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Answer :
Answer:
1. 90.49% of test takers scored less than Alison.
2.
Kamal scored in the 82th percentile.
Kamal's score was of 510.8.
3. 15.52% of test takers scored between 505 and 510 on the MCAT.
Step-by-step explanation:
Z-score:
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.
For the current version of the MCAT, the population mean is 501.1 and the population standard deviation is 10.6 points.
This means that [tex]\mu = 501.1, \sigma = 10.6[/tex]
1. Alison scored 515 on the MCAT. What is the percentage of test takers who scored less than Alison (what is the percentile rank associated with a raw score of 515)?
The proportion is the pvalue of Z when X = 515. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{515 - 501.1}{10.6}[/tex]
[tex]Z = 1.31[/tex]
[tex]Z = 1.31[/tex] has a pvalue of 0.9049
0.9049*100% = 90.49%
90.49% of test takers scored less than Alison.
2. Kamal scored higher than 82% of the test takers. What is Kamal's percentile—what is Kamal's score on the MCAT?
Kamal scored in the 82th percentile.
His score is the value of Z when Z has a pvalue of 0.82. So X when Z = 0.915.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]0.915 = \frac{X - 501.1}{10.6}[/tex]
[tex]X - 501.1 = 0.915*10.6[/tex]
[tex]X = 510.8[/tex]
Kamal's score was of 510.8.
3. What percentage of test takers scored between 505 and 510 on the MCAT?
The proportion is the pvalue of Z when X = 510 subtracted by the pvalue of Z when X = 505. So
X = 510
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{510 - 501.1}{10.6}[/tex]
[tex]Z = 0.84[/tex]
[tex]Z = 0.84[/tex] has a pvalue of 0.7995
X = 505
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{505 - 501.1}{10.6}[/tex]
[tex]Z = 0.37[/tex]
[tex]Z = 0.37[/tex] has a pvalue of 0.6443
0.7995 - 0.6443 = 0.1552
0.1552*100% = 15.52%
15.52% of test takers scored between 505 and 510 on the MCAT.
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