Question

The national high school sports database reports that players of lacrosse average 57 goals per season. The standard deviation for goals per season is 24. To qualify for the national Junior Olympics a lacrosse player must be in the top 1% of the nation.
1. How many goals must a player score in one season in order to qualify for the Junior Olympic team?

Answer 1

Answer:

$z=2.33<\frac{a-57}{24}$

And if we solve for a we got

$a=57 +2.33*24=112.92$

So the value of height that separates the bottom 99% of data from the top 1% is 112.92.

So then the player need to score 112.92 or more in order to achieve the requirement.  

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 scores of a population, and for this case we know the distribution for X is given by:

$X \sim N(57,24)$  

Where $\mu=57$ and $\sigma=24$

We can solve this problem using the z score formula given by:

$z=\frac{x-\mu}{\sigma}$

For this part we want to find a value a, such that we satisfy this condition:

$P(X>a)=0.01$   (a)

$P(X<a)=0.99$   (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 0.99 of the area on the left and 0.01 of the area on the right it's z=2.33. On this case P(Z<2.33)=0.99 and P(z>2.33)=0.051

If we use condition (b) from previous we have this:

$P(X<a)=P(\frac{X-\mu}{\sigma}<\frac{a-\mu}{\sigma})=0.99$  

$P(z<\frac{a-\mu}{\sigma})=0.99$

But we know which value of z satisfy the previous equation so then we can do this:

$z=2.33<\frac{a-57}{24}$

And if we solve for a we got

$a=57 +2.33*24=112.92$

So the value of height that separates the bottom 99% of data from the top 1% is 112.92.

So then the player need to score 112.92 or more in order to achieve the requirement.