Section 7.1 The Standard Normal Curve 295 EXAMPLE 7.11 Finding the z-score corresponding to an area in the middle Use Table A.2 to find the z-scores that bound the middle 95% of the area under the standard normal curve. Area = 0.025 Area = 0.025 Area = 0.95 −1.96 1.96 Figure 7.13 Solution Step 1: Sketch a normal curve and shade in the given area. Label the z-score on the left z1 and the z-score on the right z2. Step 2: Find the area to the left of z1. Since the area in the middle is 0.95, the area in the two tails combined is 0.05. Half of that area, or 0.025, is to the left of z1. We conclude that z1 is the z-score that has an area of 0.025 to its left. Step 3: In Table A.2, an area of 0.025 corresponds to a z-score of −1.96. Therefore z1 = −1.96. Step 4: To find z2, note that the area to the left of z2 is 0.9750. We can use Table A.2 to determine that z2 = 1.96. Alternatively, note that z1 and z2 are equidistant from the mode. Since the normal curve is symmetric, z1 must be the negative of z2. Therefore z2 = 1.96. See Figure 7.13. USING TECHNOLOGY TI-84 PLUS Finding areas under the standard normal curve The normalcdf command is used to calculate area under a normal curve. Step 1. Press 2nd, then VARS to access the DISTR menu. Select 2:normalcdf (Figure A). Step 2. Enter the left endpoint, comma, the right endpoint, comma, the mean (0 for the standard normal), comma, and the standard deviation (1 for the standard normal). ∙ When finding the area to the right of a given value, use 1E99 as the right endpoint. ∙ When finding the area to the left of a given value, use −1E99 as the left endpoint. Step 3. Press ENTER. Using the TI-84 PLUS Stat Wizards (see Appendix B for more information) Step 1. Press 2nd, then VARS to access the DISTR menu. Select 2:normalcdf (Figure A). Step 2. Enter the left endpoint in the lower field, the right endpoint in the upper field, the mean in the �� field (0 for the standard normal), and the standard deviation in the �� field (1 for the standard normal). ∙ When finding the area to the right of a given value, use 1E99 as the right endpoint. ∙ When finding the area to the left of a given value, use −1E99 as the left endpoint. Step 3. Select Paste and press ENTER to paste the command to the home screen. Press ENTER again to run the command. Figure B illustrates finding the area to the left of z = 1.26 (Example 7.2). Figure C illustrates finding the area to the right of z = −0.58 (Example 7.3). Figure D illustrates finding the area between z = −1.45 and z = 0.42. (Example 7.4). Note: The quantity 1E99 in the TI-84 PLUS calculator represents a very large number, specifically a 1 followed by 99 zeros. Figure A Figure B Figure C Figure D
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