290 Chapter 7 The Normal Distribution SUMMARY ∙ Points on the horizontal axis to the left of the mode have negative z-scores. ∙ Points on the horizontal axis to the right of the mode have positive z-scores. ∙ The mode itself has a z-score of 0. Finding areas under the standard normal curve by using Table A.2 In general, areas under the standard normal curve can be found by using Table A.2 or by using technology. Table A.2 contains z-scores and areas. Each of the four-digit numbers in the body of the table is the area to the left of a z-score. Examples 7.2–7.4 will show how to use Table A.2. EXAMPLE 7.2 Finding an area to the left of a z-score Use Table A.2 to find the area to the left of z = 1.26. Explain It Again Looking up a z-score: In Table A.2, the units and tenths digits of the z-score correspond to a row, and the hundredths digit corresponds to a column. Thus, for the z-score 1.26, we find the row corresponding to 1.2 and the column corresponding to 0.06. Solution Step 1: Sketch a normal curve, label the point z = 1.26, and shade in the area to the left of it. Note that z = 1.26 is located to the right of the mode, since it is positive. Step 2: Consult Table A.2. To look up z = 1.26, find the row containing 1.2 and the column containing 0.06. The value in the intersection of the row and column is 0.8962. This is the area to the left of z = 1.26 (see Figure 7.7). 1.26 Area = 0.8962 Figure 7.7 z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 1.0 .8413 .8438 .8461 .8485 .8508 .8531 .8554 .8577 .8599 .8621 1.1 .8643 .8665 .8686 .8708 .8729 .8749 .8770 .8790 .8810 .8830 1.2 .8849 .8869 .8888 .8907 .8925 .8944 .8962 .8980 .8997 .9015 1.3 .9032 .9049 .9066 .9082 .9099 .9115 .9131 .9147 .9162 .9177 1.4 .9192 .9207 .9222 .9236 .9251 .9265 .9279 .9292 .9306 .9319 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ An area to the left of a z-score represents the proportion of a population that is less than a given value. Sometimes we need to know the proportion of a population that is greater than a given value. In these cases we need to find the area to the right of a z-score. Since the area under the entire curve is equal to 1, we can find the area to the right of a z-score by finding the area to the left and subtracting from 1. EXAMPLE 7.3 Finding an area to the right of a z-score Use Table A.2 to find the area to the right of z = −0.58. Explain It Again Finding an area to the right of a z-score: To find the area to the right of a z-score, use Table A.2 to find the area to the left, then subtract from 1. Solution Step 1: Sketch a normal curve, label the point z = −0.58, and shade in the area to the right of it. Note that z = −0.58 is located to the left of the mode, since it is negative. Step 2: Consult Table A.2. To look up z = −0.58, find the row containing −0.5 and the column containing 0.08. The value in the intersection of the row and column is
navidi_monk_elementary_statistics_2e_ch7-9
To see the actual publication please follow the link above