Respuesta :
Answer:
[tex]P(37.5<X<38)=P(\frac{37.5-\mu}{\sigma}<\frac{X-\mu}{\sigma}<\frac{38-\mu}{\sigma})=P(\frac{37.5-37.8}{0.12}<Z<\frac{38-37.8}{0.12})=P(-2.5<z<1.67)[/tex]
And we can find this probability with this difference:
[tex]P(-2.5<z<1.67)=P(z<1.67)-P(z<-2.5) [/tex]
And in order to find these probabilities we can use tables for the normal standard distribution, excel or a calculator.
[tex]P(-2.5<z<1.67)=P(z<1.67)-P(z<-2.5)=0.952-0.0062=0.946 [/tex]
So then the percentage usable are 94,6%
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 diameters of a population, and for this case we know the distribution for X is given by:
[tex]X \sim N(37.8,0.12)[/tex]
Where [tex]\mu=37.8[/tex] and [tex]\sigma=0.12[/tex]
We are interested on this probability
[tex]P(37.5<X<38)[/tex]
And the best way to solve this problem is using the normal standard distribution and the z score given by:
[tex]z=\frac{x-\mu}{\sigma}[/tex]
If we apply this formula to our probability we got this:
[tex]P(37.5<X<38)=P(\frac{37.5-\mu}{\sigma}<\frac{X-\mu}{\sigma}<\frac{38-\mu}{\sigma})=P(\frac{37.5-37.8}{0.12}<Z<\frac{38-37.8}{0.12})=P(-2.5<z<1.67)[/tex]
And we can find this probability with this difference:
[tex]P(-2.5<z<1.67)=P(z<1.67)-P(z<-2.5) [/tex]
And in order to find these probabilities we can use tables for the normal standard distribution, excel or a calculator.
[tex]P(-2.5<z<1.67)=P(z<1.67)-P(z<-2.5)=0.952-0.0062=0.946 [/tex]
So then the percentage usable are 94,6%
