CAIE A-Level Mathematics 6.4.6 Unbiased estimates

Practise calculating unbiased estimates of population mean and variance from raw lists, totals, or summarised sample data.

Exam points

calculate x̄ as the unbiased estimate of the population mean from sample totals
rearrange the unbiased variance formula to find a missing sum

[Maximum number: 3]

In a survey, a random sample of 250 adults in Fromleigh were asked to fill in a questionnaire about their travel.

The survey included a question about the amount, x dollars, spent on travel per year. The results are summarised as follows.

n=250 \quad \Sigma x=50460 \quad \Sigma x^{2}=19854200

Find unbiased estimates of the population mean and variance of the amount spent per year on travel.

A councillor wanted to select a random sample of houses in Fromleigh. He planned to select the first house on each of the 143 streets in Fromleigh.

Estimate of mean:
$\frac{50460}{250}=201.84$
B1 Allow without units; allow 3 s.f. $\$202$ .
$\frac{250}{249}\left(\frac{19854200}{250}-\left(\frac{50460}{250}\right)^2\right)$
or
$\frac{1}{249}\left(19854200-\frac{50460^2}{250}\right)$
M1 Correct unbiased variance expression.
A1 Estimate of variance =38832.75 dollars $^2$ , or 38800 (3 s.f.). Calculation of biased variance gives M0 A0.

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6.4.6—Unbiased estimates question 1