CAIE S1 2012 November — Question 2 5 marks

Exam BoardCAIE
ModuleS1 (Statistics 1)
Year2012
SessionNovember
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicMeasures of Location and Spread
TypeFind coded sums from raw data
DifficultyModerate -0.8 This is a straightforward application of coding formulas requiring algebraic manipulation of summation notation. Students need to expand Σ(x-36) and Σ(x-36)² using standard identities, then solve for Σx and Σx². While it requires careful algebra, it's a routine textbook exercise with no conceptual difficulty or problem-solving insight needed.
Spec2.02g Calculate mean and standard deviation
2 The amounts of money, \(x\) dollars, that 24 people had in their pockets are summarised by \(\Sigma ( x - 36 ) = - 60\) and \(\Sigma ( x - 36 ) ^ { 2 } = 227.76\). Find \(\Sigma x\) and \(\Sigma x ^ { 2 }\).
Question 2:
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(\Sigma x - \Sigma 36 = -60\)M1 Expanding brackets i.e. mult by 24 and subt 60
\(\Sigma x = 24 \times 36 - 60 = 804\)A1 [2] Correct answer
OR \(\bar{x} = 36 - \frac{60}{24} = 33.5\); \(\Sigma x = 33.5 \times 24 = 804\)M1, A1 Dividing by 24 and subt from 36; Correct answer
\(\Sigma x^2 - 2.36\Sigma x + \Sigma 36^2 = 227.6\)M1 Expanding brackets with \(36\Sigma x\) and \(\Sigma 36^2\)
M1min \(\Sigma x^2 - 2\times 36\Sigma x + \Sigma 36^2 = 227.6\) seen
\(\Sigma x^2 = 27011.76\ (27000)\)A1 [3] Correct answer
OR \(\frac{227.76}{24} - (-2.5)^2 = sd^2 = 3.24\)M1 \(\frac{227.76}{24} - (\text{their coded mean})^2\) seen
\(\frac{\Sigma x^2}{24} - (33.5)^2 = 3.24\)M1 \(\frac{\Sigma x^2}{24} - (\bar{x})^2 = \text{their var if +ve seen o.e.}\)
\(\Sigma x^2 = 27011.76\ (27000)\)A1 Correct answer 
## Question 2:

| Answer/Working | Marks | Guidance |
|---|---|---|
| $\Sigma x - \Sigma 36 = -60$ | M1 | Expanding brackets i.e. mult by 24 and subt 60 |
| $\Sigma x = 24 \times 36 - 60 = 804$ | A1 [2] | Correct answer |
| OR $\bar{x} = 36 - \frac{60}{24} = 33.5$; $\Sigma x = 33.5 \times 24 = 804$ | M1, A1 | Dividing by 24 and subt from 36; Correct answer |
| $\Sigma x^2 - 2.36\Sigma x + \Sigma 36^2 = 227.6$ | M1 | Expanding brackets with $36\Sigma x$ and $\Sigma 36^2$ |
| | M1 | min $\Sigma x^2 - 2\times 36\Sigma x + \Sigma 36^2 = 227.6$ seen |
| $\Sigma x^2 = 27011.76\ (27000)$ | A1 [3] | Correct answer |
| OR $\frac{227.76}{24} - (-2.5)^2 = sd^2 = 3.24$ | M1 | $\frac{227.76}{24} - (\text{their coded mean})^2$ seen |
| $\frac{\Sigma x^2}{24} - (33.5)^2 = 3.24$ | M1 | $\frac{\Sigma x^2}{24} - (\bar{x})^2 = \text{their var if +ve seen o.e.}$ |
| $\Sigma x^2 = 27011.76\ (27000)$ | A1 | Correct answer |

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2 The amounts of money, $x$ dollars, that 24 people had in their pockets are summarised by $\Sigma ( x - 36 ) = - 60$ and $\Sigma ( x - 36 ) ^ { 2 } = 227.76$. Find $\Sigma x$ and $\Sigma x ^ { 2 }$.

\hfill \mbox{\textit{CAIE S1 2012 Q2 [5]}}
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