Respuesta :
Answer:
[tex]P(63<X<75)=P(\frac{63-\mu}{\sigma}<\frac{X-\mu}{\sigma}<\frac{75-\mu}{\sigma})=P(\frac{63-68}{3}<Z<\frac{75-68}{3})=P(-1.667<z<2.333)[/tex]
And we can find this probability with this difference:
[tex]P(-1.667<z<2.333)=P(z<2.333)-P(z<-1.667)[/tex]
And in order to find these probabilities we can using tables for the normal standard distribution, excel or a calculator.
[tex]P(-1.667<z<2.333)=P(z<2.333)-P(z<-1.667)=0.9902-0.0478=0.9424[/tex]
Step-by-step explanation:
Asuming this previous info: The Rockwell hardness of a metal is determined by impressing a hardened point into the surface of the metal and then measuring the depth of penetration of the point. Suppose the Rockwell hardness of a particular alloy is normally distributed with mean 68 and standard deviation 3.
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 hardness of a population, and for this case we know the distribution for X is given by:
[tex]X \sim N(68,3)[/tex]
Where [tex]\mu=68[/tex] and [tex]\sigma=3[/tex]
We are interested on this probability
[tex]P(63<X<75)[/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(63<X<75)=P(\frac{63-\mu}{\sigma}<\frac{X-\mu}{\sigma}<\frac{75-\mu}{\sigma})=P(\frac{63-68}{3}<Z<\frac{75-68}{3})=P(-1.667<z<2.333)[/tex]
And we can find this probability with this difference:
[tex]P(-1.667<z<2.333)=P(z<2.333)-P(z<-1.667)[/tex]
And in order to find these probabilities we can using tables for the normal standard distribution, excel or a calculator.
[tex]P(-1.667<z<2.333)=P(z<2.333)-P(z<-1.667)=0.9902-0.0478=0.9424[/tex]
