The random variable $$\(T\)$$ denotes the time, in seconds, for $$\(100 \mathrm{~m}\)$$ races run by Tania. $$\(T\)$$ is normally distributed with mean $$\(\mu\)$$ and variance $$\(\sigma^{2}\)$$. A random sample of 40 races run by Tania gave the following results. $$\( n=40 \quad \Sigma t=560 \quad \Sigma t^{2}=7850 \)$$ Calculate unbiased estimates of $$\(\mu\)$$ and $$\(\sigma^{2}\)$$. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................
Exam No：9709_w21_qp_63 Year：2021 Question No：6(a)
Answer:
est \(\mu=14\) accept \(\frac{560}{40}\)
est \(\sigma^{2}=\frac{40}{39}\left(\frac{7850}{40}-14^{2}\right)\) or \(\frac{1}{39}\left(7850-\frac{560^{2}}{40}\right)\)
\(0.25641\) or \(0.256(3 \mathrm{sf})\)
est \(\sigma^{2}=\frac{40}{39}\left(\frac{7850}{40}-14^{2}\right)\) or \(\frac{1}{39}\left(7850-\frac{560^{2}}{40}\right)\)
\(0.25641\) or \(0.256(3 \mathrm{sf})\)
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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