear combinations: E(aX+bY), Var(aX+bY) |
| Answer | Marks | Guidance |
|---|---|---|
| Mean = 500 + 3 × 142 = 926 (cents) | B1 | |
| SD = 3 × 35 = 105 (cents) | M1 A1 [3] | Or 9 × 35² seen; Accept √11025 |
| Answer | Marks | Guidance |
|---|---|---|
| Mean = 6 × '926' = 5556 (cents); 6 × 105² (= 66150); SD = √66150 = 257 (cents) (3 sf) | B1ft M1 A1 [3] | or SD = √(6 × '105'). ft their (i) |
**(i)**
Mean = 500 + 3 × 142 = 926 (cents) | B1 |
SD = 3 × 35 = 105 (cents) | M1 A1 [3] | Or 9 × 35² seen; Accept √11025
**(ii)**
Mean = 6 × '926' = 5556 (cents); 6 × 105² (= 66150); SD = √66150 = 257 (cents) (3 sf) | B1ft M1 A1 [3] | or SD = √(6 × '105'). ft their (i)
Accept √661503 The cost of hiring a bicycle consists of a fixed charge of 500 cents together with a charge of 3 cents per minute. The number of minutes for which people hire a bicycle has mean 142 and standard deviation 35.\\
(i) Find the mean and standard deviation of the amount people pay when hiring a bicycle.\\
(ii) 6 people hire bicycles independently. Find the mean and standard deviation of the total amount paid by all 6 people.
\hfill \mbox{\textit{CAIE S2 2012 Q3 [6]}}