Answer:
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 grades of a population, and for this case we know the distribution for X is given by:
For this case we have two conditions given:
or equivalently
And the best way to solve this problem is using the normal standard distribution and the z score given by:
So we can find a value from the normal standard distribution that accumulates 0.1 and 0.9 of the area in the left, for this case the two values are:
We can verify that P(Z<-1.28) =0.1[/tex] and P(Z<1.28) =0.9[/tex]
And then using the z score we have the following formulas:
(1)
(2)
If we add equations (1) and (2) we got:
We can multiply both sides of the equation by and we got:
And then we can find the standard deviation for example from equation (1) and we got:
So then the answer would be:
