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

A. 13.9 and 14.1 inches

See explanation below.

Step-by-step 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:

Where and

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:

  (a)

  (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:

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

And if we solve for a we got

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

And if we solve for a we got

So the correct answer for this case would be:

A. 13.9 and 14.1 inches

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step-by-step explanation: