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d1i1m1o1n
15 days ago
12

A curious student in a large economics course is interested in calculating the percentage of his classmates who scored lower tha

n he did on the GMAT; he scored 490. He knows that GMAT scores are normally distributed and that the average score is approximately 540. He also knows that 95% of his classmates scored between 400 and 680. Based on this information, calculate the percentage of his classmates who scored lower than he did?
Mathematics
1 answer:
zzz [4K]15 days ago
6 0

Answer: 23.89%

Detailed solution:

The empirical rule indicates that about 95% of data points lie within two standard deviations from the mean.

Given: The GMAT scores follow a normal distribution with the mean near \mu=540.

Additionally, 95% of classmates scored between 400 and 680.

Applying the empirical rule, 95% of scores are within mu\pm 2\sigma,

which corresponds to 540- 2\sigma=400  (1)

and 540+2\sigma=680  (2)

Subtracting (1) from (2) yields:

4\sigma=680-400=280\\\\\Rightarrow\ \sigma=\dfrac{280}{4}=70

Let x symbolize the scores of students.

z-score formula: z=\dfrac{x-\mu}{\sigma}

For x = 490, calculate

z=\dfrac{490-540}{70}\approx-0.71

Then, the p-value equals: P(z

This means that 23.89% of his peers scored below his score.

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