Respuesta :
Answer:
[tex]P(X<5.0)=P(\frac{X-\mu}{\sigma}<\frac{5.0-\mu}{\sigma})=P(Z<\frac{5.0-5.43}{0.54})=P(Z<-0.796)[/tex]
And we can find this probability using the normal standard distribution or excel:
[tex]P(Z<-0.796)=0.213[/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 PH of a population, and for this case we know the distribution for X is given by:
[tex]X \sim N(5.43,0.54)[/tex]
Where [tex]\mu=5.43[/tex] and [tex]\sigma=0.54[/tex]
We are interested on this probability
[tex]P(X<5.0)[/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<5.0)=P(\frac{X-\mu}{\sigma}<\frac{5.0-\mu}{\sigma})=P(Z<\frac{5.0-5.43}{0.54})=P(Z<-0.796)[/tex]
And we can find this probability using the normal standard distribution or excel:
[tex]P(Z<-0.796)=0.213[/tex]
