CAIE S1 2015 November — Question 1 3 marks

Exam BoardCAIE
ModuleS1 (Statistics 1)
Year2015
SessionNovember
Marks3
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TopicMeasures of Location and Spread
TypeCalculate variance from summary statistics
DifficultyEasy -1.2 This is a straightforward application of the variance formula using the computational form with a shifted origin. Students need to recognize that Var(t) = Σ(t-2.5)²/n - (mean - 2.5)², then take the square root. It's a direct substitution into a standard formula with minimal problem-solving required, making it easier than average.
Spec2.02g Calculate mean and standard deviation
1 The time taken, \(t\) hours, to deliver letters on a particular route each day is measured on 250 working days. The mean time taken is 2.8 hours. Given that \(\Sigma ( t - 2.5 ) ^ { 2 } = 96.1\), find the standard deviation of the times taken.
Question 1:
AnswerMarks Guidance
Answer/WorkingMarks Guidance
Coded mean \(= 0.3\) oeB1 \(\Sigma(t-2.5) = 75\), B0 until \(\div 250\)
\(\text{sd} = \sqrt{\frac{96.1}{250}-(0.3)^2}\)M1 Subst in variance formula both terms coded
\(= 0.543\)A1 (3) Correct answer
Alt: \(\Sigma(t-2.5)^2\) expanded, \(\Sigma t^2 = 2033.6\)Or B1
\(\text{sd} = \sqrt{\frac{2033.6}{250}-2.8^2}\)M1 Substituting their \(\Sigma t^2\) from expanded 3-term expression, 250 and 2.8 in variance formula
\(= 0.543\)A1 (3) 
## Question 1:

| Answer/Working | Marks | Guidance |
|---|---|---|
| Coded mean $= 0.3$ oe | B1 | $\Sigma(t-2.5) = 75$, B0 until $\div 250$ |
| $\text{sd} = \sqrt{\frac{96.1}{250}-(0.3)^2}$ | M1 | Subst in variance formula both terms coded |
| $= 0.543$ | A1 (3) | Correct answer |
| Alt: $\Sigma(t-2.5)^2$ expanded, $\Sigma t^2 = 2033.6$ | Or B1 | |
| $\text{sd} = \sqrt{\frac{2033.6}{250}-2.8^2}$ | M1 | Substituting their $\Sigma t^2$ from expanded 3-term expression, 250 and 2.8 in variance formula |
| $= 0.543$ | A1 (3) | |

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1 The time taken, $t$ hours, to deliver letters on a particular route each day is measured on 250 working days. The mean time taken is 2.8 hours. Given that $\Sigma ( t - 2.5 ) ^ { 2 } = 96.1$, find the standard deviation of the times taken.

\hfill \mbox{\textit{CAIE S1 2015 Q1 [3]}}
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