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Chapter 6: Problem 12

An incoming freshman took her college's placement exams in French and mathematics. In French, she scored 82 and in math 86 . The overall results on the French exam had a mean of 72 and a standard deviation of 8 , while the mean math score was 68 , with a standard deviation of \(12 .\) On which exam did she do better compared with the other freshmen?

Short Answer

Expert verified
She did better in math based on the z-scores.

Step by step solution

01

Understand the Problem

We need to compare the student's performance on two exams: French and Math. To determine which exam she did better on compared to others, we'll use the concept of z-scores, which measure how many standard deviations a score is from the mean of the distribution.
02

Calculate Z-Score for French

To calculate the z-score for the French exam, use the formula: \( z = \frac{X - \mu}{\sigma} \), where \( X \) is the student's score, \( \mu \) is the mean score, and \( \sigma \) is the standard deviation. Substituting in the given values for French: \( X = 82 \), \( \mu = 72 \), and \( \sigma = 8 \), we get: \[ z = \frac{82 - 72}{8} = 1.25 \].
03

Calculate Z-Score for Math

Repeat the process to calculate the z-score for the Math exam using the same formula. For Math: \( X = 86 \), \( \mu = 68 \), and \( \sigma = 12 \). Therefore, we have: \[ z = \frac{86 - 68}{12} = 1.5 \].
04

Compare Z-Scores

The z-score tells us how far each score is from the mean in terms of standard deviations. A higher z-score indicates a better performance relative to the mean of that exam. For French, the z-score is 1.25, and for Math, the z-score is 1.5. Thus, the student did better on the Math exam compared to her peers, as the z-score is higher.

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Key Concepts

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

Understanding Placement Exams
Placement exams are essential in determining where an incoming student stands academically in specific subjects.
These exams place students in the appropriate level of coursework, ensuring they are neither under-challenged nor overwhelmed.
They typically assess knowledge that should have been gained before entering college and help schools decide optimal class levels for each student.
  • Placement exams can cover a variety of subjects such as languages, mathematics, and sciences.
  • They serve as a diagnostic tool for educators to identify strengths and areas for improvement in students.
  • The outcome directs students to their starting point in a college curriculum.
Understanding how placement exams relate to overall academic performance is key for students.
While absolute scores are important, comparing scores to the performance of others is also crucial.
What is Standard Deviation?
Standard deviation is a measure of the amount of variation or dispersion in a set of values.
In the context of placement exams, it helps us understand the spread of students' scores and how any individual score compares to the group's scores.
A smaller standard deviation means scores are more closely clustered around the mean. A larger one shows that scores are spread out over a wider range of values.
  • Standard deviation provides an insight into the predictability and consistency of student performances.
  • It's used in conjunction with the mean to calculate z-scores, which shows the relative performance of individual scores in a distribution.
  • This measure is critical in educational settings, as it identifies the range within which most scores fall.
By understanding standard deviation, students and educators better interpret exam results and potential impacts on course placements.
Grasping Mean Score
The mean score is the average score of a set of numbers, calculated by adding all scores and dividing by the number of scores.
In placement exams, the mean score provides a central benchmark to which individual scores can be compared.
  • The mean helps establish what is considered a 'typical' score in a group.
  • It acts as a reference point for calculating z-scores, offering a way to assess an individual's performance relative to peers.
  • If a student scores above the mean, their performance is considered better than average.
Understanding the mean helps students view their position within a larger group context.
It is an essential component in assessing academic readiness and in planning educational paths.

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Most popular questions from this chapter

In the Normal model \(N(100,16)\), what cutoff value bounds a) the highest \(5 \%\) of all IQs? b) the lowest \(30 \%\) of the IQs? c) the middle \(80 \%\) of the IQs?

. Exercise 26 proposes modeling IQ scores with \(N(100,16) .\) What IQ would you consider to be unusually high? Explain.

A specialty foods company sells "gourmet hams" by mail order. The hams vary in size from \(4.15\) to \(7.45\) pounds, with a mean weight of 6 pounds and standard deviation of \(0.65\) pounds. The quartiles and median weights are \(5.6,6.2\), and \(6.55\) pounds. a) Find the range and the IQR of the weights. b) Do you think the distribution of the weights is symmetric or skewed? If skewed, which way? Why?

A company that manufactures rivets believes the shear strength (in pounds) is modeled by \(N(800,50)\). a) Draw and label the Normal model. b) Would it be safe to use these rivets in a situation requiring a shear strength of 750 pounds? Explain. c) About what percent of these rivets would you expect to fall below 900 pounds? d) Rivets are used in a variety of applications with varying shear strength requirements. What is the maximum shear strength for which you would feel comfortable approving this company's rivets? Explain your reasoning.

A popular band on tour played a series of concerts in large venues. They always drew a large crowd, averaging 21,359 fans. While the band did not announce (and probably never calculated) the standard deviation, which of these values do you think is most likely to be correct: \(20,200,2000\), or 20,000 fans? Explain your choice.
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