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Chapter 3: Problem 4

Clayton and Timothy took different sections of Introduction to Economics. Each section had a different final exam. Timothy scored 83 out of 100 and had a percentile rank in his class of \(72 .\) Clayton scored 85 out of 100 but his percentile rank in his class was 70 . Who performed better with respect to the rest of the students in the class, Clayton or Timothy? Explain your answer.

Short Answer

Expert verified
Timothy performed better relative to his classmates, with a higher percentile rank of 72.

Step by step solution

01

Understanding Percentile Rank

Percentile rank represents the percentage of students that scored below a particular score. For example, a percentile rank of 72 means that 72% of the students scored lower than the student.
02

Analyzing Timothy's Performance

Timothy scored 83 out of 100 and had a percentile rank of 72. This indicates that Timothy performed better than 72% of his classmates.
03

Analyzing Clayton's Performance

Clayton scored 85 out of 100 and had a percentile rank of 70. This means that Clayton performed better than 70% of his classmates.
04

Comparing Percentile Ranks

Comparing the two percentile ranks, Timothy's rank of 72 is higher than Clayton's rank of 70. This indicates that Timothy's performance was better relative to his classmates, even though Clayton scored higher numerically.

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Key Concepts

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

Understanding Statistics
Statistics is a fascinating branch of mathematics that helps us collect, analyze, and interpret data.
It's not just about numbers but about understanding what these numbers tell us. In the context of evaluations, like exam scores, statistics show us trends and comparisons between different groups and individuals.

In academic settings, scores themselves are not the only important factor. It's crucial to see how one's score compares to others. This is done using percentile ranks, which are a vital part of statistics. These ranks allow us to interpret scores in a meaningful way by placing them in a broader context rather than interpreting the scores by themselves.
Explaining Percentile Interpretation
Percentile interpretation helps us understand where a particular score stands relative to all other scores in a group. A percentile rank tells us what percentage of scores fall below a particular score.
For example, a percentile rank of 72 indicates that a student scored better than 72% of their classmates.

This method of interpreting scores is incredibly useful, especially in scenarios where different tests or different classes might have different averages or score distributions.
  • Percentiles are more about the distribution of scores than the actual scores themselves.
  • High percentile doesn't always correlate with higher absolute scores.
This understanding can help in making decisions or assessments on an individual's performance more accurately, considering the context of their peers.
Student Performance Comparison
When comparing students like Clayton and Timothy, it's not just their test scores that matter, but how they rank compared to their peers.
Even if one student scores numerically higher, the other might have a better percentile rank, indicating better performance in relation to their classmates.

In this exercise, Timothy had a percentile rank of 72, while Clayton had a rank of 70. Although Clayton scored two points more on the exam, Timothy performed better in comparison to his peers.
  • Clayton's score of 85 was higher numerically.
  • Timothy's percentile rank of 72 shows broader performance excellence.
This situation highlights why understanding both score and percentile rank is crucial in education to properly gauge a student's standing in their class.

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Most popular questions from this chapter

In your biology class, your final grade is based on several things: a lab score, scores on two major tests, and your score on the final exam. There are 100 points available for each score. However, the lab score is worth \(25 \%\) of your total grade, each major test is worth \(22.5 \%\), and the final exam is worth \(30 \%\). Compute the weighted average for the following scores: 92 on the lab, 81 on the first major test, 93 on the second major test, and 85 on the final exam.

How large is a wolf pack? The following information is from a random sample of winter wolf packs in regions of Alaska, Minnesota, Michigan, Wisconsin, Canada, and Finland (Source: The Wolf, by L. D. Mech, University of Minnesota Press). Winter pack size: \(\begin{array}{rrrrrrrrr}13 & 10 & 7 & 5 & 7 & 7 & 2 & 4 & 3 \\ 2 & 3 & 15 & 4 & 4 & 2 & 8 & 7 & 8\end{array}\) Compute the mean, median, and mode for the size of winter wolf packs.

Given the sample data \(\begin{array}{llllll}x: & 23 & 17 & 15 & 30 & 25\end{array}\) (a) Find the range. (b) Verify that \(\Sigma x=110\) and \(\Sigma x^{2}=2568\). (c) Use the results of part (b) and appropriate computation formulas to compute the sample variance \(s^{2}\) and sample standard deviation \(s\). (d) Use the defining formulas to compute the sample variance \(s^{2}\) and sample standard deviation \(s\). (c) Suppose the given data comprise the entire population of all \(x\) values. Compute the population variance \(\sigma^{2}\) and population standard deviation \(\sigma\).

What symbol is used for the arithmetic mean when it is a sample statistic? What symbol is used when the arithmetic mean is a population parameter?

If you like mathematical puzzles or love algebra, try this! Otherwise, just trust that the computational formula for the sum of squares is correct. We have a sample of \(x\) values. The sample size is \(n\). Fill in the details for the following steps. $$ \begin{aligned} \Sigma(x-\bar{x})^{2} &=\Sigma x^{2}-2 \bar{x} \Sigma x+n \bar{x}^{2} \\ &=\Sigma x^{2}-2 n \bar{x}^{2}+n \bar{x}^{2} \\ &=\Sigma x^{2}-\frac{(\Sigma x)^{2}}{n} \end{aligned} $$
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