Respuesta :
Answer:
a) [tex]P(1<X<3)=0.7036[/tex]
b) [tex]P(X<4)=0.9991[/tex]
Step-by-step explanation:
1) 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".
2) Part a
Let X the random variable that represent the duration of a safe dosage of pain relief drug, and for this case we know the distribution for X is given by:
[tex]X \sim N(1.5,0.8)[/tex]
Where [tex]\mu=1.5[/tex] and [tex]\sigma=0.8[/tex]
We are interested on this probability
[tex]P(1<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(1<X<3)=P(\frac{1-\mu}{\sigma}<\frac{X-\mu}{\sigma}<\frac{3-\mu}{\sigma})=P(\frac{1-1.5}{0.8}<Z<\frac{3-1.5}{0.8})=P(-0.625<z<1.875)[/tex]
And we can find this probability on this way:
[tex]P(-0.625<z<1.875)=P(z<1.875)-P(z<-0.625)[/tex]
And in order to find these probabilities we can find tables for the normal standard distribution, excel or a calculator.
[tex]P(-0.625<z<1.875)=P(z<1.875)-P(z<-0.625)=0.9696-0.2660=0.7036[/tex]
3) Part b
We are interested on this probability
[tex]P(X<4)[/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<4)=P(\frac{X-\mu}{\sigma}<\frac{4-\mu}{\sigma})=P(Z<\frac{4-1.5}{0.8})=P(z<3.125)[/tex]
And we can find this probability on this way:
[tex]P(z<3.125)=0.9991[/tex]
