June 2024 Paper 3 Q14
14 The annual cost of energy in 2021 for each of the 350 households in Village A can be modelled by a random variable £\(X\)
It is given that
\[\sum x = 945\,000 \qquad \sum x^2 = 2\,607\,500\,000\](a) Calculate the mean of \(X\). [1 mark]
(b) Calculate the standard deviation of \(X\). [2 marks]
(c) For households in Village B the annual cost of energy in 2021 has mean £3100 and standard deviation £325
Compare the annual cost of energy in 2021 for households in Village A and Village B. [2 marks]
| Scheme | Marks | AO |
|---|---|---|
| Obtains 2700 | B1 | 1.1b |
| (1) |
Typical solution
2700
| Scheme | Marks | AO |
|---|---|---|
| Uses the correct formula for standard deviation using their mean PI by 400 | M1 | 1.1a |
| Obtains 400 | A1 | 1.1b |
| (2) |
Typical solution
\[\sqrt{\frac{2\,607\,500\,000}{350} - 2700^2} = 400\]| Scheme | Marks | AO |
|---|---|---|
| Concludes correctly in context for the means Comparison must include the word such as ‘on average’ or ‘typically’ etc Follow through the correct comparison of 3100 with their mean from part 14(a) | E1F | 2.4 |
| Concludes correctly in context for the standard deviations Comparison must include the word such as ‘varies’, ‘spread’ ‘disperse’ ‘variation’ or ‘consistent’ etc Follow through the correct comparison of 325 with their standard deviation from part 14(b) Do not allow comparison that includes ‘range’ or ‘variety’ | E1F | 2.4 |
| (2) | ||
| (5 marks) |
Typical solution
The cost of energy for households in Village A is less on average
The variation in cost of energy is higher in Village A.