What Percentage of Washers Falls Outside the Tolerance Range?

[ElBell]

Hi Everyone,

Question:

Mean is 0.252, standard deviation is 0.003, permitted tolerance of 0.246 to 0.258. What percentage of washers will be discarded.

Attempt:

I am so stuck. I have looked at my text and online for hours and know that it should be fairly easy but I am getting stuck with all the equations and what the symbols mean.

Can anybody please shed some light on this topic?

Any help would be very much appreciated

 

[Dickfore]

You need to find the following:

$$p = P(\{0.246 \le X \le 0.258\}) = P\left(\{\frac{0.246 - 0.252}{0.003} \le Z \le \frac{0.258 - 0.252}{0.003}\}\right)$$

where:

$$Z \equiv \frac{X - \mu}{\sigma}$$

is a normalized random variable with a $\mathcal{N}(0, 1)$. The probability of an event:

$$P(\{a \le Z \le b\}) = P(\{Z \le b\}) - P(\{Z \le b\}) = \Phi(b) -\Phi(a)$$

and the function $\Phi(z)$ is tabulated in any standard Statistics Table.

The ratio p that you would find is actually the ratio of accepted machines. What is the ratio of discarded machines? How can you convert it into percent?

 

[ZQrn]

Hi Everyone,

Question:

Mean is 0.252, standard deviation is 0.003, permitted tolerance of 0.246 to 0.258. What percentage of washers will be discarded.

Attempt:

I am so stuck. I have looked at my text and online for hours and know that it should be fairly easy but I am getting stuck with all the equations and what the symbols mean.

Can anybody please shed some light on this topic?

Any help would be very much appreciated

Okay, let me walk you through this:

What a mean is in a normal distribution is the expectancy value, basically see it like this 'If we repeat the experiment infinite times, the average value we expect is the mean', like, when we throw up a coin 'infinitely many' times, we averagely get heads 0.5 of the time, the expectancy value for head is thus 0.5. In a normal distribution, the mean is always the same as the expectancy value, but this needn't be true in other distributions.

The standard deviation is the expectancy value of how much we expect it to deviate from the mean. Basically the average deviation from the mean. For instance, say for sake of argument that human males are on average 1.80 m in height and the standard deviation is 5 cm. We then expect after meassuring a very large sample of human males that on average, they are 5 removed from 180 cm, as much below and above. This is thus distance, which cannot be negative, and not difference which can be negative.

Okay, the easiest to do this is to calculate what percentage makes it, and just get the complement to see which percentage gets discarded. A washer either makes it or gets discarded, it's binary situation, so we can use the complement rule.

So we have to know how many washers fall between [0.246, 0.258] when the average is 0.252 and we expect of a random washer to be 0.003 from the average, right?

Well, that number as a ratio of all washers, a percentage if you like, happens to be the surface area under our normal distribution from 0.246 to 0.258. This is what defines the normal distribution, the probability that a random item is between x and y is the surface area under the graph between x and y. Indeed, it's called normal distribution because the surface area under the entire graph is always exactly 1, even though it runs unbounded in both positive and negative. We say the integral converges if we take the limits to infinity, at some graphs, it diverges towards positive or minus infinity rather than a constant finite value.

So, assuming that f(mean,deviation) is our curve. As a normal distribution is determined completely by its mean and its standard deviation (given the symbols mu and sigma respectively), we have to integrate f(0.252,0.003) from 0.246 to 0.258.

http://en.wikipedia.org/wiki/Normal_distribution#Probability_density_function

And that page there explains better than I can hope to how to transform our normal curve into our cumulative normal curve, which is the primitive function we need to integrate it. (Most graphic calculators will do this for you)

Edit: remember, mean is given by mu, ($\mu$) standard deviation by sigma ($\sigma$)