OCR S2 2011 January — Question 1 4 marks

Exam BoardOCR
ModuleS2 (Statistics 2)
Year2011
SessionJanuary
Marks4
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Mark schemeDownload PDF ↗
TopicMeasures of Location and Spread
TypeStandard unbiased estimates calculation
DifficultyEasy -1.2 This is a straightforward application of standard formulas for unbiased estimates requiring only direct substitution of given values into memorized formulas (mean = Σx/n, variance = [Σx² - (Σx)²/n]/(n-1)). It involves minimal calculation and no problem-solving or conceptual understanding beyond formula recall.
Spec5.05b Unbiased estimates: of population mean and variance
1 A random sample of nine observations of a random variable is obtained. The results are summarised as $$\Sigma x = 468 , \quad \Sigma x ^ { 2 } = 24820 .$$ Calculate unbiased estimates of the population mean and variance.
AnswerMarks Guidance
\(\mu = \bar{x} = \frac{468}{9} = 52\)B1 52 stated
Correct method for biased estimatorM1 Multiply by 9/8
\(\frac{24820}{9} - 52^2 = [53.78]\)M1 [if single formula, allow M0 M1 if wrong but divisor 8 seen anywhere]
\(\sigma^2 = \frac{9}{8} \times 53.78 = 60.5\)A1 Answer 60.5 or exact equivalent 
$\mu = \bar{x} = \frac{468}{9} = 52$ | B1 | 52 stated
Correct method for biased estimator | M1 | Multiply by 9/8
$\frac{24820}{9} - 52^2 = [53.78]$ | M1 | [if single formula, allow M0 M1 if wrong but divisor 8 seen anywhere]
$\sigma^2 = \frac{9}{8} \times 53.78 = 60.5$ | A1 | Answer 60.5 or exact equivalent
1 A random sample of nine observations of a random variable is obtained. The results are summarised as

$$\Sigma x = 468 , \quad \Sigma x ^ { 2 } = 24820 .$$

Calculate unbiased estimates of the population mean and variance.

\hfill \mbox{\textit{OCR S2 2011 Q1 [4]}}
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