CAIE S1 2013 November — Question 3 5 marks

Exam BoardCAIE
ModuleS1 (Statistics 1)
Year2013
SessionNovember
Marks5
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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 for sums. Given n=18, mean, and Σx², students use Σ(x-5) = Σx - 5n and Σ(x-5)² = Σx² - 10Σx + 25n. Requires only algebraic manipulation of standard results with no conceptual difficulty or problem-solving insight.
Spec2.02f Measures of average and spread2.02g Calculate mean and standard deviation
3 Swati measured the lengths, \(x \mathrm {~cm}\), of 18 stick insects and found that \(\Sigma x ^ { 2 } = 967\). Given that the mean length is \(\frac { 58 } { 9 } \mathrm {~cm}\), find the values of \(\Sigma ( x - 5 )\) and \(\Sigma ( x - 5 ) ^ { 2 }\).
Question 3:
AnswerMarks Guidance
Answer/WorkingMark Guidance
\(\Sigma(x-5) = 116 - 18 \times 5 = 26\)M1, A1 Obtaining \(\Sigma x\) and subtracting \(18 \times 5\); correct answer
\(\dfrac{\Sigma(x-5)^2}{18} - \left(\dfrac{26}{18}\right)^2 = \dfrac{967}{18} - \left(\dfrac{58}{9}\right)^2\)M1, M1 Subst in correct var formula all coded vals; subst in correct var formula all uncoded
\(\Sigma(x-5)^2 = 257\)A1 [5] Correct answer
*OR* coded mean \(= \frac{58}{9} - 5 = 1.444\); \(\Sigma(x-5) = 1.444 \times 18 = 26\)M1, A1 Subtracting 5 from true mean and mult by 18; correct answer
\(\Sigma(x-5)^2 = \Sigma x^2 - 10\Sigma x + 25 \times 18 = 967 - 1160 + 450 = 257\)M1, A1, A1 Expanding \(\Sigma(x-5)^2\) 3 terms needed; any 2 terms correct; correct answer 
## Question 3:

| Answer/Working | Mark | Guidance |
|---|---|---|
| $\Sigma(x-5) = 116 - 18 \times 5 = 26$ | M1, A1 | Obtaining $\Sigma x$ and subtracting $18 \times 5$; correct answer |
| $\dfrac{\Sigma(x-5)^2}{18} - \left(\dfrac{26}{18}\right)^2 = \dfrac{967}{18} - \left(\dfrac{58}{9}\right)^2$ | M1, M1 | Subst in correct var formula all coded vals; subst in correct var formula all uncoded |
| $\Sigma(x-5)^2 = 257$ | A1 **[5]** | Correct answer |
| *OR* coded mean $= \frac{58}{9} - 5 = 1.444$; $\Sigma(x-5) = 1.444 \times 18 = 26$ | M1, A1 | Subtracting 5 from true mean and mult by 18; correct answer |
| $\Sigma(x-5)^2 = \Sigma x^2 - 10\Sigma x + 25 \times 18 = 967 - 1160 + 450 = 257$ | M1, A1, A1 | Expanding $\Sigma(x-5)^2$ 3 terms needed; any 2 terms correct; correct answer |

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3 Swati measured the lengths, $x \mathrm {~cm}$, of 18 stick insects and found that $\Sigma x ^ { 2 } = 967$. Given that the mean length is $\frac { 58 } { 9 } \mathrm {~cm}$, find the values of $\Sigma ( x - 5 )$ and $\Sigma ( x - 5 ) ^ { 2 }$.

\hfill \mbox{\textit{CAIE S1 2013 Q3 [5]}}
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