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

## Question:

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.

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