A machining operation produces bearings with diameters that...

A machining operation produces bearings with diameters that are normally distributed with mean 3.0005 inches and standard deviation. 0010 inch. Specifications require the bearing diameters to lie in the interval $3.000 \pm .0020$ inches. Those outside the interval are considered scrap and must be remachined. With the existing machine setting, what fraction of total production will be scrap? a. Answer the question, using Table 4, Appendix 3. b. Applet Exercise obtain the answer, using the applet Normal Probabilities.

A machining operation produces bearings with diameters that are normally distributed with mean 3.0005 inches and standard deviation. 0010 inch. Specifications require the bearing diameters
to lie in the interval $3.000 \pm .0020$ inches. Those outside the interval are considered scrap and must be remachined. With the existing machine setting, what fraction of total production will be scrap?
a. Answer the question, using Table 4, Appendix 3.
b. Applet Exercise obtain the answer, using the applet Normal Probabilities.

Key Concepts

Normal Distribution

A normal distribution is a continuous probability distribution characterized by a symmetric bell-shaped curve, where most observations cluster around the mean and the spread is determined by the standard deviation. It is commonly used to model processes and measurements in manufacturing and quality control because many natural variations tend to follow this pattern.

Standardization (Z-Score Calculation)

Standardization involves converting an observation from a normal distribution into a z-score by subtracting the mean and dividing by the standard deviation. This transformation allows for the use of standard normal distribution tables or software to calculate probabilities, making it easier to assess how far an observation deviates from the mean in standardized terms.

Probability Calculation over Intervals

In quality control, it is often necessary to calculate the probability that a measurement falls outside a specified tolerance interval. This involves using the cumulative distribution function (CDF) or standard normal tables to find the probability that a standardized measurement (z-score) is either below the lower limit or above the upper limit, then summing these probabilities to determine the overall defect rate or scrap fraction.

Quality Control and Process Capability

Quality control uses statistical methods to monitor, control, and improve manufacturing processes. Process capability assesses whether a process can produce output within specified limits. By comparing the distribution of produced measurements to the acceptable tolerance interval, manufacturers determine the fraction of production that meets quality standards versus the proportion considered scrap.

Transcript

00:01 Welcome to numerate.

00:02 So in the current problem we are given that there is a machining operation going on, which keeps on producing the bearings with measurements that follows normal distribution.

00:22 So x will be the measurement of diameters, right? so we will write x our variable diameters in inches of bearings produced by the machining operator blah blah now it is also mentioned that x follows normal distribution mean mu to be 3 .005 inches and the standard deviation to be 0 .00 h.

01:31 All right.

01:33 So, what will be probability that we will ask first we understand these bearings are not all perfect some of them are good some of them are bad so the bad ones that get scrapped or you know they get for remachining those will be the only acceptance area for x will be the interval 3 .00000 plus minus 3 .0 point 0 0 0000 plus minus 3 0 0 0 0 0 0 0 0000 okay which means x if it belongs to this region 3 .00 minus 0 .002 and 3 .00 plus 0 .002 we will accept.

02:39 That is if it becomes from 0 .998 to 3 .00.

02:53 So this is where we accept all the bearings.

02:58 Now, they are asking that what percentage or what probability using whatever approach will be accepted.

03:07 And because whatever approach we take, be it athlete or be it the appendix 3 tables, it will always be the same.

03:15 Now, we then will try to have a, have, we will try to visualize how our normal table will be.

03:24 So if you see, the mean is at 3 .005.

03:28 That means in the center of the data, we must at 3 .00005.

03:34 Then we know the next point will be at mu plus sigma.

03:39 So this point should be this value plus the sigma value.

03:43 Means 3 .0015.

03:48 The next point will be at mu plus two sigma.

03:52 That will be 3 .0025.

03:55 The next point will be mu plus 3 sigma.

04:00 Correct? so that will be 3 .0035.

04:05 So how would that normal table look? let's have a view.

04:08 I have already kept the diagram ready for you.

04:11 So if you look at this current.

04:16 You can see that the center is at 3 .005 correct over here then 3 .0015, 2 .0025, 2 .0025, 3 .0035.

04:30 And on the same note, the left side and say.

04:33 Now what is our acceptance region? our acceptance region is 2 .998 to 3 .002...

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