2.04d Normal approximation to binomial

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SPS SPS FM Statistics 2021 September Q4
4 marks Moderate -0.8
In a town, 54% of the residents are female and 46% are male. A random sample of 200 residents is chosen from the town. Using a suitable approximation, find the probability that more than half the sample are female. [4 marks]
OCR H240/02 2018 December Q15
9 marks Moderate -0.3
A fair dice is thrown 1000 times and the number, \(X\), of throws on which the score is 6 is noted.
    1. State the distribution of \(X\). [1]
    2. Explain why a normal distribution would be an appropriate approximation to the distribution of \(X\). [1]
  2. Use a normal distribution to find two positive integer values, \(a\) and \(b\), such that \(\text{P}(a < X < b) \approx 0.4\). [5]
  3. For your two values of \(a\) and \(b\), use the distribution of part (a)(i) to find the value of \(\text{P}(a < X < b)\), correct to 3 significant figures. [2]
OCR H240/02 2017 Specimen Q12
5 marks Challenging +1.2
The table shows information for England and Wales, taken from the UK 2011 census.
Total populationNumber of children aged 5-17
56 075 9128 473 617
A random sample of 10 000 people in another country was chosen in 2011, and the number, \(m\), of children aged 5-17 was noted. It was found that there was evidence at the 2.5% level that the proportion of children aged 5-17 in the same year was higher than in the UK. Unfortunately, when the results were recorded the value of \(m\) was omitted. Use an appropriate normal distribution to find an estimate of the smallest possible value of \(m\). [5]
Pre-U Pre-U 9795/2 Specimen Q11
12 marks Standard +0.3
  1. State briefly the conditions under which the binomial distribution \(\text{B}(n, p)\) may be approximated by a normal distribution. [2]
  2. A multiple-choice test has \(50\) questions. Each question has four possible answers. A student passes the test if answering \(36\%\) or more of the questions correctly. Using a suitable distributional approximation, estimate the probability that a student who selects answers to all the questions randomly will pass the test. [5]
  3. A test similar to that in part (ii) has \(N\) questions instead of \(50\) questions. Estimate the least value of \(N\) so that the probability that a student gets \(36\%\) or more of the questions correct, by selecting answers to all questions randomly, will be less than \(0.01\). (A continuity correction is not required in this part of the question.) [5]