Respuesta :
Answer:
a) [tex]P(X<3)=P(\frac{X-\mu}{\sigma}<\frac{3-\mu}{\sigma})=P(Z<\frac{3-5.1}{0.9})=P(z<-2.33)[/tex]
And we can find this probability using the normal standard table and we got:
[tex]P(z<-2.33)=0.010[/tex]
b) [tex]P(X>7)=P(\frac{X-\mu}{\sigma}>\frac{7-\mu}{\sigma})=P(Z>\frac{7-5.1}{0.9})=P(z>2.11)[/tex]
And we can find this probability using the complement rule and the normal standard table and we got:
[tex]P(z>2.11)=1-P(Z<2.11) = 1-0.983 = 0.017[/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".
Solution to the problem
Let X the random variable that represent the variable of interest of a population, and for this case we know the distribution for X is given by:
[tex]X \sim N(5.1,0.9)[/tex]
Where [tex]\mu=5.1[/tex] and [tex]\sigma=0.9[/tex]
Part a
We are interested on this probability
[tex]P(X<3)[/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<3)=P(\frac{X-\mu}{\sigma}<\frac{3-\mu}{\sigma})=P(Z<\frac{3-5.1}{0.9})=P(z<-2.33)[/tex]
And we can find this probability using the normal standard table and we got:
[tex]P(z<-2.33)=0.010[/tex]
Part b
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{7-5.1}{0.9})=P(z>2.11)[/tex]
And we can find this probability using the complement rule and the normal standard table and we got:
[tex]P(z>2.11)=1-P(Z<2.11) = 1-0.983 = 0.017[/tex]
