Shading a normal lower tail, the symmetric upper boundary, and the probability between the two boundaries

Shading a normal lower tail, the symmetric upper boundary, and the probability between the two boundaries

6 marks

The length of a leaf picked from a hedge, \(L\) cm, follows a normal distribution, \(L\sim\mathrm{N}(64,\ 3^2)\).

  1. Shade, on the sketch of the distribution of \(L\), the area equal to \(\mathrm{P}(L<58)\).[2]
  2. The constant \(d\) satisfies \(\mathrm{P}(L>d)=\mathrm{P}(L<58)\). Find \(d\).[2]
  3. Taking \(\mathrm{P}(L<58)=0.023\) (to \(2\) significant figures), find \(\mathrm{P}(58<L<d)\).[2]

Solution:

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Answer:(a) region to the left of \(58\) shaded (b) \(d=70\) (c) \(0.954\)

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