Respuesta :
Answer:
a) [tex]P(X<7)=P(\frac{X-\mu}{\sigma}<\frac{7-\mu}{\sigma})=P(Z<\frac{7-12}{2})=P(z<-2.5)[/tex]
And we can find this probability using the normal standard distirbution or excel and we got:
[tex]P(z<-2.5)=0.0062[/tex]
b) [tex]P(7<X<12)=P(\frac{7-\mu}{\sigma}<\frac{X-\mu}{\sigma}<\frac{12-\mu}{\sigma})=P(\frac{7-12}{2}<Z<\frac{12-12}{2})=P(-2.5<z<0)[/tex]
And we can find this probability with this difference:
[tex]P(-2.5<z<0)=P(z<0)-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<0)=P(z<0)-P(z<-2.5)=0.5-0.0062=0.494[/tex]
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".
a) less than 7 months.
Let X the random variable that represent the length of life of an instrument of a population, and for this case we know the distribution for X is given by:
[tex]X \sim N(12,2)[/tex]
Where [tex]\mu=12[/tex] and [tex]\sigma=2[/tex]
We are interested on this probability
[tex]P(X<7)[/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(X<7)=P(\frac{X-\mu}{\sigma}<\frac{7-\mu}{\sigma})=P(Z<\frac{7-12}{2})=P(z<-2.5)[/tex]
And we can find this probability using the normal standard distirbution or excel and we got:
[tex]P(z<-2.5)=0.0062[/tex]
b) between 7 and 12 months.
[tex]P(7<X<12)=P(\frac{7-\mu}{\sigma}<\frac{X-\mu}{\sigma}<\frac{12-\mu}{\sigma})=P(\frac{7-12}{2}<Z<\frac{12-12}{2})=P(-2.5<z<0)[/tex]
And we can find this probability with this difference:
[tex]P(-2.5<z<0)=P(z<0)-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<0)=P(z<0)-P(z<-2.5)=0.5-0.0062=0.494[/tex]
