Reports on a student's ACT, SAT, or MCAT usually give the percentile as well as the actual score. The percentile is just the cumulative proportion stated as a percent: the percent of all scores that were lower than this one. In a certain year, the total MCAT scores were close to Normal with mean 25.2 and standard deviation 6.5. William scored 30. What was his percentile?

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".

Let X the random variable that represent the scors of a population, and for this case we know the distribution for X is given by:

[tex]X \sim N(25.2,6.5)[/tex]

Where [tex]\mu=25.2[/tex] and [tex]\sigma=6.5[/tex]

2) Solution to the problem

We are interested in find the percentile for the score of 30 obtained by William. The best way to solve this problem is using the z score formula given by:

[tex]z=\frac{X-\mu}{\sigma}[/tex]

And z follows a normal standard distribution. We can find the z scor for the score X=30 like this:

[tex]z=\frac{30-25.2}{6.5}=0.738[/tex]

And now we can find the probability using excel or a table, on this way:

[tex]P(Z<0.738)=0.7697[/tex]

And that means the 76.97 percentile since we have on the left of the score of 30, 0.7697 of the area on the left and 0.2303 of the area on the right.

So then the percentile for William is 76.97.