Across The USA, Results For These Exams Are Normally Distrib

Across the USA, results for these exams are normally distributed

Review the student post and evaluate their solutions for using the .7 (Empirical) Rule to determine the percentage of GRE scores between 350 and 650. Are the student’s calculations correct? If yes, note this and if not correct them with an example.

Next, explain to the student why 50% of the scores are above 500 and why 50% are below (approximately).

Review the student’s GRE score choice from number 4 above. Are the student calculations correct? Include the student’s calculations in your response and note any issues if discovered. Then, offer the student a second example using any other value between 300 and 500. Be sure to explain all the steps in your example to the student and to show all work.

Paper For Above instruction

Understanding the distribution of GRE scores across the United States is essential for interpreting test results and assessing student performance. The assumption that GRE scores are normally distributed reflects that most scores cluster around the average, with fewer scores appearing as they deviate significantly from the mean. This pattern results from the central limit theorem, which states that the sum of a large number of independent variables tends to follow a normal distribution, especially when the variables represent diverse and random factors influencing test performance.

When creating a histogram of all GRE scores, the resulting shape would indeed resemble a bell curve—symmetric and unimodal. The highest point of the histogram would coincide with the mean score, signifying that most test-takers score near this value. This symmetry implies that the distribution has only one mode (peak). Additionally, the histogram should be balanced on both sides, indicating minimal skewness—either to the right (positive skew) or to the left (negative skew). Such a bell-shaped curve confirms the normal distribution characteristic of standardized test scores, validating the use of statistical tools like the empirical rule for analysis.

Applying the empirical rule, with a mean of 500 and a standard deviation of 75, reveals that approximately 95% of scores range from 350 to 650—spanning within two standard deviations from the mean. This is because, according to the empirical rule, about 68% of data falls within one standard deviation (425 to 575), and about 95% within two (350 to 650). Moreover, about 50% of students are expected to score above 500, the mean, and an equivalent percentage below, assuming a symmetric distribution.

A score below 275, which is three standard deviations below the mean (since 3×75 = 225; 500 - 225 = 275), comprises approximately 0.15% of scores in the distribution—making it a significant outlier. Such a score indicates performance markedly below average, and statistically, it is considered a rare event. Therefore, a score of 275 suggests a test result far from the typical range and can be classified as significantly different from the mean, warranting further investigation into possible reasons or factors influencing such an outcome.

For the second example, consider a GRE score of 600. Calculating the z-score involves subtracting the mean from the score and dividing by the standard deviation: (600 - 500) / 75 = 100 / 75 ≈ 1.333. Using the z-table, a z-value of 1.33 corresponds to a cumulative probability of approximately 0.9082, meaning about 90.82% of students score below 600 and about 9.18% score higher. This example demonstrates the practical use of the z-score and the z-table for interpreting individual scores within the context of the overall distribution.

References

• Triola, M. F. (2018). Elementary Statistics (13th ed.). Pearson Education.
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• Moivre, A. (1733). The Doctrine of Chances. (Historical source on probability distributions)
• Hogg, R. V., Tanis, E. A., & Zimmerman, D. (2019). Probability and Statistical Inference (9th ed.). Pearson.
• U.S. News & World Report. (2020). Understanding GRE Score Distributions. https://www.usnews.com
• Educational Testing Service. (2021). GRE Quantitative Reasoning Score Conversion. ETS.org.
• Wasserman, L. (2004). All of Statistics: A Concise Course in Statistical Inference. Springer.
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