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Section 6.2 Finding Area under a Normal Distribution HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Objectives o Find areas under the standard normal distribution. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.2: Finding Area to the Left of a Positive z-Value Using a Cumulative Normal Table Find the area under the standard normal curve to the left of z = 1.37. Solution To read the table, we must break the given z-value (1.37) into two parts: one containing the first decimal place (1.3) and the other containing the second decimal place (0.07). So, in Table B from Appendix A, look across the row labeled 1.3 and down the column labeled 0.07. The row and column intersect at 0.9147. Thus, the area under the standard normal curve to the left of z = 1.37 is 0.9147. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.2: Finding Area to the Left of a Positive z-Value Using a Cumulative Normal Table (cont.) z 1.0 1.1 1.2 1.3 1.4 1.5 0.05 0.8531 0.8749 0.8944 0.9115 0.9265 0.9394 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. 0.06 0.8554 0.8770 0.8962 0.9131 0.9279 0.9406 0.07 0.8577 0.8790 0.8980 0.9147 0.9292 0.9418 0.08 0.8599 0.8810 0.8997 0.9162 0.9306 0.9429 0.09 0.8621 0.8830 0.9015 0.9177 0.9319 0.9441 Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.2: Finding Area to the Left of a Positive z-Value Using a Cumulative Normal Table (cont.) HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.3: Finding Area to the Left of a Negative z-Value Using a Table or a TI-83/84 Plus Calculator Find the area under the standard normal curve to the left of z = −2.03. Solution The first part of the z-value is -2.0 and the second part is 0.03. This time, use Table A from Appendix A since the z-value is negative; look across the row labeled -2.0 and down the column labeled 0.03. The row and column intersect at 0.0212. Thus, the area under the normal curve to the left of z = −2.03 is 0.0212. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.3: Finding Area to the Left of a Negative zValue Using a Table or a TI-83/84 Plus Calculator (cont.) z -2.2 -2.1 -2.0 -1.9 -1.8 -1.7 0.04 0.0125 0.0162 0.0207 0.0262 0.0329 0.0409 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. 0.03 0.0129 0.0166 0.0212 0.0268 0.0336 0.0418 0.02 0.0132 0.0170 0.0217 0.0274 0.0344 0.0427 0.01 0.0136 0.0174 0.0222 0.0281 0.0351 0.0436 0 0.0139 0.0179 0.0228 0.0287 0.0359 0.0446 Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.3: Finding Area to the Left of a Negative zValue Using a Table or a TI-83/84 Plus Calculator (cont.) HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.3: Finding Area to the Left of a Negative zValue Using a Table or a TI-83/84 Plus Calculator (cont.) To obtain the solution using a TI-83/84 Plus calculator, perform the following steps. • Press and then to access the DISTR menu. • Choose option 2:normalcdf(. • Enter lower bound, upper bound, m, s. Note: If you want to find area under the standard normal curve, as in this example, then you do not need to enter m or s. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.3: Finding Area to the Left of a Negative zValue Using a Table or a TI-83/84 Plus Calculator (cont.) • Since we are asked to find the area to the left of z, the lower bound is -∞. We cannot enter -∞ into the calculator, so we will enter a very small value for the lower endpoint, such as -1099. This number appears as -1E99 when entered correctly into the calculator. To enter -1E99, press . HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.3: Finding Area to the Left of a Negative zValue Using a Table or a TI-83/84 Plus Calculator (cont.) Enter normalcdf(-1E99,-2.03), as shown in the screen shot. The area is approximately 0.0212. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.4 : Finding Area to the Right of a Positive z-Value Using a Cumulative Normal Table Find the area under the standard normal curve to the right of z = 1.37. Solution • Method 1: From Example 6.2, we know that the area under the standard normal curve to the left of z = 1.37 is 0.9147. So, the area under the standard normal curve to the right of z = 1.37 is 1 − 0.9147 = 0.0853. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.4 : Finding Area to the Right of a Positive z-Value Using a Cumulative Normal Table (cont.) HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.4 : Finding Area to the Right of a Positive z-Value Using a Cumulative Normal Table (cont.) • Method 2: We can look up z = −1.37 in Table A from Appendix A, which also gives us 0.0853. z -1.6 -1.5 -1.4 -1.3 -1.2 -1.1 0.09 0.0455 0.0559 0.0681 0.0823 0.0985 0.1170 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. 0.08 0.0465 0.0571 0.0694 0.0838 0.1003 0.1190 0.07 0.0475 0.0582 0.0708 0.0853 0.1020 0.1210 0.06 0.0485 0.0594 0.0721 0.0869 0.1038 0.1230 0.05 0.0495 0.0606 0.0735 0.0885 0.1056 0.1251 Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.5: Finding Area to the Right of a Negative z-Value Using a Table or a TI-83/84 Plus Calculator Find the area under the standard normal curve to the right of z = -0.90. Solution If we look up z = -0.90 in Table A from Appendix A, we see that an area of 0.1841 lies to the left of z. Then, subtract this area from 1 to find the amount of area to the right of z. Hence, an area of 1 - 0.1841 = 0.8159 lies to the right of z. Note that this is the same area you find by simply looking up z = 0.90 in Table B from Appendix A. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.5: Finding Area to the Right of a Negative zValue Using a Table or a TI-83/84 Plus Calculator (cont.) HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.5: Finding Area to the Right of a Negative zValue Using a Table or a TI-83/84 Plus Calculator (cont.) To obtain the solution using a TI-83/84 Plus calculator, perform the following steps. • Press and then to access the DISTR menu. • Choose option 2:normalcdf(. • Because the calculator will not allow us to enter ∞ for the upper endpoint, we will choose a sufficiently large number, such as 1099. This number appears as 1û99 when entered correctly into the calculator. To enter 1û99 into the calculator, press . HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.5: Finding Area to the Right of a Negative zValue Using a Table or a TI-83/84 Plus Calculator (cont.) Thus, to obtain the answer, we enter normalcdf(-0.90,1û99) into the calculator, as shown in the screenshot, and find that the area to the right of z = -0.90 is indeed approximately 0.8159. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.6: Finding Area between Two z-Values Using Tables or a TI-83/84 Plus Calculator Find the area under the standard normal curve between z1 = -1.68 and z2 = 2.00. Solution First, look up the area to the left of z1 = -1.68, which is 0.0465. Second, look up the area to the left of z2 = 2.00, which is 0.9772. Finally, subtract: 0.9772 - 0.0465 = 0.9307. Thus, the area between the two z-values is 0.9307. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.6: Finding Area between Two z-Values Using Tables or a TI-83/84 Plus Calculator (cont.) Enter normalcdf(-1.68,2.00), as shown in the screenshot on next slide, to find that the area between z1 = -1.68 and z2 = 2.00 is approximately 0.9308. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.6: Finding Area between Two z-Values Using Tables or a TI-83/84 Plus Calculator (cont.) Notice that this is a slightly different answer than what we obtained using the tables. The reason is that the table values have been rounded in the intermediate steps and the calculator values have not. For this reason, calculator solutions may vary slightly from those obtained by using the tables. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.7: Finding Area between Two z-Values Using a TI-83/84 Plus Calculator Find the area under the standard normal curve between z1 = 1.50 and z2 = 2.75. Solution Let’s use a TI-83/84 Plus calculator to find this area. We want to use a lower bound of 1.50 and an upper bound of 2.75. Enter normalcdf(1.50,2.75) into the calculator, which gives a value of approximately 0.0638 as shown in the screenshot on next slide. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.7: Finding Area between Two z-Values Using a TI-83/84 Plus Calculator (cont.) Therefore, we see that an area of approximately 0.0638 lies between z1 = 1.50 and z2 = 2.75. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.7: Finding Area between Two z-Values Using a TI-83/84 Plus Calculator (cont.) HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.8: Finding Area in the Tails for Two z-Values Using a TI-83/84 Plus Calculator Find the total of the areas under the standard normal curve to the left of z1 = −2.50 and to the right of z2 = 3.00. Solution There are two areas that we must find. The total we are interested in is the sum of these two areas. Let’s begin by finding the area to the left of z1 = −2.50. Enter normalcdf(-1û99,-2.50) to find that the area to the left of z1 = −2.50 is approximately 0.006210, as shown in the screenshot on next slide. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.8: Finding Area in the Tails for Two z-Values Using a TI-83/84 Plus Calculator (cont.) Next, we need to find the area to the right of z2 = 3.00. Enter normalcdf(3.00,1û99) to find that the area to the right of z2 = 3.00 is approximately 0.001350, as shown in the screenshot. Thus, the total area in the tails is the sum of the two areas, 0.006210 + 0.001350 ≈ 0.0076. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.8: Finding Area in the Tails for Two z-Values Using a TI-83/84 Plus Calculator (cont.) HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.8: Finding Area in the Tails for Two z-Values Using a TI-83/84 Plus Calculator (cont.) Note an alternative method for finding this area that is particularly clever. By definition, we know that the total area under the curve equals 1. Using this fact, the area in the tails can be obtained by finding the area between z1 = −2.50 and z2 = 3.00 and then subtracting that area from 1. This method can even be entered in one step into your calculator as shown below and in the screenshot on next slide. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.8: Finding Area in the Tails for Two z-Values Using a TI-83/84 Plus Calculator (cont.) 1Þnormalcdf(-2.50,3.00) ≈ 0.0076 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.9: Finding Area in the Tails for Two z-Values Using a TI-83/84 Plus Calculator Find the total of the areas under the standard normal curve to the left of z1 = -1.23 and to the right of z2 = 1.23. Solution Notice that the absolute values of z1 and z2 are the same. Thus, the areas in the two tails of the distribution will be the same because of the symmetric property of the standard normal curve. So, to find the total area in the two tails, we only need to look up the area in the left tail and multiply that by 2. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.9: Finding Area in the Tails for Two z-Values Using a TI-83/84 Plus Calculator (cont.) By entering normalcdf(-1û99,-1.23), we find that the area to the left of z1 = -1.23 is approximately 0.109349, as shown in the screenshot. Multiply this area by 2 in order to obtain the combined area in the tails. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.9: Finding Area in the Tails for Two z-Values Using a TI-83/84 Plus Calculator (cont.) Thus, (0.109349)(2) 0.2187. So, the total area in the two tails is approximately 0.2187. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Finding Area under a Normal Distribution Using the Cumulative Normal Distribution Tables to Find Areas under the Standard Normal Curve HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Finding Area under a Normal Distribution Using the Cumulative Normal Distribution Tables to Find Areas under the Standard Normal Curve (cont.) HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Finding Area under a Normal Distribution Using a TI-83/84 Plus Calculator to Find Areas under the Standard Normal Curve HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Finding Area under a Normal Distribution Using a TI-83/84 Plus Calculator to Find Areas under the Standard Normal Curve (cont.) HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.10: Interpreting Probability for the Standard Normal Distribution as an Area under the Curve Interpret P(z −2.67). Solution P(z −2.67) stands for the probability that z is less than or equal to -2.67. This is equal to the area under the standard normal curve to the left of z = −2.67. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.10: Interpreting Probability for the Standard Normal Distribution as an Area under the Curve (cont.) HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.10: Interpreting Probability for the Standard Normal Distribution as an Area under the Curve (cont.) Note that the probability that z is less than a value is the same as the probability that z is less than or equal to that value, or symbolically, P(z −2.67) = P(z < −2.67). HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.11 : Finding Probabilities for the Standard Normal Distribution Using Tables or a TI-83/84 Plus Calculator Find the following probabilities using the cumulative normal distribution tables or a TI-83/84 Plus calculator. a. P(z < 1.45) b. P(z −1.37) c. P(1.25 < z < 2.31) d. P(z < −2.5 or z > 2.5) e. P(z < −4.01) f. P(z 3.98) HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.11 : Finding Probabilities for the Standard Normal Distribution Using Tables or a TI-83/84 Plus Calculator (cont.) Solution For each part of this example, the solution first describes the method for finding the answer using the cumulative normal distribution tables and then explains how to find the answer using a TI-83/84 Plus calculator. a. P(z < 1.45) is the area under the standard normal curve to the left of z = 1.45. Look up z = 1.45 in the cumulative normal table for positive z-values, Table B. The area is 0.9265. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.11 : Finding Probabilities for the Standard Normal Distribution Using Tables or a TI-83/84 Plus Calculator (cont.) HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.11 : Finding Probabilities for the Standard Normal Distribution Using Tables or a TI-83/84 Plus Calculator (cont.) By entering normalcdf(-1û99,1.45), we find that the area to the left of z = 1.45 is approximately 0.9265, as shown in the screenshot. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.11 : Finding Probabilities for the Standard Normal Distribution Using Tables or a TI-83/84 Plus Calculator (cont.) b. P(z −1.37) is the area under the standard normal curve to the right of z = −1.37. Use the symmetry property of the standard normal curve and look up z = 1.37 in the cumulative normal table for positive z-values, Table B. The area to the right of z = −1.37 is 0.9147. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.11 : Finding Probabilities for the Standard Normal Distribution Using Tables or a TI-83/84 Plus Calculator (cont.) HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.11 : Finding Probabilities for the Standard Normal Distribution Using Tables or a TI-83/84 Plus Calculator (cont.) By entering normalcdf(-1.37,1û99), we find that the area to the right of z = −1.37 is approximately 0.9147, as shown in the screenshot. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.11 : Finding Probabilities for the Standard Normal Distribution Using Tables or a TI-83/84 Plus Calculator (cont.) c. P(1.25 < z < 2.31) is the area under the standard normal curve between z1 = 1.25 and z2 = 2.31. Look up each value in the cumulative normal table for positive z-values, Table B. The area to the left of z1 = 1.25 is 0.8944. The area to the left of z2 = 2.31 is 0.9896. The area between z1 = 1.25 and z2 = 2.31 is the difference between these two areas. Thus, the area is 0.9896 − 0.8944 = 0.0952. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.11 : Finding Probabilities for the Standard Normal Distribution Using Tables or a TI-83/84 Plus Calculator (cont.) HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.11 : Finding Probabilities for the Standard Normal Distribution Using Tables or a TI-83/84 Plus Calculator (cont.) By entering normalcdf(1.25,2.31), we find that the area between z1 = 1.25 and z2 = 2.31 is approximately 0.0952, as shown in the screenshot. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.11 : Finding Probabilities for the Standard Normal Distribution Using Tables or a TI-83/84 Plus Calculator (cont.) d. P(z < −2.5 or z > 2.5) is the sum of the areas to the left of z1 = −2.5 and to the right of z2 = 2.5. Since the normal curve is symmetric, these two areas are the same; therefore, look up the area to the left of z = −2.50 in the cumulative normal table for negative z-values, Table A, and multiply that area by 2. The area to the left of z = −2.50 is 0.0062. Thus, the area we are interested in is (0.0062)(2) = 0.0124. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.11 : Finding Probabilities for the Standard Normal Distribution Using Tables or a TI-83/84 Plus Calculator (cont.) HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.11 : Finding Probabilities for the Standard Normal Distribution Using Tables or a TI-83/84 Plus Calculator (cont.) Enter 1Þnormalcdf(-2.5,2.5). We find that the sum of the areas to the left of z1 = −2.5 and to the right of z2 = 2.5 is approximately 0.0124, as shown in the screenshot. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.11 : Finding Probabilities for the Standard Normal Distribution Using Tables or a TI-83/84 Plus Calculator (cont.) e. P(z < −4.01) is the area under the standard normal curve to the left of z = −4.01. Notice that z = −4.01 is not in the cumulative normal table for negative z-values, Table A. It is smaller than the z-values in the table, which means that it is further to the left. Thus, the area under the curve is smaller than all of the areas listed in the table. Therefore, P(z < −4.01) is approximately 0.0000. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.11 : Finding Probabilities for the Standard Normal Distribution Using Tables or a TI-83/84 Plus Calculator (cont.) HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.11 : Finding Probabilities for the Standard Normal Distribution Using Tables or a TI-83/84 Plus Calculator (cont.) By entering normalcdf(-1û99,-4.01), we find that the area to the left of z = −4.01 is approximately 0.00003, as shown in the screenshot. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.11 : Finding Probabilities for the Standard Normal Distribution Using Tables or a TI-83/84 Plus Calculator (cont.) f. P(z 3.98) is the area under the standard normal curve to the left of z = 3.98. Notice that z = 3.98 is not in the cumulative normal table for positive z-values, Table B. It is larger than the z-values in the table, which means that it is further to the right. Thus, the area under the curve is larger than all of the areas listed in the table. Therefore, P(z 3.98) is approximately 1.0000. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.11 : Finding Probabilities for the Standard Normal Distribution Using Tables or a TI-83/84 Plus Calculator (cont.) HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6.11 : Finding Probabilities for the Standard Normal Distribution Using Tables or a TI-83/84 Plus Calculator (cont.) By entering normalcdf(-1û99,3.98),we find that the area to the left of z = 3.98 is approximately 1.0000, as shown in the screenshot. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved.