Area under the Normal Curve using Printed Tables

Transcription

1 Area under the Normal Curve using Printed Tables Example problem curve the left of. Area = Probability = or 91.47% curve the right of. Area = Probability = or 8.53% curve between and. Area = Probability = or 25.90% Look in row 1.30 and column 0.07 where it says is the area between z=0 and z= more is in the left half (where z is negative) tal area Look in row 1.30 and column 0.07 where it says is the area between z=0 and z=1.37 The area the right of z=1.37 is: tal area in the right half area between z=0 and z=1.37 = area the right of z=1.37 First, find area the left of z=1.85: Row 1.80 Column 0.05 says Add for the left half get tal Then find area the left of z=0.55: Row 0.50 Column 0.05 says Add for the left half get tal Then subtract find the area in between those two z values: minus equals net area between.

2 Find the tal area under the normal curve. You should know this very basic special fact, that the tal area under a probability distribution is always exactly precisely = 1. What is the area under the right half of the normal curve? and What is the area under the left half of the normal curve? You should know these very special facts about the Normal Distribution: it is symmetric, is in the middle, half of the area (0.5) is the left and half of the area (0.5) is the right. No table is needed. This is fact that you simply have know. No table is needed. These are facts that you simply have know.

3 curve the left of. Area = Probability = or 2.17% curve the right of. Area = Probability = or 97.83% curve between and. Area = Probability = or 26.82% curve between and. Area = Probability = or 94.94% The table only has positive z-values. Because of symmetry, the area the left of z=-2.02 Is the same as the area the right of z= Look in row 2.00 column 0.02 find Area between z=0 and z=2.02 (positive) is Therefore area the right of z=+2.02 (positive) is minus equals The table has only positive z-values. Because of symmetry, the area between z=0 and z=+2.02 Is the same as the area between z=-2.02 and z=0. Look in row 2.00 column 0.02 find The tal area between z=-2.02 and z=0 is Add for the right half of the area = tal. Because of symmetry, the area between z=-1.38 and z=-0.38 Is the same as the area between z=+0.38 and z= Look in row 1.30 column 0.08 find Look in row 0.30 column 0.08 find Subtract minus get area Or, if you added the each, minus = , the same answer again. The right half: Look in row 2.20 column find The left half: Because of symmetry, the area between z=-1.79 and z=0 is the same as the area between z=0 and z= Look in row 1.70 column 0.09 find Add plus = tal area

4 50% of the area? 50% = shaded in middle divided by 2 = % of the area? 68% = shaded in middle divided by 2 is Look in the body of the table find the value closest It is Read outward see you re in row 0.60 column 0.07, So the z-score is By symmetry, the other end is at z= Look in the body of the table find the value closest It is Read outward see you re in row 0.90 column So the z-score is 0.99 By symmetry, the other end is at z= This is why the Empirical Rule says In a Normal Distribution, about 68% of the data lies within 1 standard deviation of the mean.

5 95% of the area? 95% = shaded in middle divided by 2 is % of the area? 99.7% = divided by 2 is This is why the Empirical Rule says In a Normal Distribution, about 95% of the data lies within 2 standard deviations of the mean. Look in the body of the table find the value closest It is in the table, this is unusual! Read outward see you re in row 1.90 column So the z-score is 1.96 By symmetry, the other end is at z= Look in the body of the table find the value closest It is in the table twice. That happens with the extreme tails of the normal distribution. Read outward see you re in row 2.90 column 0.06 or column So the z-score is 2.96 or 2.97 This is why the Empirical Rule says In a Normal Distribution, about 95% of the data lies within 2 standard deviations of the mean. By symmetry, the other end is at z=-2.96 or z=-2.97 It turns out that with the help of computers or calculars we can determine that the 2.97 version is really more accurate.

6 What z-score separates the p 10% of the data from the botm 90%? 90% = area is the left. 10% = area is the right What score separates the botm third of the data from the p two-thirds? 1/3 the left. You can use the fraction in TI-84 or in Excel. The table gives areas in the right half. The tal area the left is Subtract the left half get area in the right half. Look in the body of the table find the value closest It is Read outward see you re in row 1.20 column The z-score of interest is therefore 1.28 By symmetry, the z-score that separates the botm 1/3 of the area is the negative of the z-score that separates the p 1/3 of the area. In the right half, we want the z-score that separates of the area the left from the of the area the right. Look in the body of the table find the value closest It is Read out find you re in row 0.40 column Therefore the z-value is But we want the mirror image of that, so our z-score is