The heights, $$\(h\)$$ centimetres, of a random sample of 100 fully grown animals of a certain species were measured. The results are summarised below. $$\( n=100 \quad \Sigma h=7570 \quad \Sigma h^{2}=588050 \)$$ Find unbiased estimates of the population mean and variance. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................
Exam No：9709_s21_qp_62 Year：2021 Question No：6(a)
Answer:
\(\operatorname{est}(\mu)=\frac{7570}{100}(=75.7)\)
\(\operatorname{est}\left(\sigma^{2}\right)=\frac{100}{99}\left(\frac{\sum h^{2}}{100}-{ }^{\prime} 75.7^{\prime^{2}}\right)\) or \(\frac{1}{99}\left(588050-\frac{7570^{2}}{100}\right)\)
\(=\frac{100}{99}\left(\frac{588050}{100}-{ }^{\prime} 75.7^{\prime 2}\right)[=151.525]\)
\(=152(3 \mathrm{sf})\)
\(\operatorname{est}\left(\sigma^{2}\right)=\frac{100}{99}\left(\frac{\sum h^{2}}{100}-{ }^{\prime} 75.7^{\prime^{2}}\right)\) or \(\frac{1}{99}\left(588050-\frac{7570^{2}}{100}\right)\)
\(=\frac{100}{99}\left(\frac{588050}{100}-{ }^{\prime} 75.7^{\prime 2}\right)[=151.525]\)
\(=152(3 \mathrm{sf})\)
Knowledge points:
6.4.6 calculate unbiased estimates of the population mean and variance from a sample, using either raw or summarised data (Only a simple understanding of the term ‘unbiased’ is required, e.g. that although individual estimates will vary the process gives an accurate result ‘on average’.)
Solution:
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