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## Chapter Six Normal Distributions

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**Understandable StatisticsSeventh EditionBy Brase and**BrasePrepared by: Lynn SmithGloucester County College Chapter Six Normal Distributions**Properties of The Normal Distribution** The curve is bell-shaped with the highest point over the mean, .**Properties of The Normal Distribution** The curve is symmetrical about a vertical line through .**Properties of The Normal Distribution** The curve approaches the horizontal axis but never touches or crosses it.**Properties of The Normal Distribution** – The transition points between cupping upward and downward occur above + and – .**The Normal Density Function**This formula generates the density curve which gives the shape of the normal distribution.**The Empirical Rule**Approximately 68% of the data values lie is within one standard deviation of the mean. 68% One standard deviation from the mean.**The Empirical Rule**Approximately 95% of the data values lie within two standard deviations of the mean. 95% Two standard deviations from the mean.**The Empirical Rule**Almost all (approximately 99.7%) of the data values will be within three standard deviations of the mean. 99.7% Three standard deviations from the mean.**Application of the Empirical Rule**The life of a particular type of light bulb is normally distributed with a mean of 1100 hours and a standard deviation of 100 hours. What is the probability that a light bulb of this type will last between 1000 and 1200 hours? Approximately 68%**Control Chart**a statistical tool to track data over a period of equally spaced time intervals or in some sequential order**Statistical Control**A random variable is in statistical control if it can be described by the same probability distribution when it is observed at successive points in time.**To Construct a Control Chart**• Draw a center horizontal line at . • Draw dashed lines (control limits) at and . • The values of and may be target values or may be computed from past data when the process was in control. • Plot the variable being measured using time on the horizontal axis.**Control Chart** 1 2 3 4 5 6 7**Control Chart** 1 2 3 4 5 6 7**Out-Of-Control Warning Signals**I One point beyond the 3 level II A run of nine consecutive points on one side of the center line at target III At least two of three consecutive points beyond the 2 level on the same side of the center line.**Is the Process in Control?** 1 2 3 4 5 6 7**Is the Process in Control?** 1 2 3 4 5 6 7 8 9 10 11 12 13**Is the Process in Control?** 1 2 3 4 5 6 7**Is the Process in Control?** 1 2 3 4 5 6 7**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.**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.**Calculating z-scores**Mean delivery time = 25 minutes Standard deviation = 2 minutes Convert 29.7 minutes to a z score.**Interpreting z-scores**Mean delivery time = 25 minutes Standard deviation = 2 minutes Interpret a z score of 1.6. The delivery time is 28.2 minutes.**Standard Normal Distribution:** = 0 = 1 -1 1 0 Values are converted to z scores where z =**Importance of the Standard Normal Distribution:**Standard Normal Distribution: 1 Any Normal Distribution: 0 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. z 0**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 . z2 0 z1**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 . z1 z2 0**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. z 0**Use of the Normal Probability Table**a. P(z < 1.24) = ______ b. P(0 < z < 1.60) = _______ c. P( - 2.37 < z < 0) = ______ .8925 .4452 .4911**Normal Probability**.9974 d. P( - 3 < z < 3 ) = ________ e. P( - 2.34 < z < 1.57 ) = _____ f. P( 1.24 < z < 1.88 ) = _______ .9322 .0774**Normal Probability**.2254 g. P( - 2.44 < z < - 0.73 ) = _______ h. P( z < 1.64 ) = __________ i . P( z > 2.39 ) = _________ .9495 .0084**Normal Probability**j. P ( z > - 1.43 ) = __________ k. P( z < - 2.71 ) = __________ .9236 .0034**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. a. ___________b. less than 30 minutes. b. __________ c. less than 22.7 minutes. c. __________ .3413 .9938 .1251**Inverse Normal Distribution**Finding z scores when probabilities (areas) are given**Find the indicated z score:**Find the indicated z score: .8907 0 z = 1.23**z 0**Find the indicated z score: .6331 .3669 z = – 0.34**Find the indicated z score:**.3560 .8560 0 z = 1.06**Find the indicated z score:**.4792 .0208 – 2.04 z = 0**Find the indicated z score:**.4900 0 z = 2.33**Find the indicated z score:**.005 z = 0 – 2.575**Find the indicated z score:**= .005 A B – z 0 z 2.575 or 2.58 If area A + area B = .01, z = __________