Question

A history teacher assigns letter grades on a test according to the following "Percentile" scheme:

A: Top 10%
B: Scores below the top 10% and above the bottom 60%
C: Scores below the top 40% and above the bottom 20%C: Scores below the top 40% and above the bottom 20%
D: Scores below the top 20% and above the bottom 10%
F: Bottom 10%
Scores on the test are normally distributed with a mean of 67 and a standard deviation of 13.2.
Find the numerical limits for each letter grade.

Answer 1

To determine the numerical limits for letter grades based on percentile ranks in a normally distributed set of scores with a mean of 67 and a standard deviation of 13.2, use the normal distribution function to find the specific test scores that correspond to the percentile cutoffs for each grade category.

Step-by-step explanation:

To find the numerical limits for each letter grade in a normal distribution, we need to identify the percentile ranks associated with each grade category according to the history teacher's grading scheme. The test scores are given as normally distributed with a mean (average) of 67 and a standard deviation of 13.2.

Steps to calculate the numerical limits for each grade:

1. Identify the percentile rank for each letter grade category:
   
2. Use the normal distribution function to find the z-scores corresponding to the above percentile ranks.
3. Convert z-scores into test scores using the formula: X = μ + z(σ), where X is the test score, μ is the mean, and σ is the standard deviation.

Calculating the score limits for each grade using the normal distribution involves using statistical tools like tables, computers or calculators that can provide the value of the percentile rank corresponding to a particular z-score (such as the TI-83, 83+, 84, 84+ Calculator). By using the invNorm function on a TI calculator for example, you can find the exact scores that correspond to these percentile cutoffs.