Respuesta :
Answer:
a) [tex]P(411<X<649)=P(\frac{411-\mu}{\sigma}<\frac{X-\mu}{\sigma}<\frac{649-\mu}{\sigma})=P(\frac{411-530}{119}<Z<\frac{649-530}{119})=P(-1<z<1)[/tex]
And we can find this probability with this difference:
[tex]P(-1<z<1)=P(z<1)-P(z<-1)[/tex]
And in order to find these probabilities we can use tables for the normal standard distribution, excel or a calculator.
[tex]P(-1<z<1)=P(z<1)-P(z<-1)=0.841-0.159=0.683[/tex]
And the percentage would be 68.3%
b) For this case we can find this with the complement rule and with the result of the part a and we got:
[tex] P(X<411 \cup X>649) = 1-P(411< X< 649) = 1-0.683 = 0.317[/tex]
And that represent 31.7%
c) [tex]P(X>768)=P(\frac{X-\mu}{\sigma}>\frac{768-\mu}{\sigma})=P(Z>\frac{768-530}{119})=P(z>2)[/tex]
And we can find this probability with the complment rule like this:
[tex]P(z>2)=1-P(z<2)[/tex]
And in order to find these probabilities we can use tables for the normal standard distribution, excel or a calculator.
[tex]P(z>2)=1-P(z<2)= 1-0.977=0.02275 [/tex]
And the percentage would be 2.3%
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".
Part a
Let X the random variable that represent the standardized test's
Where [tex]\mu=530[/tex] and [tex]\sigma=119[/tex]
We are interested on this probability
[tex]P(411<X<649)[/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(411<X<649)=P(\frac{411-\mu}{\sigma}<\frac{X-\mu}{\sigma}<\frac{649-\mu}{\sigma})=P(\frac{411-530}{119}<Z<\frac{649-530}{119})=P(-1<z<1)[/tex]
And we can find this probability with this difference:
[tex]P(-1<z<1)=P(z<1)-P(z<-1)[/tex]
And in order to find these probabilities we can use tables for the normal standard distribution, excel or a calculator.
[tex]P(-1<z<1)=P(z<1)-P(z<-1)=0.841-0.159=0.683[/tex]
And the percentage would be 68.3%
Part b
For this case we can find this with the complement rule and with the result of the part a and we got:
[tex] P(X<411 \cup X>649) = 1-P(411< X< 649) = 1-0.683 = 0.317[/tex]
And that represent 31.7%
Part c
[tex]P(X>768)[/tex]
Using the z score we got:
[tex]P(X>768)=P(\frac{X-\mu}{\sigma}>\frac{768-\mu}{\sigma})=P(Z>\frac{768-530}{119})=P(z>2)[/tex]
And we can find this probability with the complment rule like this:
[tex]P(z>2)=1-P(z<2)[/tex]
And in order to find these probabilities we can use tables for the normal standard distribution, excel or a calculator.
[tex]P(z>2)=1-P(z<2)= 1-0.977=0.02275 [/tex]
And the percentage would be 2.3%
