Respuesta :
Answer:
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:
[tex]X \sim N(14,0.1)[/tex]
Where [tex]\mu=14[/tex] and [tex]\sigma=0.1[/tex]
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:
[tex]P(X>a)=0.84[/tex] (a)
[tex]P(X<a)=0.16[/tex] (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:
[tex]P(X<a)=P(\frac{X-\mu}{\sigma}<\frac{a-\mu}{\sigma})=0.16[/tex]
[tex]P(z<\frac{a-\mu}{\sigma})=0.16[/tex]
But we know which value of z satisfy the previous equation so then we can do this:
[tex]z=-0.994<\frac{a-14}{0.1}[/tex]
And if we solve for a we got
[tex]a=14 -0.994*0.1=13.9[/tex]
And since the distribution is symmetrical for the upper limit we can use z = 0.994 and we have:
[tex]z=0.994<\frac{a-14}{0.1}[/tex]
And if we solve for a we got
[tex]a=14 +0.994*0.1=14.1[/tex]
So the correct answer for this case would be:
A. 13.9 and 14.1 inches
