Question 1 - A-Level Maths

CAIE S2 2022 November — Question 1 3 marks

Exam Board	CAIE
Module	S2 (Statistics 2)
Year	2022
Session	November
Marks	3
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Measures of Location and Spread
Type	Standard unbiased estimates calculation
Difficulty	Easy -1.2 This is a straightforward calculation requiring only direct application of standard formulas for unbiased estimates of mean (sample mean) and variance (using n-1 divisor). It involves basic arithmetic with given data and no problem-solving or conceptual insight beyond formula recall.
Spec	5.05b Unbiased estimates: of population mean and variance

1 The heights, in metres, of a random sample of 10 mature trees of a certain variety are given below. \(\begin{array} { l l l l l l l l l l } 5.9 & 6.5 & 6.7 & 5.9 & 6.9 & 6.0 & 6.4 & 6.2 & 5.8 & 5.8 \end{array}\) Find unbiased estimates of the population mean and variance of the heights of all mature trees of this variety.

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Question 1:

Answer	Marks	Guidance
Answer	Marks	Guidance
\(\frac{62.1}{10} = 6.21\)	B1	OE
\([\Sigma x^2 = 387.05]\) \(\frac{10}{9}\left(\frac{\text{their } '387.05'}{10} - (\text{their } '6.21')^2\right)\) or \(\frac{1}{9}\left(\frac{\text{their } '387.05'}{10} - \frac{(\text{their } '6.21')^2}{10}\right)\)	M1	Can be implied. Accept alternative methods (e.g. working mean of 6). Biased 0.1409 M0.
\(= 0.157\) (3 sf) or \(\frac{1409}{9000}\)	A1
3

## Question 1:

| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{62.1}{10} = 6.21$ | **B1** | OE |
| $[\Sigma x^2 = 387.05]$ $\frac{10}{9}\left(\frac{\text{their } '387.05'}{10} - (\text{their } '6.21')^2\right)$ or $\frac{1}{9}\left(\frac{\text{their } '387.05'}{10} - \frac{(\text{their } '6.21')^2}{10}\right)$ | **M1** | Can be implied. Accept alternative methods (e.g. working mean of 6). Biased 0.1409 M0. |
| $= 0.157$ (3 sf) or $\frac{1409}{9000}$ | **A1** | |
| | **3** | |

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1 The heights, in metres, of a random sample of 10 mature trees of a certain variety are given below.\\
$\begin{array} { l l l l l l l l l l } 5.9 & 6.5 & 6.7 & 5.9 & 6.9 & 6.0 & 6.4 & 6.2 & 5.8 & 5.8 \end{array}$

Find unbiased estimates of the population mean and variance of the heights of all mature trees of this variety.\\

\hfill \mbox{\textit{CAIE S2 2022 Q1 [3]}}

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Q1 3 Q2 8 Q3 7 Q4 8 Q5 7 Q6 7 Q7 10