Answer:
And if we solve for a we got
So the value of height that separates the bottom 10% (Fastest 10%) of data from the top 90% is 393.2.
so if the time is 393.2 or lower a boy would earn a certificate of recognition from the fitness association
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 time for this event of a population, and for this case we know the distribution for X is given by:
Where 
and 
NOTE: For this case the fastest 10% is on the lower tail of the distribution
For this part we want to find a value a, such that we satisfy this condition:
(a)
(b)
Both conditions are equivalent on this case. We can use the z score again in order to find the value a.
As we can see on the figure attached the z value that satisfy the condition with 0.1 of the area on the left and 0.9 of the area on the right it's z=-1.28. On this case P(Z<-1.28)=0.1 and P(z>-1.28)=0.9
If we use condition (b) from previous we have this:
But we know which value of z satisfy the previous equation so then we can do this:
And if we solve for a we got
So the value of height that separates the bottom 10% (Fastest 10%) of data from the top 90% is 393.2.
so if the time is 393.2 or lower a boy would earn a certificate of recognition from the fitness association
