style="font-size:15px;"> 3 P(−1.21≤Z≤2.28)=Φ(1.81)−Φ(−1.21)=0.9887−0.1131=0.87564 P(0.67≤Z≤6.28)=Φ(6.28)−Φ(0.67)=1−0.7486=0.2514
5 P(−4.21≤Z≤0.84)=Φ(0.84)−0=0.7995
The pth percentile of the standard normal can be found by looking up the cumulative probability p100 inside of the standard normal tables given in Appendix A and then working backward to find the value of Z. Unfortunately, in some cases the value of p100 will not be listed inside the table, and in this case, the value closest to p100 inside of the table should be used for finding the approximate value of the pth percentile.
Example 2.37
Determine the 90th percentile of a standard normal distribution.
Solutions To find the 90th percentile, the first step is to find 90100=0.90, or the value closest to 0.90, in the cumulative standard normal table. The row of the standard normal table containing the value 0.90 is given in Table 2.11. Since 0.8997 is the value closest to 0.90 that occurs in the row labeled 1.2 and the column labeled 0.08, the 90th percentile of a standard normal distribution is roughly z=1.2+0.08=1.28.
Table 2.11 An Excerpted Section of the Cumulative Normal Table
| Φ(z)=P(Z≤z) | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| z | 0.00 | 0.01 | 0.02 | 0.03 | 0.04 | 0.05 | 0.06 | 0.07 | 0.08 | 0.09 |
| ⋮ | ||||||||||
| 1.2 | 0.8849 | 0.8869 | 0.8888 | 0.8907 | 0.8925 | 0.8944 | 0.8962 | 0.8980 | 0.8997 | 0.9015 |
| ⋮ | ||||||||||
The most commonly used percentiles of the standard normal are given in Table 2.12, and a more complete list of the percentiles of the standard normal distribution is given in Table A3. Note that for p < 50 the percentiles of a standard normal are negative, for p = 50 the 50th percentile is the median, and for p > 50 the percentiles of a standard normal are positive.
Table 2.12 Selected Percentiles of a Standard Normal Distribution
| p | 1 | 5 | 10 | 20 | 25 | 50 |
| pth percentile | −2.33 | −1.645 | −1.28 | −0.84 | −0.67 | 0.00 |
| p | 75 | 80 | 90 | 95 | 99 | 99.9 |
| pth percentile | 0.67 | 0.84 | 1.28 | 1.645 | 2.33 | 3.09 |
When X is a normal distribution with μ≠0 or σ≠1, the distribution of X is called a non-standard normal distribution. The standard normal distribution is the reference distribution for all normal distributions because all of the probabilities and percentiles for a non-standard normal can be determined from the standard normal distribution. In particular, the following relationships between a non-standard normal with mean µ and standard deviation σ and the standard normal can be used to convert a non-standard normal value to a Z-value and vice versa.
THE RELATIONSHIPS BETWEEN A STANDARD NORMAL AND A NON-STANDARD NORMAL
1 If X is a non-standard normal with mean µ and standard deviation σ, then Z=(X−μ)/σ.
2 If Z is a standard normal, then X=σ⋅Z+μ is a non-standard normal with mean µ and standard deviation σ.
Note that the value of a non-standard normal X can be converted into a Z-value and vice versa. The first equation shows that centering and scaling the values of X converts them to Z-values, and the second equation shows how to convert a Z-value into an X-value. The relationship between a standard normal and a non-standard normal is shown in Figure 2.29.
Figure 2.29 The correspondence between the values of a standard normal and a non-standard normal.
To determine the cumulative probability for the value of a non-standard normal, say x, convert the value of x to its corresponding z value using z=z−μ/σ; then determine the cumulative probability for this z value. That is,
To compute probabilities other than a cumulative probability for a non-standard normal, note that the probability of being in any region can also be computed from the cumulative probabilities associated with the standard normal. The rules for computing the probabilities associated with a non-standard normal are given below.
NON-STANDARD NORMAL PROBABILITIES
If X is a non-standard normal variable with mean µ and standard deviation σ, then
1 P(X≥x)=1−P(X≤x) =1−P(Z≤x−μσ)
2 P(a≤X≤b)=P(X≤b)−P(X≤b) =P(Z≤b−μσ)−P(Z≤a−μσ)
Note that each of the probabilities associated with a non-standard normal distribution is based on the process of converting an x value to a z value using the formula Z=(x−μ)/σ. The reason why the standard normal can be used for computing every probability concerning a non-standard normal is that there is a one-to-one correspondence between the Z and X values (see Figure 2.29).
Example 2.38
Suppose X has a non-standard normal distribution with mean µ = 880 and standard deviation σ = 140. The probability that X is between 700 and 1000 is represented by the area shown in Figure 2.30.
Figure 2.30 P(700≤X≤1000).
Converting
