6-1 The Standard Normal Distribution 249 is the same: There is a correspondence between area and probability. In Figure 6-4 we show that for a standard normal distribution, the area under the density curve is equal to 1. In Figure 6-4, we use “z Score” as a label for the horizontal axis, and this is common for the standard normal distribution, defined as follows. 1 2 3 0 z Score Area 5 1 21 22 23 FIGURE 6-4 Standard Normal Distribution DEFINITION The standard normal distribution is a probability distribution with these properties: • The distribution is a normal distribution, so it is bell-shaped as in Figure 6-4. • The population parameter of the mean has the specific value of m = 0. • The population parameter of the standard deviation has the specific value of s = 1. • The total area under its density curve is equal to 1 (as in Figure 6-4). Finding Probabilities When Given z Scores It is not easy to manually find areas in Figure 6-4, but we can find areas (or probabilities) for many different regions in Figure 6-4 by using technology, or we can also use Table A-2 (in Appendix A). Key features of the different methods are summarized in Table 6-1, which follows. (StatCrunch provides options for a cumulative left region, a cumulative right region, or the region between two boundaries.) Because calculators and software generally give more accurate results than Table A-2, we strongly recommend using technology. (When there are discrepancies, answers in Appendix D will generally include results based on technology as well as answers based on Table A-2.) If using Table A-2, it is essential to understand these points: 1. Table A-2 is designed only for the standard normal distribution, which is a normal distribution with a mean of 0 and a standard deviation of 1. 2. Table A-2 is on two pages, with the left page for negative z scores and the right page for positive z scores. 3. Each value in the body of the table is a cumulative area from the left up to a vertical boundary above a specific z score. 4. When working with a graph, avoid confusion between z scores and areas. z score: Distance along the horizontal scale of the standard normal distribution (corresponding to the number of standard deviations above or below the mean); refer to the leftmost column and top row of Table A-2. Area: Region under the curve; refer to the values in the body of Table A-2. 5. The part of the z score denoting hundredths is found across the top row of Table A-2.
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