Non-normal population sample mean (CLT)

Questions asking about the sample mean from a single non-normal population (binomial or other), requiring the Central Limit Theorem to approximate the sample mean distribution as normal.

4 questions · Standard +0.5

5.05a Sample mean distribution: central limit theorem
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CAIE S2 2022 June Q1
3 marks Moderate -0.5
1 The number of characters in emails sent by a particular company is modelled by the distribution \(\mathrm { N } \left( 1250,480 ^ { 2 } \right)\). Find the probability that the mean number of characters in a random sample of 100 emails sent by the company is more than 1300 .
CAIE S2 2006 June Q3
6 marks Standard +0.8
3 Random samples of size 120 are taken from the distribution \(\mathrm { B } ( 15,0.4 )\).
  1. Describe fully the distribution of the sample mean.
  2. Find the probability that the mean of a random sample of size 120 is greater than 6.1.
CAIE S2 2007 June Q1
5 marks Challenging +1.2
1 The random variable \(X\) has the distribution \(\mathrm { B } ( 10,0.15 )\). Find the probability that the mean of a random sample of 50 observations of \(X\) is greater than 1.4.
OCR S3 2009 January Q6
13 marks Standard +0.3
6 A mathematics examination is taken by 29 boys and 26 girls. Experience has shown that the probability that any boy forgets to bring a calculator to the examination is 0.3 , and that any girl forgets is 0.2 . Whether or not any student forgets to bring a calculator is independent of all other students. The numbers of boys and girls who forget to bring a calculator are denoted by \(B\) and \(G\) respectively, and \(F = B + G\).
  1. Find \(\mathrm { E } ( F )\) and \(\operatorname { Var } ( F )\).
  2. Using suitable approximations to the distributions of \(B\) and \(G\), which should be justified, find the smallest number of spare calculators that should be available in order to be at least \(99 \%\) certain that all 55 students will have a calculator.