Compare approximation methods

5 The random variable \(W\) has the distribution \(\mathbf { B } ( 30 , p )\).

1. Use the exact binomial distribution to calculate \(\mathbf { P } ( W = 10 )\) when \(p = 0.4\).
2. Find the range of values of \(p\) for which you would expect that a normal distribution could be used as an approximation to the distribution of \(W\).
3. Use a normal approximation to calculate \(\mathrm { P } ( W = 10 )\) when \(p = 0.4\).

6 At a tourist car park, a survey is made of the regions from which cars come.

1. It is given that \(40 \%\) of cars come from the London region. Use a suitable approximation to find the probability that, in a random sample of 32 cars, more than 17 come from the London region. Justify your approximation.
2. It is given that \(1 \%\) of cars come from France. Use a suitable approximation to find the probability that, in a random sample of 90 cars, exactly 3 come from France.

6. The probability that a sunflower plant grows over 1.5 metres high is 0.25 . A random sample of 40 sunflower plants is taken and each sunflower plant is measured and its height recorded.

1. Find the probability that the number of sunflower plants over 1.5 m high is between 8 and 13 (inclusive) using
   1. a Poisson approximation,
   2. a Normal approximation.
2. Write down which of the approximations used in part (a) is the most accurate estimate of the probability. You must give a reason for your answer.