91Ó°ÊÓ

A college senior who took the Graduate Record Examination exam scored 620 on the Verbal Reasoning section and 670 on the Quantitative Reasoning section. The mean score for Verbal Reasoning section was 462 with a standard deviation of \(119,\) and the mean score for the Quantitative Reasoning was 584 with a standard deviation of 151. Suppose that both distributions are nearly normal. (a) Write down the short-hand for these two normal distributions. (b) What is her \(\mathrm{Z}\) score on the Verbal Reasoning section? On the Quantitative Reasoning section? Draw a standard normal distribution curve and mark these two Z scores. (c) What do these \(Z\) scores tell you? (d) Relative to others, which section did she do better on? (e) Find her percentile scores for the two exams. (f) What percent of the test takers did better than her on the Verbal Reasoning section? On the Quantitative Reasoning section? (g) Explain why simply comparing her raw scores from the two sections would lead to the incorrect conclusion that she did better on the Quantitative Reasoning section. (h) If the distributions of the scores on these exams are not nearly normal, would your answers to parts (b) - (f) change? Explain your reasoning.

Step by step solution

01

Normal Distributions Shorthand

For the Verbal Reasoning section, the scores are normally distributed with a mean of 462 and a standard deviation of 119. This can be represented as: \( N(462, 119) \). Similarly, the Quantitative Reasoning scores are normally distributed with a mean of 584 and a standard deviation of 151, represented as \( N(584, 151) \).

02

Calculate Z-Scores

The \( Z \)-score is calculated using the formula \( Z = \frac{X - \mu}{\sigma} \), where \( X \) is the score, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.For Verbal Reasoning: \[ Z_{Verbal} = \frac{620 - 462}{119} \approx 1.33 \]For Quantitative Reasoning:\[ Z_{Quantitative} = \frac{670 - 584}{151} \approx 0.57 \]

03

Interpret Z-Scores

A \( Z \)-score tells us how many standard deviations an individual score is from the mean. * For the Verbal Reasoning section, a Z-score of 1.33 means she scored 1.33 standard deviations above the mean.* For the Quantitative Reasoning section, a Z-score of 0.57 means she scored 0.57 standard deviations above the mean.

04

Determine Better Performance

Comparing the two Z-scores, her score on the Verbal Reasoning is 1.33 standard deviations above the mean, which is better than her 0.57 standard deviations above the mean in Quantitative Reasoning.

05

Calculate Percentile Scores

Percentile scores can be found using a standard normal distribution table. * Verbal Z-score of 1.33 corresponds to approximately the 91st percentile. * Quantitative Z-score of 0.57 corresponds to approximately the 72nd percentile.

06

Percent Doing Better Than Her

Calculate what percent scored better than her by subtracting the percentiles from 100%. * On Verbal Reasoning, approximately 9% scored better (100% - 91%). * On Quantitative Reasoning, approximately 28% scored better (100% - 72%).

07

Misleading Raw Score Comparison

Raw scores alone do not reflect how well scores perform relative to the distribution. The Z-scores account for mean and variance, showing that Verbal Reasoning performance (1.33) is better than Quantitative Reasoning (0.57) despite the higher raw score in Quantitative.

08

Impact of Non-Normality on Results

If the distributions are not normal, Z-scores and percentile ranks might not be accurate, as these are based on the assumption of normality. The accuracy of conclusions in parts (b) to (f) would decrease if distributions were skewed or had outliers.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Normal Distribution

The Graduate Record Examination (GRE) scores mentioned in the problem follow a normal distribution. This is a common model in statistics that helps to predict how values are dispersed across a range. A normal distribution, also known as a bell curve due to its shape, is symmetrical about the mean. For any normal distribution, a few critical aspects give it its defined shape. These are the mean and the standard deviation.

• The mean is the central point of the distribution, showing where the data clusters.
• The standard deviation measures the spread of data around the mean.

For example, the Verbal Reasoning section has a mean (\(\mu\)) of 462 and a standard deviation (\(\sigma\)) of 119. In shorthand notation, it is expressed as \(N(462, 119)\). Similarly, the Quantitative Reasoning section is \(N(584, 151)\). This notation helps succinctly describe the distribution each set of scores follows. Understanding this concept is vital, as it forms the foundation for further statistical analysis like z-scores and percentiles.

Z-Scores

Z-scores are a fundamental concept in statistics, particularly in understanding standardized relationships between individual scores and the overall distribution. The z-score indicates how far and in what direction, an individual's score is from the mean, measured in standard deviations. The formula for calculating a z-score is: \[Z = \frac{X - \mu}{\sigma}\]where \(X\) is the individual's score, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.

• For the Verbal Reasoning score of 620, using the mean of 462 and standard deviation of 119, her z-score is approximately 1.33. This means her score is 1.33 standard deviations above the mean.
• For the Quantitative Reasoning score of 670, with a mean of 584 and a standard deviation of 151, her z-score is about 0.57. This indicates her score is 0.57 standard deviations above the mean.

These z-scores allow us to make direct comparisons even if the underlying distributions have different means and standard deviations, highlighting performance relative to peers.

Percentile Rank

Percentile rank is a statistical measure used to evaluate the relative standing of a value within a data set. It shows the percentage of scores that fall below a particular score. Knowing someone's percentile rank can help contextualize their performance in terms of how they compare to the rest of the test-takers.
For example, if you are told that a student's score is at the 91st percentile, this means they have scored better than 91% of the participants.

• For the Verbal Reasoning section, the calculated z-score of 1.33 translates to approximately the 91st percentile.
• For the Quantitative Reasoning section, the z-score of 0.57 corresponds to the 72nd percentile.

By knowing where a student stands in terms of percentile, one can understand how competitive their score is. It's crucial for students aiming to improve their scores, as higher percentile ranks also relate to better potential opportunities.

Statistical Analysis

Statistical analysis plays a crucial role when examining scores and performances. It helps translate raw data into meaningful insights. For test scores like those from the GRE, statistical analysis provides context by considering the distribution, mean, and variability of the data.

• Z-scores are a tool for statistical analysis, letting us transform a raw score into a standard score. This illustrates how far a score varies from the average.
• Percentile ranks, derived from z-scores, give interpretable insight into how a score stacks up against others. This is particularly useful for competitive exams like the GRE, where relative performance matters.
• Finally, comparing scores using statistical methods can prevent misleading conclusions. For example, even though a student's quantitative score is higher numerically, their performance relative to the mean is better in verbal reasoning.

Such analysis answers not just what the scores are but what they mean in a broader context. It provides a clearer picture and enables well-informed decisions regarding performance, growth areas, and potential outcomes.