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Originally, Dr. Smith decides to curve his students' exam grades as follows. • Students whose scores are at or above the 90* percentile will receive an A. • Students whose scores are in the 80*_89 percentiles will receive a B. • Students whose scores are in the 70"_79" percentiles will receive a C. • Students whose scores are in the 60"_69 percentiles will receive a D. • Students whose scores are below the 60 percentile will receive an F. 3. Find the z-scores that correspond to the following percentiles. • 90" percentile • 80" percentile • 70* percentile • 60" percentile 4. Using that information, find the exam scores that correspond to the curved grading scale. Assume that the exam scores range from 0 to 100. (Round to the nearest whole Number)

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Answer:

Step-by-step explanation:

Finding z-scores for Percentiles

We can't pinpoint exact scores for each percentile without knowing the distribution of the exam grades (normal distribution is often assumed for curving). However, we can find the z-scores that correspond to each percentile using a standard normal distribution table or calculator.

Here's the approach:

Look up percentiles in a standard normal distribution table: Find the cumulative area (percentage) less than each percentile (90th, 80th, 70th, 60th) in the table.

Z-scores correspond to cumulative area: The z-score value you find will be the z-score that separates the given percentile from the lower portion of the distribution.

Estimated z-scores (may vary slightly depending on the table):

90th percentile: z ≈ 1.28

80th percentile: z ≈ 0.85

70th percentile: z ≈ 0.52

60th percentile: z ≈ 0.25

Finding Curved Grades Scores (Assuming Normal Distribution)

Now, we can estimate the exam scores corresponding to the curved grading scale using the z-scores and assuming a normal distribution of exam grades (scores range from 0 to 100):

Standard deviation (SD): A normal distribution requires a mean (average) and standard deviation (SD). Since the scores range from 0 to 100, we can assume a mean of 50 (center of the range) and estimate the SD based on the z-scores.

SD estimation: The z-scores define how many standard deviations a specific score falls from the mean (50). For example, the 90th percentile (z ≈ 1.28) is 1.28 standard deviations above the mean.

Calculate exam scores: We can use the z-score formula to estimate the exam scores for each percentile:

Score = Mean + (z-score * Standard Deviation)

Estimated Scores (rounded to nearest whole number):

90th percentile: Score ≈ 50 + (1.28 * SD) ≈ 50 + (1.28 * ~15) ≈ 79 (assuming SD around 15)

80th percentile: Score ≈ 50 + (0.85 * SD) ≈ 50 + (0.85 * ~15) ≈ 63

70th percentile: Score ≈ 50 + (0.52 * SD) ≈ 50 + (0.52 * ~15) ≈ 58

60th percentile: Score ≈ 50 + (0.25 * SD) ≈ 50 + (0.25 * ~15) ≈ 54

Important Note:

This is an estimation assuming a normal distribution and a specific standard deviation. The actual scores may vary depending on the real distribution of the grades.

If the professor has access to the class data (grades distribution), they can use a more precise method to determine the cut-off scores for each letter grade.

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