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Chapter 3: Problem 54

Grading Managers. In Exercise 3.44, we saw that Ford Motor Company once graded its managers in such a way that the top \(10 \%\) received an A grade, the bottom \(10 \%\) a C, and the middle \(80 \%\) a B. Let's suppose that performance scores follow a Normal distribution. How many standard deviations above and below the mean do the \(A / B\) and \(B / C\) cutoffs lie? (Use the standard Normal distribution to answer this question.)

Short Answer

Expert verified
The A/B cutoff is 1.28 SD above the mean; the B/C cutoff is -1.28 SD below the mean.

Step by step solution

01

Understanding the Problem

We are given a normal distribution and need to find the cutoff points in terms of standard deviations from the mean for grading categories: A, B, and C. Specifically, we need to determine the cutoff for the top 10% and the bottom 10% of the distribution.
02

Use Standard Normal Distribution

The performance scores are assumed to follow a standard normal distribution (mean = 0, standard deviation = 1). Our task is to find the z-scores that correspond to the 10th and 90th percentiles.
03

Locate the Percentiles

Consult the standard normal distribution table or use a calculator to find the z-scores. - The 90th percentile (top 10%) corresponds to a z-score of approximately 1.28. - The 10th percentile (bottom 10%) corresponds to a z-score of approximately -1.28.
04

Conclude the Cutoffs

Therefore, the cutoff for an A grade (above the 90th percentile) is 1.28 standard deviations above the mean, and the cutoff for a C grade (below the 10th percentile) is -1.28 standard deviations below the mean.

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Key Concepts

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

Standard Deviation
Standard deviation is a core concept in statistics that measures the amount of variation or dispersion in a set of data. When you deal with a normal distribution, understanding standard deviation helps you understand how scores are spread around the mean. Here are some key points:
  • A smaller standard deviation indicates that the data points are closer to the mean.
  • A larger standard deviation suggests a wide range of values.
  • In a normal distribution, about 68% of values fall within one standard deviation from the mean, 95% within two, and 99.7% within three standard deviations.

Standard deviation is especially useful when comparing two different datasets to see which one has more variability.
Z-score
The Z-score, or standard score, is a numerical measurement used in statistics that describes a value's relationship to the mean of a group of values, measured in terms of standard deviations. Here's how it works:
  • A Z-score of 0 indicates the score is exactly at the mean.
  • A positive Z-score means the value is above the mean, while a negative score indicates it is below the mean.
  • Z-scores enable comparisons between scores from different distributions by converting them into a common scale.
  • To find a Z-score, use the formula: \( Z = \frac{(X - \mu)}{\sigma} \), where \( X \) is the value, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.

By calculating Z-scores, you determine how far and in what direction, a point deviates from the average, making it easier to interpret or compare results.
Percentiles
Percentiles are values that divide a dataset into 100 equal parts, making it easier to interpret positions or rankings of numbers within the dataset. Here's what you should know:
  • The 50th percentile is the median, which is the middle value.
  • If you score in the 90th percentile, you performed better than 90% of the other values.
  • Percentiles are often used in education to assess student performance.

When dealing with normal distributions, particular percentiles can be associated with specific Z-scores. This relationship allows you to deduce how high or low a score is relative to others in the group.
Grading Distribution
Grading distribution, particularly in standardized settings, often uses concepts of normal distribution, percentiles, and Z-scores to ensure consistency and fairness. Here鈥檚 how this can be applied:
  • Grades are often assigned by quantifying how many standard deviations a student's performance is from the mean.
  • In our example, grades are divided into top 10%, middle 80%, and bottom 10% based on performance.
  • The standard normal distribution helps educators set cutoff points for these grades by using Z-scores associated with particular percentiles (e.g., 90th and 10th percentiles for A and C grades respectively).

This method helps in consistently evaluating different batches of students, thereby providing a structure where performance is reflected not just by the raw scores but their standardized position within the group.

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Most popular questions from this chapter

The scores of adults on an IQ test are approximately Normally distributed, with mean 100 and standard deviation 15. Alysha scores 135 on such a test. Her \(z\)-score is about a. \(1.33 .\) b. \(2.33\) c. \(6.33 .\)

Perfect SAT Scores. It is possible to score higher than 1600 on the combined mathematics and evidence-based reading and writing portions of the SAT, but scores 1600 and above are reported as 1600 . The distribution of SAT scores (combining Mathematics and Reading) in 2019 was close to Normal, with mean 1059 and standard deviation 210. What proportion of SAT scores for these two parts were reported as 1600? (That is, what proportion of SAT scores were actually 1600 or higher?)

Upper Arm Lengths. Anthropomorphic data are measurements on the human body that can track growth and weight of infants and children and evaluate changes in the body that occur over the adult life span. The resulting data can be used in areas as diverse as ergonomics and clothing design. The upper arm length of males over 20 years old in the United States is approximately Normal with mean \(39.1\) centimeters \((\mathrm{cm})\) and standard deviation \(5.0 \mathrm{~cm}\). Draw a Normal curve on which this mean and standard deviation are correctly located. (Hint: Draw an unlabeled Normal curve, locate the points where the curvature changes, then add number labels on the horizontal axis.)

The distribution of hours of sleep per weeknight among college students is found to be Normally distributed, with a mean of \(6.5\) hours and a standard deviation of 1 hour. The percentage of college students that sleep at least eight hours per weeknight is about a. \(95 \%\) b. \(6.7 \%\) c. \(2.5 \%\)

Where Are the Quartiles? How many standard deviations above and below the mean do the quartiles of any Normal distribution lie? (Use the standard Normal distribution to answer this question.)
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