6. Normal Distribution and Continuous Random Variables
Standard Normal Distribution
3:57 minutesProblem 5.2.24
Textbook Question
In Exercises 21–24, a control chart is shown. Each chart has horizontal lines drawn at the mean mu, at mu ±2sigma, and at mu±3sigma. Determine whether the process shown is in control or out of control. Explain.
An engine part has been designed to have a diameter of 55 millimeters. The standard deviation of the process is 0.001 millimeter.
Verified step by step guidance1
Step 1: Understand the control chart. The chart shows the mean (μ = 55 mm), upper control limits (μ + 2σ and μ + 3σ), and lower control limits (μ - 2σ and μ - 3σ). The standard deviation (σ) is given as 0.001 mm.
Step 2: Calculate the control limits. Use the formulas for the upper and lower control limits: μ ± 2σ and μ ± 3σ. For example, μ + 2σ = 55 + (2 × 0.001) = 55.002 mm, and μ - 2σ = 55 - (2 × 0.001) = 54.998 mm. Similarly, calculate μ ± 3σ.
Step 3: Analyze the data points on the control chart. Observe whether the data points fall within the control limits (μ ± 3σ). If all points are within these limits, the process is in control. If any point falls outside these limits, the process is out of control.
Step 4: Check for patterns or trends. Even if all points are within the control limits, look for systematic patterns (e.g., consecutive points above or below the mean) that might indicate a potential issue with the process.
Step 5: Conclude whether the process is in control or out of control. Based on the observations, determine if the process is stable and consistent or if corrective actions are needed.
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