Areas Under Any Normal Curve:
Converting Normal Distributions to Standard Normal:
In many applied situations, the original normal curve is not the standard normal
curve. Generally, there will not be a table of areas available for the original normal
curve. This does not mean that we cannot find the probability that a measurement x will
fall in an interval from a to b. What we must do is convert original measurements x, a,
and b to z values.
Given that x has a normal distribution with μ = 4 and σ = 1, find P(1.1 < x < 2.4)
for an x value selected at random.
Solution: 1) Convert the intervals 1.1 and 2.4 into z values. Comment on
whether you feel the values will be positive or negative. (Hint: If an interval is above the
mean it will be positive!)
2) Sketch the areas corresponding to the desired probabilities under
the x curve and under the z curve.
3) See where the areas lie, then add or subtract to find the answer.
Complete text p. 381 examples (#’s 1 – 10 and word problems).
Finding z or x, Given a Probability:
Often, we need to find z or x values that correspond to a given area under the
normal curve. This situation arises when we want to specify a guarantee period so that a
given percentage of total products produced by a company lasts as long as the duration of
the guarantee period.
*** When we use Table 6 in Appendix II to find a z value corresponding to a given area,
we usually use the nearest area value rather than a mean of the two values. When the z
value corresponding to an area is larger than, say, 2, the standard convention is to use the
z values corresponding to the next larger area.
Find the z value so that 15% of the area under the standard normal curve lies to
the left of z.
Solution: 1) Draw a standard normal curve and shade the region so that 15%
of the area lies to the left of z.
2) Calculate the area that lies between this value and 0 by
subtracting 50% and 15%.
3) Use Table 6, Appendix II to find the z value so that 35% of the
area under the standard normal curve is between 0 and z. Look for the value closest to
4) Use the symmetry of the standard normal curve to find the z
vale so that 15% of the area lies to the left of z. That means should z be positive or
Complete text p. 381 (#’s 11 – 20 and word problems).