A sieve for blueberries

Topic: Functions of random variables and limit theorems · more questions on this topic
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Blueberry diameters are \operatorname{N}(1, 0.1^2), in centimetres. A sieve of diameter d classifies a berry as large when its diameter exceeds d.

  1. Choose d so that 30% of berries are large. Use the standard normal quantile z_{0.7}\approx 0.524, that is the point with \Phi(z_{0.7})=0.7.

  2. With that choice of d, approximate the probability that fewer than 280 of 1000 independent berries are large.

  1. A berry is large with probability \mathbb{P}(D>d)=1-\Phi((d-1)/0.1). Setting this to 0.3 requires (d-1)/0.1=z_{0.7}=0.524, so d=1.0524 cm.

  2. The number of large berries is N\sim\operatorname{Bin}(1000,0.3), with mean 300 and variance 210. By the central limit theorem with a continuity correction,

\mathbb{P}(N<280)=\mathbb{P}(N\le279)\approx\Phi\Bigl(\frac{279.5-300}{\sqrt{210}}\Bigr)=\Phi(-1.41)=0.079.

The exact binomial value is 0.078.