A6 APPENDIX A Alternative Presentation of the Standard Normal Distribution Finding Area Under the Standard Normal Curve Find the area under the standard normal curve between z = -1.5 and z = 1.25. SOLUTION The area under the standard normal curve between z = -1.5 and z = 1.25 is shown. 1.25 0 z −1.5 Area = 0.4332 + 0.3944 Area = 0.4332 Area = 0.3944 From the Standard Normal Table (0-to-z), the area corresponding to z = -1.5 is 0.4332 and the area corresponding to z = 1.25 is 0.3944. To find the area between these two z-scores, add the resulting areas. Area = 0.4332 + 0.3944 = 0.8276 Note that when you use technology, your answers may differ slightly from those found using the Standard Normal Table. For instance, when finding the area between z = -1.5 and z = 1.25 on a TI-84 Plus, you get the result shown at the left. Interpretation So, about 82.76% of the area under the curve falls between z = -1.5 and z = 1.25. TRY IT YOURSELF 4 Find the area under the standard normal curve between z = -2.165 and z = -1.35. Answer: Page A43 Because the normal distribution is a continuous probability distribution, the area under the standard normal curve to the left of a z-score gives the probability that z is less than that z-score. For instance, in Example 2, the area to the left of z = -0.99 is 0.1611. So, P1z 6 -0.992 = 0.1611, which is read as “the probability that z is less than -0.99 is 0.1611.” The table shows the probabilities for Examples 3 and 4. Area Probability Example 3 To the right of z = 1.06: 0.1446 P1z 7 1.062 = 0.1446 Example 4 Between z = -1.5 and z = 1.25: 0.8276 P1-1.5 6 z 6 1.252 = 0.8276 Recall from Section 2.4 that values lying more than two standard deviations from the mean are considered unusual. Values lying more than three standard deviations from the mean are considered very unusual. So, a z-score greater than 2 or less than -2 is unusual. A z-score greater than 3 or less than -3 is very unusual. You are now ready to continue Section 5.1 on page 242 with the section exercises. EXAMPLE 4 TI-84 PLUS normalcdf(-1.5,1.25) 0.8275429323
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