May 2006 Question 2  The head circumference of 3-year-old boys is known to be Normally distributed with mean 49.7cm and standard deviation 1.6cm.  Find.

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May 2006 Question 2  The head circumference of 3-year-old boys is known to be Normally distributed with mean 49.7cm and standard deviation 1.6cm.  Find the probability of having a head (A) over 51.5cm (B) between 48cm and 51.5cm  4 3-year-old boys are selected. Find the probability that exactly one of them has a head size between 48cm and 51.5cm

May 2006 Question 2 Solution X~N(49.7,1.6 2 ) (i)(A) P(X>51.5)=P(Z>51.5)=P(Z> ) P(Z>1.125)=1-P(Z<1.125) =1-Φ(1.125) = =0.1304

(i)(B) P(48≤X≤51.5)=P(X≤51.5)-P(X≤48) =P(Z≤1.125)-P(Z≤ ) = Φ(1.125)- Φ(-1.063) = (1- Φ(1.063)) = ( ) = =0.7257

(ii) P(48≤X≤51.5)= n=4 Therefore X~Bi(4,0.7257) P(X=1)= =0.0599

Key Points  Look carefully at the question.  See if you can use anything you’ve calculated before.  Just because the binomial distribution is not on the S2 syllabus directly, doesn’t mean they won’t ask you to use it.