Respuesta :
Answer:
[tex]P(X>220)=P(\frac{X-\mu}{\sigma}>\frac{220-\mu}{\sigma})=P(Z>\frac{220-160}{55})=P(Z>1.09)[/tex]
And we can find this probability on this way using the complement rule and the standard normal table or excel:
[tex]P(Z>1.09)=1-P(z<1.09)=1-0.862=0.138 [/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 number of gallons for a population, and for this case we know the distribution for X is given by:
[tex]X \sim N(160,55)[/tex]
Where [tex]\mu=160[/tex] and [tex]\sigma=55[/tex]
We are interested on this probability
[tex]P(X>220)[/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>220)=P(\frac{X-\mu}{\sigma}>\frac{220-\mu}{\sigma})=P(Z>\frac{220-160}{55})=P(Z>1.09)[/tex]
And we can find this probability on this way using the complement rule and the standard normal table or excel:
[tex]P(Z>1.09)=1-P(z<1.09)=1-0.862=0.138 [/tex]
