Question

The distribution of a sample of the outside diameters of PVC pipes approximates a normal distribution. The mean is 14.0 inches, and the standard deviation is 0.1 inches. About 68% of the outside diameters lie between what two amounts? A. 13.9 and 14.1 inches B. 13.0 and 15.0 inches C. 13.8 and 14.2 inches D. 13.5 and 14.5 inches

Answer 1

A. 13.9 and 14.1 inches

See explanation below.

Explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".

Solution to the problem

Let X the random variable that represent the outside diameters of a population, and for this case we know the distribution for X is given by:

`X \sim N(14,0.1)`

Where
`\mu=14` and
`\sigma=0.1`

If we want the middle 68% of the data we need to have on the tails 16% on each one

For this part we want to find a value a, such that we satisfy this condition:

`P(X>a)=0.84` (a)

`P(X<a)=0.16` (b)

Both conditions are equivalent on this case. We can use the z score again in order to find the value a.

As we can see on the figure attached the z value that satisfy the condition with 0.16 of the area on the left and 0.84 of the area on the right it's z=-0.994. On this case P(Z<-0.994)=0.16 and P(z>-0.994)=0.84

If we use condition (b) from previous we have this:

`P(X<a)=P((X-\mu)/(\sigma)<(a-\mu)/(\sigma))=0.16`

`P(z<(a-\mu)/(\sigma))=0.16`

But we know which value of z satisfy the previous equation so then we can do this:

`z=-0.994<(a-14)/(0.1)`

And if we solve for a we got

`a=14 -0.994*0.1=13.9`

And since the distribution is symmetrical for the upper limit we can use z = 0.994 and we have:

`z=0.994<(a-14)/(0.1)`

And if we solve for a we got

`a=14 +0.994*0.1=14.1`

So the correct answer for this case would be:

A. 13.9 and 14.1 inches