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The Normal Distribution 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 ( x )2 P( x) 2 2 e 2 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% x 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% x 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 IIIAt least two of three consecutive points beyond the 2 level on the same side of the center line. Probability of a False Alarm Warning Signal Probability of false alarm I Point beyond 3 0.003 II Nine conscecutive points on same side of III At least 2/3 points beyond 2 0.004 0.004 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

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