Exact binomial then normal approximation (same context, different n)

Questions where both parts use the same probability/context but part (i)/(a) uses exact binomial for small n and part (ii)/(b) uses normal approximation for large n within a single scenario.

28 questions · Moderate -0.0

2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities2.04d Normal approximation to binomial
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Edexcel S2 Q6
18 marks Standard +0.3
When a park is redeveloped, it is claimed that 70\% of the local population approve of the new design. Assuming this to be true, find the probability that, in a group of 10 residents selected at random,
  1. 6 or more approve, [3 marks]
  2. exactly 7 approve. [3 marks]
A conservation group, however, carries out a survey of 20 people, and finds that only 9 approve.
  1. Use this information to carry out a hypothesis test on the original claim, working at the 5\% significance level. State your conclusion clearly. [5 marks]
If the conservationists are right, and only 45\% approve of the new park,
  1. use a suitable approximation to the binomial distribution to estimate the probability that in a larger survey, of 500 people, less than half will approve. [7 marks]
Edexcel S2 Q7
18 marks Standard +0.3
In an orchard, all the trees are either apple or pear trees. There are four times as many apple trees as pear trees. Find the probability that, in a random sample of 10 trees, there are
  1. equal numbers of apple and pear trees, [3 marks]
  2. more than 7 apple trees. [3 marks]
In a sample of 60 trees in the orchard,
  1. find the expected number of pear trees. [1 mark]
  2. Calculate the standard deviation of the number of pear trees and compare this result with the standard deviation of the number of apple trees. [2 marks]
  3. Find the probability that exactly 35 in the sample of 60 trees are pear trees. [4 marks]
  4. Find an approximate value for the probability that more than 15 of the 60 trees are pear trees. [5 marks]
Edexcel S2 Q6
14 marks Standard +0.3
A small opinion poll shows that the Trendies have a \(10\%\) lead over the Oldies. The poll is based on a survey of 20 voters, in which the Trendies got 11 and the Oldies 9. The Oldies spokesman says that the result is consistent with a \(10\%\) lead for the Oldies, whilst the Trendies spokesperson says that this is impossible.
  1. At the \(5\%\) significance level, test which is right, stating your null hypothesis carefully. [6 marks]
  2. If it is indeed true that the Trendies are supported by \(55\%\) of the population, use a suitable approximation to find the probability that in a random sample of 200 voters they would obtain less than half of the votes. [8 marks]