# Specifications for a part for a DVD player state that the part should weigh between 25.5 and 26.5...

## Question:

Specifications for a part for a DVD player state that the part should weigh between 25.5 and 26.5 ounces. The process that produces the parts has a mean of 26.0 ounces and a standard deviation of .22 ounce. The distribution of output is normal.. a. What percentage of parts will not meet the weight specs? (Round your "z" value and final answer to 2 decimal places. b. Within what values will 99.74 percent of sample means of this process fall, if samples of n = 12 are taken and the process is in control (random)?

## Process Capability

A process is considered capable if a large share of its output is in accordance with specifications. The share is usually determined as six standard deviations of the process. To increase the process capability, one needs to decrease its variance.

a. Finding the percentage of parts not meeting specification

The first step is to find the z-value which shows how many standard deviations of the process fall withing the specification limits. We can find it as

{eq}z=\frac{\text{Upper Specification}-\text{Lower Specification}}{\text{Standard Deviation}} \\ =\frac{26.5-25.5}{0.22} = 4.55 {/eq}

This z-value means that the output above z/2 and below -z/2 will not meet the specifications. Now we can find the probability that a random part will not meet the specifications as

{eq}p=\text{NORM.S.DIST}(-\frac{z}{2})+1-\text{NORM.S.DIST}(\frac{z}{2})\\ =2\times \text{NORM.S.DIST}(-\frac{z}{2})\\ =2\times \text{NORM.S.DIST}(-\frac{4.55}{2}) = 0.0229 {/eq}

Hence, {eq}2.29\% {/eq} of parts will not meet the specs.

b. Finding the Limits for the Mean

We find the z-value corresponding to 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}\pm z\times\frac{\text{Standard Deviation}}{\sqrt{n}} \\ =26\pm 2.79\times\frac{0.22}{\sqrt{12}}\\ =26 \pm 0.18 {/eq}

Hence, the means will fall in the range between 25.82 and 26.18