Motorola used the normal distribution to determine the...

Motorola used the normal distribution to determine the probability of defects and the number of defects expected in a production process. Assume a production process produces items with a mean weight of 10 ounces. Calculate the probability of a defect and the expected number of defects for a 1000 -unit production run in the following situations. a. The process standard deviation is $.15,$ and the process control is set at plus or minus one standard deviation. Units with weights less than 9.85 or greater than 10.15 ounces will be classified as defects. b. Through process design improvements, the process standard deviation can be reduced to $.05 .$ Assume the process control remains the same, with weights less than 9.85 or greater than 10.15 ounces being classified as defects. c. What is the advantage of reducing process variation, thereby causing process control limits to be at a greater number of standard deviations from the mean?

Motorola used the normal distribution to determine the probability of defects and the number of defects expected in a production process. Assume a production process produces items with a mean weight of 10 ounces. Calculate the probability of a defect and the expected number of defects for a 1000 -unit production run in the following situations.
a. The process standard deviation is $.15,$ and the process control is set at plus or minus one standard deviation. Units with weights less than 9.85 or greater than 10.15 ounces will be classified as defects.
b. Through process design improvements, the process standard deviation can be reduced to $.05 .$ Assume the process control remains the same, with weights less than 9.85 or greater than 10.15 ounces being classified as defects.
c. What is the advantage of reducing process variation, thereby causing process control limits to be at a greater number of standard deviations from the mean?

Key Concepts

Normal Distribution

A probability distribution that is symmetric about the mean, depicting that data near the mean are more frequent in occurrence than data far from the mean. This distribution is commonly used in quality control and process analysis because it accurately models natural variations in production processes.

Standard Deviation

A statistical measure that quantifies the amount of variability or dispersion in a set of data values. In the context of manufacturing, the standard deviation indicates the typical deviation of individual measurements from the process mean, which is critical in assessing the consistency of production.

Process Control Limits

Thresholds set at specific distances from the process mean (often in terms of standard deviations) that define acceptable operating conditions. Items measured outside these limits are considered defects, and these control limits help in monitoring and controlling process performance.

Defect Probability Calculation

The method of determining the likelihood that an item produced by a process will fall outside the defined control limits by using the properties of the normal distribution. This calculation is essential for estimating the number of defective items in a production run.

Process Variation Improvement

The practice of reducing the variability within a process, typically by lowering the standard deviation. This reduction leads to fewer defects because a higher percentage of produced items fall within the acceptable range defined by the control limits, thus improving overall quality and consistency.

Transcript

00:01 All right, so we're assuming that a production process, potentially mandated by motorola, makes batches of a certain item in a thousand unit batches, and the mean weight is 10 ounces.

00:17 Excuse me.

00:19 We are also given the criteria for what's considered a defect.

00:24 If x is just a random variable for our weight, then defects are where x, x, is less than 9 .85 or x is greater than 10 .15.

00:45 And for part a and b, we're given two standard deviations.

00:52 For a, we're given 0 .15 ounces.

00:56 Part b, we're given 0 .05 ounces.

01:00 And we're supposed to find the probability of a defect and the expected number of defects per 1 ,000 unit batch, is what i'm going to call them.

01:09 Okay, so part a.

01:13 Oh, yeah, it's also worth mentioning we are under assumption that these are normally distributed in terms of the weight of each product.

01:25 All right, so let's find the probability.

01:33 X is this, which is just going to be one minus the probability that x is between 9 .85 and 10 .15.

01:51 Because those are the units that are okay.

01:56 Since it's normally distributed, we can use a z score, but instead of actually putting stuff into the z score formula, let's think about what a z score means.

02:05 It means the number of standard deviations away from the mean.

02:07 Well, let's see.

02:09 9 .85 is one standard deviation less than the mean, and 10 .15 is one standard deviation greater than the mean.

02:21 So our z scores are a negative 1 in 1.

02:25 So this is going to equal 1 minus probability that z is less than or equal to 1 minus the probability that z is less than or equal to negative 1...

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