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.
2.4.3 Z Scores
The result of converting a non-standard normal value, a raw value, to a Z-value is a Z score. A Z score is a measure of the relative position a value has within its distribution. In particular, a Z score simply measures how many standard deviations a point is above or below the mean. When a Z score is negative the raw value lies below the mean of its distribution, and when a Z score is positive the raw value lies above the mean. Z scores are unitless measures of relative standing and provide a meaningful measure of relative standing only for mound-shaped distributions. Furthermore, Z scores can be used to compare the relative standing of individuals in two mound-shaped distributions.
Example 2.41
The weights of men and women both follow mound-shaped distributions with different means and standard deviations. In fact, the weight of a male adult in the United States is approximately normal with mean µ = 180 and standard deviation σ = 30, and the weight of a female adult in the United States is approximately normal with mean µ = 145 and standard deviation σ = 15. Given a male weighing 215 lb and a female weighing 170 lb, which individual weighs more relative to their respective population?
The answer to this question can be found by computing the Z scores associated with each of these weights to measure their relative standing. In this case,
and
Since the female’s weight is 1.67 standard deviations from the mean weight of a female and the male’s weight is 1.17 standard deviations from the mean weight of a male, relative to their respective populations a female weighing 170 lb is heavier than a male weighing 215 lb.
Glossary
Absolute RiskThe absolute risk of a condition or disease is the probability that an individual develops the condition or disease.Binomial Probability ModelThe binomial probability model is a probability model for a discrete random variable that counts the number of successes in n independent trials of a chance experiment having only two possible outcomes.Chance ExperimentA task where the outcome cannot be predetermined is called a random experiment or a chance experiment.Conditional ProbabilityThe conditional probability of the event A given that the event B has occurred is denoted by P(A|B) and is defined as
Continuous VariableA quantitative variable is a continuous variable when the variable can take on any value in one or more intervals.Discrete VariableA quantitative variable is a discrete variable when there are either a finite or a countable number of possible values for the variable.DistributionThe distribution of a variable explicitly describes how the values of the variable are distributed in terms of percentages.EventAn event is a subcollection of the outcomes in the sample space is associated with a chance experiment.Explanatory VariableAn explanatory variable is a variable that is believed to cause changes in the response variable.Independent EventsTwo events A and B are independent when P(A|B)=P(A) or P(B|A)=P(B).Interquartile RangeThe Interquartile range of a population is the distance between the 25th and 75th percentiles and will be denoted by IQR.MeanThe mean of a variable X measured on a population consisting of N units is
MedianThe median of a population is the 50th percentile of the possible values of the variable X and will be denoted by μ~.ModeThe mode of a population is the most frequent value of the variable X in the population and will be denoted by M.Multivariate VariableA collection of variables that will be measured on each unit is called a multivariate variable.Negative Predictive ValueIn a diagnostic test, the negative predictive value (NPV) is the probability of a correct negative test result, P(−|not D).Nominal VariableA qualitative variable is called a nominal variable when the values of the variable have no intrinsic ordering.Non-standard NormalA non-standard normal is any normal distribution that does not have a standard normal distribution (i.e., either μ≠ or σ≠1).OddsThe odds of an event A is odds(A)=P(A)1−P(A).Odds RatioThe odds ratio for a disease is the ratio of the odds of the disease when the risk factor is present to the odds when the risk factor is absent.
Ordinal VariableA qualitative variable is called an ordinal
