| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2016 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Measures of Location and Spread |
| Type | Calculate mean from coded sums |
| Difficulty | Moderate -0.8 This question tests standard formulas for coded data (mean from Σ(x-c), variance from coded sums) and updating means with additional data points. All three parts involve direct application of well-known formulas with straightforward algebra—no problem-solving insight required, just routine manipulation of summary statistics. |
| Spec | 2.02g Calculate mean and standard deviation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(1845/9\ (=205)\), \(c = 2205 - 205 = 2000\) | M1, A1 | Accept \((1845 \pm \text{anything})/9\) |
| OR: \(\Sigma x = 2205 \times 9\ (=19845)\); \(\Sigma x - \Sigma c = 1845\); \(\Sigma c = 19845 - 1845 = 18000\); \(c = 2000\) | M1, A1 [2] | For \(2205 \times 9\) seen |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\text{var} = \dfrac{477450}{9} - 205^2 = 11025\) | M1, A1 | For \(\dfrac{477450}{9} - (\text{their coded mean})^2\) |
| OR: \(\text{var} = \dfrac{43857450}{9} - 2205^2 = 11025\) | M1, A1 [2] | For their \(\Sigma x^2/9 - 2205^2\) where \(\Sigma x^2\) obtained from expanding \(\Sigma(x-c)^2\) with \(2c\Sigma x\) seen |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| new total \(= 2120.5 \times 10 = 21205\); new price \(= 21205 - 19845 = 1360\) | M1, A1 [2] | Attempt at new total |
## Question 4:
### Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $1845/9\ (=205)$, $c = 2205 - 205 = 2000$ | **M1**, **A1** | Accept $(1845 \pm \text{anything})/9$ |
| OR: $\Sigma x = 2205 \times 9\ (=19845)$; $\Sigma x - \Sigma c = 1845$; $\Sigma c = 19845 - 1845 = 18000$; $c = 2000$ | **M1**, **A1** [2] | For $2205 \times 9$ seen |
### Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\text{var} = \dfrac{477450}{9} - 205^2 = 11025$ | **M1**, **A1** | For $\dfrac{477450}{9} - (\text{their coded mean})^2$ |
| OR: $\text{var} = \dfrac{43857450}{9} - 2205^2 = 11025$ | **M1**, **A1** [2] | For their $\Sigma x^2/9 - 2205^2$ where $\Sigma x^2$ obtained from expanding $\Sigma(x-c)^2$ with $2c\Sigma x$ seen |
### Part (iii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| new total $= 2120.5 \times 10 = 21205$; new price $= 21205 - 19845 = 1360$ | **M1**, **A1** [2] | Attempt at new total |
---4 The monthly rental prices, $\$ x$, for 9 apartments in a certain city are listed and are summarised as follows.
$$\Sigma ( x - c ) = 1845 \quad \Sigma ( x - c ) ^ { 2 } = 477450$$
The mean monthly rental price is $\$ 2205$.\\
(i) Find the value of the constant $c$.\\
(ii) Find the variance of these values of $x$.\\
(iii) Another apartment is added to the list. The mean monthly rental price is now $\$ 2120.50$. Find the rental price of this additional apartment.
\hfill \mbox{\textit{CAIE S1 2016 Q4 [6]}}