Variance estimation from probability

Questions that work backwards from a given probability about the sample mean to estimate the population variance or standard deviation.

2 questions · Standard +0.8

5.04a Linear combinations: E(aX+bY), Var(aX+bY)5.05a Sample mean distribution: central limit theorem
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OCR S2 2012 June Q2
6 marks Standard +0.3
2
  1. For the continuous random variable \(V\), it is known that \(\mathrm { E } ( V ) = 72.0\). The mean of a random sample of 40 observations of \(V\) is denoted by \(\bar { V }\). Given that \(\mathrm { P } ( \bar { V } < 71.2 ) = 0.35\), estimate the value of \(\operatorname { Var } ( V )\).
  2. Explain why you need to use the Central Limit Theorem in part (i), and why its use is justified.
OCR S2 2013 June Q3
9 marks Challenging +1.2
3 The mean of a sample of 80 independent observations of a continuous random variable \(Y\) is denoted by \(\bar { Y }\). It is given that \(\mathrm { P } ( \bar { Y } \leqslant 157.18 ) = 0.1\) and \(\mathrm { P } ( \bar { Y } \geqslant 164.76 ) = 0.7\).
  1. Calculate \(\mathrm { E } ( Y )\) and the standard deviation of \(Y\).
  2. State
    1. where in your calculations you have used the Central Limit Theorem,
    2. why it was necessary to use the Central Limit Theorem,
    3. why it was possible to use the Central Limit Theorem.