1. Intro to Stats and Collecting Data

Intro to Stats

1. Intro to Stats and Collecting Data

Intro to Stats

2:02 minutes

Problem 14.2.3

Textbook Question

Control Limits In constructing a control chart for the proportions of defective dimes, it is found that the lower control limit is -0.00325. How should that value be adjusted?

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Understand that control limits in a control chart are used to determine the range of acceptable variation in a process. The lower control limit (LCL) should not be negative, as proportions cannot be negative.

Recognize that the LCL is calculated using the formula: LCL = p̂ - z * sqrt((p̂(1-p̂))/n), where p̂ is the sample proportion, z is the z-score corresponding to the desired confidence level, and n is the sample size.

Since the calculated LCL is negative, it indicates that the process variation is too small or the sample size is too small. In practice, a negative LCL is adjusted to zero because a proportion cannot be less than zero.

Adjust the LCL to zero to reflect the realistic lower bound for proportions. This adjustment ensures that the control chart accurately represents the process capability.

Re-evaluate the control chart with the adjusted LCL to ensure that it accurately reflects the process stability and identify any out-of-control signals based on the new limits.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Control Limits

Control limits are boundaries set within a control chart to determine the acceptable range of variation in a process. They are calculated using statistical methods and help identify when a process is out of control. Typically, control limits are set at three standard deviations from the process mean, indicating the expected range of variation.

Proportions of Defective Items

Proportions of defective items refer to the fraction or percentage of items in a sample that do not meet quality standards. In statistical quality control, this measure is used to monitor and control the quality of a production process. It is essential to ensure that the proportion remains within the control limits to maintain process stability.

Adjustment of Control Limits

Adjusting control limits involves recalculating them to ensure they are meaningful and applicable. If a control limit is negative, as in the case of a lower control limit for proportions, it should be adjusted to zero because proportions cannot be negative. This adjustment ensures the control chart accurately reflects the process's capability and quality standards.

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Textbook Question

Quarters. In Exercises 9–12, refer to the accompanying table of weights (grams) of quarters minted by the U.S. government. This table is available in Data Set 44 “Weights of Minted Quarters” in Appendix B.

Quarters: R Chart Treat the five measurements from each day as a sample and construct an R chart. What does the result suggest?

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Textbook Question

Quarters: xbar Chart Treat the 5 measurements from each day as a sample and construct an xbar chart. What does the result suggest?

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Textbook Question

Identify three specific criteria for determining when a process is out of statistical control.

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Textbook Question

Notation The control chart for Exercise 1 shows a value of p_bar = 0.0975. What does that value denote, and how is it obtained? What do UCL and LCL indicate?

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Textbook Question

Minting Quarters Specifications for a quarter require that it be 8.33% nickel and 91.67% copper; it must weigh 5.670 g and have a diameter of 24.26 mm and a thickness of 1.75 mm; and it must have 119 reeds on the edge. A quarter is considered to be defective if it deviates substantially from those specifications. A production process is monitored, defects are recorded and the accompanying control chart is obtained. Does this process appear to be within statistical control? If not, identify any out-of-control criteria that are satisfied. Is the manufacturing process deteriorating?

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Textbook Question

Control Charts for p. In Exercises 5–12, use the given process data to construct a control chart for p. In each case, use the three out-of-control criteria listed near the beginning of this section and determine whether the process is within statistical control. If it is not, identify which of the three out-of-control criteria apply.

Euro Coins Consider a process of minting coins with a value of one euro. Listed below are the numbers of defective coins in successive batches of 10,000 coins randomly selected on consecutive days of production.

32 21 25 19 35 34 27 30 26 33

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Textbook Question

Find the values of LCL and UCL for an xbar chart.

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Textbook Question

In Exercises 5–8, use the following two control charts that result from testing batches of newly manufactured aircraft altimeters, with 100 in each batch. The original sample values are errors (in feet) obtained when the altimeters are tested in a pressure chamber that simulates an altitude of 6000 ft. The Federal Aviation Administration requires an error of no more than 40 ft at that altitude.

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Is the process variation within statistical control? Why or why not?

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