| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2011 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Measures of Location and Spread |
| Type | Standard combined mean and SD |
| Difficulty | Moderate -0.8 This is a straightforward application of combined mean and standard deviation formulas. Students need to calculate Fairwheel's mean/SD from summary statistics, then use standard formulas to combine two datasets. All steps are routine with no problem-solving required, making it easier than average but not trivial due to the multi-step calculation and potential for arithmetic errors. |
| Spec | 2.02f Measures of average and spread2.02g Calculate mean and standard deviation |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \((3.6 \times 9 + 64) / 24 = 4.02\) years | M1, A1 [2] | Mult by 9, adding 64 then dividing by 24; Correct answer |
| (ii) \(\frac{\sum x_i^2}{9} - 3.6^2 = 1.925^2\) | M1 | Attempt to find \(\sum x_i^2\) using correct variance formula |
| \(\sum x_i^2 = 150\) | A1 | Correct \(\sum x_i^2\) |
| \(\frac{150.0 + 352}{24} = 4.017^2 = 4.780\) | M1 | Using 352 + their 150 in incorrect variance formula |
| \(\text{sd} = 2.19\) | A1 [4] | Correct answer |
**(i)** $(3.6 \times 9 + 64) / 24 = 4.02$ years | M1, A1 [2] | Mult by 9, adding 64 then dividing by 24; Correct answer
**(ii)** $\frac{\sum x_i^2}{9} - 3.6^2 = 1.925^2$ | M1 | Attempt to find $\sum x_i^2$ using correct variance formula
$\sum x_i^2 = 150$ | A1 | Correct $\sum x_i^2$
$\frac{150.0 + 352}{24} = 4.017^2 = 4.780$ | M1 | Using 352 + their 150 in incorrect variance formula
$\text{sd} = 2.19$ | A1 [4] | Correct answer
---1 Red Street Garage has 9 used cars for sale. Fairwheel Garage has 15 used cars for sale. The mean age of the cars in Red Street Garage is 3.6 years and the standard deviation is 1.925 years. In Fairwheel Garage, $\Sigma x = 64$ and $\Sigma x ^ { 2 } = 352$, where $x$ is the age of a car in years.\\
(i) Find the mean age of all 24 cars.\\
(ii) Find the standard deviation of the ages of all 24 cars.
\hfill \mbox{\textit{CAIE S1 2011 Q1 [6]}}