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A new version of the Medical College Admissions Test (MCAT) was introduced in spring 2015 and is intended to shift the focus from what applicants know to how well they can use what they know. One result of the change is that the scale on which the exam is graded was modified, with the total score of the four sections on the test ranging from 472 to 528. In spring 2015, the mean score was 500.0 with a standard deviation of 10.6.
1) Assuming that the MCAT scores are normally distributed, use Table to find the median and the first and third quartiles of the MCAT scores.
A) Find the median of the MCAT scores.
B) Find the first quartile of the MCAT scores.
C) Find the third quartile of the MCAT scores.
D) Find the interquartile range of the MCAT scores.
2) Which interval contains the central 80% of the MCAT scores?
a. 486.432 to 513.568.
b. 472 to 528.
c. 491.096 to 508.904.
d. 480 to 520.

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Answer:

Kindly check explanation

Step-by-step explanation:

Assume a normal distribution with:

Mean (m) = 500

Standard deviation (sd) = 10.6

A) The median of the MCAT scores :

For a normal distribution, the Median = mean = 50% = 0.5

Zscore at 0.5 = - 0.0

-0.0 = (x - m) / sd

-0.0 = (x - 500) / 10.6

0 = x - 500

x = 500

B) the first quartile (Q1) score = 25% = 0.25

Using :

Zscore at 25% = 0.25 = -0.675

Zscore = (X - m) / sd

-0.675 = (X - 500) / 10.6

-7.155 = x - 500

x = - 7.155 + 500

X = 492.845

C.) the third quartile (Q3) score = 75% = 0.75

Using :

Zscore at 75% = 0.25 = 0.675

Zscore = (X - m) / sd

0.675 = (X - 500) / 10.6

7.155 = x - 500

x = 7.155 + 500

X = 507.155

D.) The interquartile range :

Q3 - Q1

= 507.155 - 492. 845

= 14.31

2.) Interval which contains 80% of MCAT Scores:

Zscore of the (100-80)% / 2 at the extremes ; = 20%/2 = 0.1 ; 0.1 = - 1.28

Interval:

(-1.28 * sd) + mean and (1.28 * sd) + mean

(- 1.28 * 10.6) + 500 and (1.28 * 10.6) + 500

-13.568 + 500 and 13.568 + 500

486.432 and 513.568

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