Porisma

Worked solution

Find cutoffs from left-tail and right-tail areas

If X∼N(100,62)X\sim N(100,6^2), find

(i) aa such that P(X<a)=0.8907P(X<a)=0.8907 and

(ii) bb such that P(X>b)=0.770P(X>b)=0.770.

CJC · Lecture · Example 2

(i) Left area 0.8907

The standard deviation is 6, not 36. The area left of a exceeds one half, so expect a>100a>100.

  1. Press 2 nd VARS, select 3:invNorm, then ENTER.
  2. In the source’s guided interface, enter area 0.8907, μ100, σ6, choose LEFT, then highlight Paste and press ENTER.
  3. Press ENTER to calculate invNorm(0.8907,100,6,LEFT).

The source screen returns 107.3815561, so a=107\boxed{a=107} (3 s.f.). The source identifies this tail-selector interface as calculator OS 5.2.2 or higher and requires STAT WIZARDS ON. A calculator using only left cumulative probabilities can use the same area 0.8907.

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(ii) Right area 0.770

The right area exceeds one half, so the cutoff must be below the mean: b<100b<100.

Select 3:invNorm from 2 nd VARS, enter area 0.770, μ100, σ6, choose RIGHT, then Paste, ENTER and ENTER again. The source command invNorm(0.770,100,6,RIGHT) displays 95.56691888, giving b=95.6\boxed{b=95.6} (3 s.f.).

With a left-area-only interface, first complement: P(X<b)=1−0.770=0.230P(X<b)=1-0.770=0.230, then use invNorm(0.230,100,6). Both routes describe the same cutoff.

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Compare your sketch

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Two normal curves with mean 100 and standard deviation 6. Left area 0.8907 ends at a≈107.3816, to the right of the mean. Right area 0.770 starts at b≈95.5669, to the left of the mean. Select LEFT and RIGHT respectively.
Two normal curves with mean 100 and standard deviation 6. Left area 0.8907 ends at a≈107.3816, to the right of the mean. Right area 0.770 starts at b≈95.5669, to the left of the mean. Select LEFT and RIGHT respectively.
Two normal curves with mean 100 and standard deviation 6. Left area 0.8907 ends at a≈107.3816, to the right of the mean. Right area 0.770 starts at b≈95.5669, to the left of the mean. Select LEFT and RIGHT respectively.

Common mistakes

Enter σ=6, not the variance 36. LEFT and RIGHT describe where the given area lies; a right-tail area 0.770 produces a cutoff below the mean. Sketch the area before selecting the tail.

Read the hint