The GMAT Scores Of All Examinees Who Took That Test This Yea

The Gmat Scores Of All Examinees Who Took That Test This Year Produce

The GMAT scores of all examinees who took that test this year produce a distribution that is approximately normal with a mean of 430 and a standard deviation of 35. The probability that the score of a randomly selected examinee is between 400 and 480, rounded to three decimal places, is: The probability that the score of a randomly selected examinee is less than 370, rounded to three decimal places, is: The probability that the score of a randomly selected examinee is more than 530, rounded to three decimal places, is:

Paper For Above instruction

The GMAT (Graduate Management Admission Test) is a standardized exam widely used for admission to business schools worldwide. The scores on the GMAT follow a roughly normal distribution, which allows for the application of statistical principles such as the empirical rule and standard normal distribution calculations. In this analysis, we aim to determine specific probabilities associated with GMAT scores, given the mean and standard deviation of the distribution.

Understanding the Distribution of GMAT Scores

The problem states that GMAT scores in a specific year are approximately normally distributed with a mean (μ) of 430 and a standard deviation (σ) of 35. This implies that most scores cluster around the mean, with a decreasing likelihood as scores deviate further from it. The properties of the normal distribution enable us to calculate probabilities associated with certain score ranges using z-scores, which standardize individual scores relative to the distribution.

Calculations of Probabilities

To facilitate these calculations, we convert raw scores into z-scores using the formula:

z = (X - μ) / σ

where X is the raw score, μ is the mean, and σ is the standard deviation.

Probability that a score falls between 400 and 480

First, calculate the z-scores for scores 400 and 480:

  • For X = 400:
  • z = (400 - 430) / 35 = -30 / 35 ≈ -0.857
  • For X = 480:
  • z = (480 - 430) / 35 = 50 / 35 ≈ 1.429

Next, using standard normal distribution tables or a calculator, find the probabilities corresponding to these z-scores:

  • P(Z
  • P(Z

To find the probability that a score is between 400 and 480, subtract the lower cumulative probability from the higher:

P(400

Rounded to three decimal places, the probability is 0.728.

Probability that a score is less than 370

Calculate the z-score for X = 370:

z = (370 - 430) / 35 = -60 / 35 ≈ -1.714

Find the probability corresponding to z = -1.714:

P(Z

Thus, the probability that a randomly selected examinee scores less than 370 is approximately 0.043 after rounding to three decimal places.

Probability that a score is more than 530

Calculate the z-score for X = 530:

z = (530 - 430) / 35 = 100 / 35 ≈ 2.857

Find the probability for Z

P(Z

Therefore, the probability that a score exceeds 530 is:

P(X > 530) = 1 - P(Z

Rounded to three decimal places, it is 0.002.

Conclusion

This statistical analysis demonstrates the utility of the normal distribution approximation in assessing probabilities related to GMAT scores. Recognizing the distribution parameters allows for precise probability calculations for various score ranges, aiding test takers, educators, and policymakers in understanding performance patterns.

References

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