Answer :
Answer:
The median of the MCAT scores was of 500.
The first quartile of MCAT scores was of 492.845.
The third quartile of MCAT scores was of 507.155.
Step-by-step explanation:
When the distribution is normal, we use the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
The mean score was 500.0 with a standard deviation of 10.6.
This means that [tex]\mu = 500, \sigma = 10.6[/tex]
Median:
In a normal distribution, the median is the same as the mean, so the median of the MCAT scores was of 500.
First quartile:
This is the 100*(1/4) = 25th percentile, which is X when Z has a pvalue of 0.25. So X when Z = -0.675.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-0.675 = \frac{X - 500}{10.6}[/tex]
[tex]X - 500 = -0.675*10.6[/tex]
[tex]X = 492.845[/tex]
The first quartile of MCAT scores was of 492.845.
Third quartile:
This is the 100*(3/4) = 75th percentile, which is X when Z has a pvalue of 0.75. So X when Z = 0.675.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]0.675 = \frac{X - 500}{10.6}[/tex]
[tex]X - 500 = 0.675*10.6[/tex]
[tex]X = 507.155[/tex]
The third quartile of MCAT scores was of 507.155.