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Continuous random variables and the normal distribution — Question 334

Original QCE Vault practice · 6 marks

Q334 · Practice questionTechnology-activeComplex unfamiliar6 marks

QUESTION 334 (6 marks)

A cutting machine produces lengths XX mm with normal distribution of adjustable mean μ\mu and fixed standard deviation 33 mm. A piece is accepted if 100≤X≤110100\leq X\leq110. Let P(μ)P(\mu) be its acceptance probability. You are given
P′(μ)=132π[e−(100−μ)2/18−e−(110−μ)2/18].P^{\prime}(\mu)=\frac1{3\sqrt{2\pi}}\left[e^{-(100-\mu)^2/18}-e^{-(110-\mu)^2/18}\right].
a)
Determine the mean that maximises the acceptance probability, with justification.
[3 marks]
b)
Determine the maximum acceptance probability to four decimal places.
[2 marks]
c)
For 500500 independently produced pieces at this setting, determine the expected number rejected, to the nearest whole piece.
[1 mark]
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