Unbiased estimator from raw data

Questions that provide raw data values and require calculating unbiased estimates of population mean and/or variance by first computing the necessary summary statistics.

2 questions · Moderate -0.7

5.05b Unbiased estimates: of population mean and variance
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Edexcel S3 2013 June Q7
13 marks Moderate -0.5
  1. Lambs are born in a shed on Mill Farm. The birth weights, \(x \mathrm {~kg}\), of a random sample of 8 newborn lambs are given below.
$$\begin{array} { l l l l l l l l } 4.12 & 5.12 & 4.84 & 4.65 & 3.55 & 3.65 & 3.96 & 3.40 \end{array}$$
  1. Calculate unbiased estimates of the mean and variance of the birth weight of lambs born on Mill Farm. A further random sample of 32 lambs is chosen and the unbiased estimates of the mean and variance of the birth weight of lambs from this sample are 4.55 and 0.25 respectively.
  2. Treating the combined sample of 40 lambs as a single sample, estimate the standard error of the mean. The owner of Mill Farm researches the breed of lamb and discovers that the population of birth weights is normally distributed with standard deviation 0.67 kg .
  3. Calculate a \(95 \%\) confidence interval for the mean birth weight of this breed of lamb using your combined sample mean.
Edexcel S3 2015 June Q3
11 marks Moderate -0.8
3. A nursery has 16 staff and 40 children on its records. In preparation for an outing the manager needs an estimate of the mean weight of the people on its records and decides to take a stratified sample of size 14 .
  1. Describe how this stratified sample should be taken. The weights, \(x \mathrm {~kg}\), of each of the 14 people selected are summarised as $$\sum x = 437 \text { and } \sum x ^ { 2 } = 26983$$
  2. Find unbiased estimates of the mean and the variance of the weights of all the people on the nursery's records.
  3. Estimate the standard error of the mean. The estimates of the standard error of the mean for the staff and for the children are 5.11 and 1.10 respectively.
  4. Comment on these values with reference to your answer to part (c) and give a reason for any differences.