SECTION 5.1 Introduction to Normal Distributions and the Standard Normal Distribution 237 The Standard Normal Distribution There are infinitely many normal distributions, each with its own mean and standard deviation. The normal distribution with a mean of 0 and a standard deviation of 1 is called the standard normal distribution. The horizontal scale of the graph of the standard normal distribution corresponds to z@scores. In Section 2.5, you learned that a z@score is a measure of position that indicates the number of standard deviations a value lies from the mean. Recall that you can transform an x@value to a z@score using the formula z = Value - Mean Standard deviation = x - m s . Round to the nearest hundredth. The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. The total area under its normal curve is 1. 0 1 2 3 z −3 −2 −1 Area = 1 Standard Normal Distribution DEFINITION When each data value of a normally distributed random variable x is transformed into a z@score, the result will be the standard normal distribution. After this transformation takes place, the area that falls in the interval under the nonstandard normal curve is the same as that under the standard normal curve within the corresponding z@boundaries. In Section 2.4, you learned to use the Empirical Rule to approximate areas under a normal curve when the values of the random variable x corresponded to -3, -2, -1, 0, 1, 2, or 3 standard deviations from the mean. Now, you will learn to calculate areas corresponding to other x@values. After you use the formula above to transform an x@value to a z@score, you can use the Standard Normal Table (Table 4 in Appendix B). The table lists the cumulative area under the standard normal curve to the left of z for z@scores from -3.49 to 3.49. As you examine the table, notice the following. 1. The cumulative area is close to 0 for z@scores close to z = -3.49. 2. The cumulative area increases as the z@scores increase. 3. The cumulative area for z = 0 is 0.5000. 4. The cumulative area is close to 1 for z@scores close to z = 3.49. Properties of the Standard Normal Distribution In addition to using the table, you can use technology to find the cumulative area that corresponds to a z-score. For instance, the next example shows how to use the Standard Normal Table and a TI-84 Plus to find the cumulative area that corresponds to a z@score. Study Tip Because every normal distribution can be transformed to the standard normal distribution, you can use z@scores and the standard normal curve to find areas (and therefore probabilities) under any normal curve. Study Tip It is important that you know the difference between x and z. The random variable x is sometimes called a raw score and represents values in a nonstandard normal distribution, whereas z represents values in the standard normal distribution.
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