Читать книгу Applied Biostatistics for the Health Sciences - Richard J. Rossi - Страница 75

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 the X values to Z-values leads to the corresponding probability, P(−1.29≤Z≤0.86), for the standard normal shown in Figure 2.31.

Figure 2.31 The Z region corresponding to 700≤X≤1000.

Example 2.39

The distribution of IQ scores is approximately normal with µ = 100 and σ = 15. Using this normal distribution to model the distribution of IQ scores,

1 an IQ score of 112 corresponds to a Z-value of

2 the probability of having an IQ score of 112 or less is

3 the probability of having an IQ score between 90 and 120 is

4 the probability of having an IQ score of 150 or higher is

Example 2.40

In the article “Distribution of LDL particle size in a population-based sample of children and adolescents and relationship with other cardiovascular risk factors” published in Clinical Chemistry (Stan et al., 2005), the authors reported the results of a study on the peak particle size of low-density lipoprotein (LDL) in children and adolescents. It is known that smaller more dense particles (≤255 Å) of LDL are associated with cardiovascular disease.

The distribution of peak particle was reported to be approximately normal with mean particle size µ = 262 Å and standard deviation σ = 4 Å. Based on this study, the probability that a child or adolescent will have a peak particle size of less than 255 Å is

Thus, there is only a 4% chance that a child or adolescent will have peak particle size less than 255 Å.