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Specifications for a part for a DVD player state that the part should weigh between 25.1 and 26.1...

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Specifications for a part for a DVD player state that the part should weigh between 25.1 and 26.1 ounces. The process that produces the parts has a mean of 25.6 ounces and a standard deviation of .24 ounce. The distribution of output is normal. Use Table-A.

a. What percentage of parts will not meet the weight specs? (Round your "z" value and final answer to 2 decimal places. Omit the "%" sign in your response.) Percentage of parts %

b. Within what values will 99.74 percent of sample means of this process fall, if samples of n = 14 are taken and the process is in control (random)? (Round your answers to 2 decimal places.) Lower value ounces, Upper value ounces

Process Capability and Statistical Process Control

While the process capability is used to determine whether a process is capable of delivering output within the specifications, the statistical process control checks whether there are any special causes of variability affecting the process.

Answer and Explanation:

a. What percentage of parts will not meet the specs?

We start with finding the z-value which shows how many standard deviations of the process fall within the specs limits. We find it as

{eq}z=\frac{\text{Mean}-\text{Lower Specification}}{\text{St. Dev.}} \\ =\frac{25.6-25.1}{0.24} = 2.08 {/eq}

This z-value means that the output above z and below -z does not meet the specifications. Now we can find the probability that a random part will not meet the specs as

{eq}\text{NORM.S.DIST}(-z)+1-\text{NORM.S.DIST}(z)\\ =\text{NORM.S.DIST}(-2.08)+1- \text{NORM.S.DIST}(2.08) \\ = 0.0375 {/eq}

Therefore, {eq}3.75 {/eq} percent of parts will not meet the specs.

b. Control Limits for the Sample Means

We find the z-value for the probability of 0.9974 as

{eq}\text{NORM.S.INV}(0.9974)=2.79 {/eq}

Now we can find the range as

{eq}\text{Mean Value}\pm z \times \frac{\text{St. Dev.}}{\sqrt{n}} \\ =25.6 \pm 2.79 \times \frac{0.24}{\sqrt{14}}\\ =25.6 \pm 0.18 {/eq}

Therefore, the lower value is 25.42 ounces and the upper value is 25.78 ounces


Learn more about this topic:

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Process Capability: Definition & Elements

from Business 112: Operations Management

Chapter 4 / Lesson 5
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