The random variable x has a normal distribution with standard deviation 25. It is known that the probability that x exceeds 150 is .90. Find the mean

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:

Where and

We know the following condition:

For this case we can use the z score formula given by:

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:

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:

And if we solve for the mean we got:

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