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Z Score • The z value or z score tells the number of standard deviations the original measurement is from the mean. • The z value is in standard units. Formula for z score x z Calculating z-scores The amount of time it takes for a pizza delivery is approximately normally distributed with a mean of 25 minutes and a standard deviation of 2 minutes. Convert 21 minutes to a z score. x 21 25 z 2.00 2 Calculating z-scores Mean delivery time = 25 minutes Standard deviation = 2 minutes Convert 29.7 minutes to a z score. x 29.7 25 z 2.35 2 Interpreting z-scores Mean delivery time = 25 minutes Standard deviation = 2 minutes Interpret a z score of 1.6. x z 1.6( 2 ) 25 28 .2 The delivery time is 28.2 minutes. Standard Normal Distribution: =0 =1 -1 0 1 Values are converted to z scores wherexz = Importance of the Standard Normal Distribution: Standard Normal Distribution: Any Normal Distribution: 0 1 Areas will be equal. 1 Use of the Normal Probability Table (Table 5) - Appendix II Entries give the probability that a standard normally distributed random variable will assume a value to the left of a given negative z-score. Use of the Normal Probability Table (Table 5a) - Appendix II Entries give the probability that a standard normally distributed random variable will assume a value to the left of a given positive z value. To find the area to the left of z = 1.34 _________________________________ ____z … 0.03 0.04 0.05 ..… _________________________________ ____ . . 1.2 … .8907 .8925 .8944 …. 1.3 … .9082 .9099 .9115 …. 1.4 … .9236 .9251 .9265 …. Patterns for Finding Areas Under the Standard Normal Curve To find the area to the left of a given negative z : Use Table 5 (Appendix II) directly. z 0 Patterns for Finding Areas Under the Standard Normal Curve To find the area to the left of a given positive z: Use Table 5 a (Appendix II) directly. 0 z Patterns for Finding Areas Under the Standard Normal Curve To find the area between z values on either side of zero: Subtract area to left of z1 from area to left of z2 . z1 0 z2 Patterns for Finding Areas Under the Standard Normal Curve To find the area between z values on the same side of zero: Subtract area to left of z1 from area to left of z2 . 0 z1 z2 Patterns for Finding Areas Under the Standard Normal Curve To find the area to the right of a positive z value or to the right of a negative z value: Subtract from 1.0000 the area to the left of the given z. Area under entire curve = 1.000. 0 z Use of the Normal Probability Table a. .8925 P(z < 1.24) = ______ b. .4452 P(0 < z < 1.60) = _______ c. .4911 P( - 2.37 < z < 0) = ______ Normal Probability d. .9974 P( - 3 < z < 3 ) = ________ e. .9322 P( - 2.34 < z < 1.57 ) = _____ f. .0774 P( 1.24 < z < 1.88 ) = _______ Normal Probability g. .2254 P( - 2.44 < z < - 0.73 ) = _______ h. .9495 P( z < 1.64 ) = __________ i. .0084 P( z > 2.39 ) = _________ Normal Probability j. .9236 P ( z > - 1.43 ) = __________ k. .0034 P( z < - 2.71 ) = __________ Application of the Normal Curve The amount of time it takes for a pizza delivery is approximately normally distributed with a mean of 25 minutes and a standard deviation of 2 minutes. If you order a pizza, find the probability that the delivery time will be: a. between 25 and 27 minutes. .3413 a. ___________ b. less than 30 minutes. .9938 b. __________ c. less than 22.7 minutes. .1251 c. __________