SECTION 5.2 Normal Distributions: Finding Probabilities 251 Extending Concepts Control Charts Statistical process control (SPC) is the use of statistics to monitor and improve the quality of a process, such as manufacturing an engine part. In SPC, information about a process is gathered and used to determine whether a process is meeting all of the specified requirements. One tool used in SPC is a control chart. When individual measurements of a variable x are normally distributed, a control chart can be used to detect processes that are possibly out of statistical control. Three warning signals that a control chart uses to detect a process that may be out of control are listed below. (1) A point lies beyond three standard deviations of the mean. (2) There are nine consecutive points that fall on one side of the mean. (3) At least two of three consecutive points lie more than two standard deviations from the mean. In Exercises 21–24, a control chart is shown. Each chart has horizontal lines drawn at the mean m, at m { 2s, and at m { 3s. Determine whether the process shown is in control or out of control. Explain. 21. A gear has been designed to have a diameter of 3 inches. The standard deviation of the process is 0.2 inch. 12345678910 1 2 3 4 Observation number Diameter (in inches) Gears 22. A nail has been designed to have a length of 4 inches. The standard deviation of the process is 0.12 inch. 2 4 6 8 10 12 3.75 4.00 4.25 4.50 Observation number Length (in inches) Nails 23. A liquid-dispensing machine has been designed to fill bottles with 1 liter of liquid. The standard deviation of the process is 0.1 liter. 2 4 6 8 10 12 0.5 1.0 1.5 Observation number Liquid dispensed (in liters) Liquid Dispenser 24. An engine part has been designed to have a diameter of 55 millimeters. The standard deviation of the process is 0.001 millimeter. 2 4 6 8 10 12 55.0000 55.0025 55.0050 54.9975 Diameter (in millimeters) Observation number Engine Part
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