Answer:
[tex] \mu= 150 +1.28*25 =182[/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 a population, and for this case we know the distribution for X is given by:
[tex]X \sim N(\mu,25)[/tex]
Where [tex]\mu[/tex] and [tex]\sigma=25[/tex]
We know the following condition:
[tex] P(X>150) = 0.9[/tex]
For this case we can use the z score formula given by:
[tex]z=\frac{x-\mu}{\sigma}[/tex]
And we can find a z score that accumulates 0.9 of the area on the left and 0.1 on the right and this value is: [tex] z= -1.28[/tex]
Becuase P(Z<-1.28) =0.1 and P(Z>-1.28) = 0.9
So then if we use the z score formula we got:
[tex] \frac{150-\mu}{25} = -1.28[/tex]
And if we solve for the mean we got:
[tex] \mu= 150 +1.28*25 =182[/tex]
