Calculate probabilities using sample mean distribution

Question requires calculating specific probabilities or finding parameters by working backwards from given probabilities involving the sample mean distribution.

6 questions · Moderate -0.2

5.05a Sample mean distribution: central limit theorem
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CAIE S2 2016 June Q1
4 marks Moderate -0.5
1 The length of time, in minutes, taken by people to complete a task has mean 53.0 and standard deviation 6.2. Find the probability that the mean time taken to complete the task by a random sample of 50 people is more than 51 minutes.
CAIE S2 2014 November Q5
10 marks Standard +0.3
5 The number of hours that Mrs Hughes spends on her business in a week is normally distributed with mean \(\mu\) and standard deviation 4.8. In the past the value of \(\mu\) has been 49.5.
  1. Assuming that \(\mu\) is still equal to 49.5 , find the probability that in a random sample of 40 weeks the mean time spent on her business in a week is more than 50.3 hours. Following a change in her arrangements, Mrs Hughes wishes to test whether \(\mu\) has decreased. She chooses a random sample of 40 weeks and notes that the total number of hours she spent on her business during these weeks is 1920.
  2. (a) Explain why a one-tail test is appropriate.
    (b) Carry out the test at the 6\% significance level.
    (c) Explain whether it was necessary to use the Central Limit theorem in part (ii) (b).
Edexcel S3 2018 Specimen Q7
5 marks Standard +0.3
  1. A fair six-sided die is labelled with the numbers \(1,2,3,4,5\) and 6
    (b) Find an approximation for the probability that the mean of the 40 scores is less than 3 \includegraphics[max width=\textwidth, alt={}, center]{0434a6c1-686a-449d-ba16-dbb8e60288e8-24_204_714_237_251}
CAIE S2 2023 November Q1
3 marks Moderate -0.5
A random variable \(X\) has the distribution N(410, 400). Find the probability that the mean of a random sample of 36 values of \(X\) is less than 405. [3]
Edexcel S3 2005 June Q2
Moderate -0.3
A sample of size 5 is taken from a population that is normally distributed with mean 10 and standard deviation 3. Find the probability that the sample mean lies between 7 and 10. (Total 6 marks)
Edexcel S3 2009 June Q4
5 marks Moderate -0.5
A sample of size 8 is to be taken from a population that is normally distributed with mean 55 and standard deviation 3. Find the probability that the sample mean will be greater than 57. [5]