SECTION 5.1 Introduction to Normal Distributions and the Standard Normal Distribution 241 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 z −1.5 0 Area = 0.0668 Area = 0.8944 − 0.0668 From the Standard Normal Table, the area to the left of z = 1.25 is 0.8944 and the area to the left of z = -1.5 is 0.0668. So, the area between z = -1.5 and z = 1.25 is Area = 0.8944 - 0.0668 = 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 on a TI-84 Plus, you get the result shown at the right. Interpretation About 82.76% of the area under the curve falls between z = -1.5 and z = 1.25. TRY IT YOURSELF 6 Find the area under the standard normal curve between z = -2.165 and z = -1.35. Answer: Page A39 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 4, 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 5 and 6. (You will learn more about finding probabilities in the next section.) Area Probability Example 5 To the right of z = 1.06: 0.1446 P1z 7 1.062 = 0.1446 Example 6 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. EXAMPLE 6 TI-84 PLUS normalcdf(-1.5,1.25,0,1) 0.8275429323
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