In one history class, 34 students took an exam. The mean of the exam was 52 with
a standard deviation of 13. If the instructor is going to assume data are
normally distributed and assign grades on a curve, then she will assign an A to
the top 10% of the class, B's to the next 15%, and C's students in the next 60%
of the students. What is the minimum score for A? What is the grade range for a
B? What is the grade range for a C?
To find the minimum score for an A, we need to determine the score that corresponds to the top 10% of the class.
Step 1: Convert the percentage into a Z-score using the standard normal distribution table. The top 10% corresponds to a Z-score of 1.28 (approximately).
Step 2: Use the Z-score formula to find the corresponding raw score:
Raw score = (Z-score * standard deviation) + mean
Raw score = (1.28 * 13) + 52
Raw score ≈ 68.64
So, the minimum score for an A is approximately 68.64.
Now let's find the grade range for a B, which includes the next 15% of the class.
Step 1: Convert the percentage into a Z-score using the standard normal distribution table. The next 15% corresponds to a Z-score of 0.68 (approximately).
Step 2: Use the Z-score formula to find the lower bound of the grade range:
Lower bound = (Z-score * standard deviation) + mean
Lower bound = (0.68 * 13) + 52
Lower bound ≈ 60.84
Step 3: Repeat step 2 to find the upper bound:
Upper bound = (Z-score * standard deviation) + mean
Upper bound = (1.28 * 13) + 52
Upper bound ≈ 68.64
So, the grade range for a B is approximately 60.84 to 68.64.
Finally, let's find the grade range for a C, which includes the next 60% of the class.
Step 1: Convert the percentage into a Z-score using the standard normal distribution table. The next 60% corresponds to a Z-score of -0.25 (approximately).
Step 2: Use the Z-score formula to find the lower bound of the grade range:
Lower bound = (Z-score * standard deviation) + mean
Lower bound = (-0.25 * 13) + 52
Lower bound ≈ 48.25
Step 3: Repeat step 2 to find the upper bound:
Upper bound = (Z-score * standard deviation) + mean
Upper bound = (0.68 * 13) + 52
Upper bound ≈ 60.84
So, the grade range for a C is approximately 48.25 to 60.84.
To find the minimum score for an A, we need to determine which score represents the top 10% of the class. Similarly, for the grade range of a B and C, we need to identify the scores that correspond to the top 25% and top 85% of the class, respectively.
1. Finding the minimum score for an A:
To determine the minimum score for an A, we need to find the score that correlates to the top 10% of the class. Since the data is assumed to be normally distributed, we can use the Z-scores to find this value.
Step 1: Find the Z-score corresponding to the top 10%.
The top 10% corresponds to the area under the curve to the left of the Z-score. We can use a Z-table or a calculator to find this value. In this case, we want to find the Z-score that leaves a 10% area to the left.
Step 2: Convert the Z-score to a raw score.
Once we have the Z-score, we can convert it to a raw score using the formula z = (x - μ) / σ, where x is the raw score, μ is the mean, and σ is the standard deviation.
Step 3: Calculate the minimum score for an A.
Use the formula x = (Z * σ) + μ to find the minimum score for an A.
2. Finding the grade range for a B:
To determine the grade range for a B, we need to find the scores corresponding to the top 10% to 25% of the class. Again, we can use Z-scores to find these values.
Step 1: Find the Z-scores corresponding to the top 10% and top 25%.
Similarly to step 1 above, find the Z-scores that capture the desired percentile ranges.
Step 2: Convert the Z-scores to raw scores.
Using the same formula as in Step 2 above, convert the Z-scores to raw scores.
Step 3: Calculate the grade range for a B.
Determine the minimum and maximum scores for a B by using the formula x = (Z * σ) + μ for both the top 10% and top 25% Z-scores.
3. Finding the grade range for a C:
To determine the grade range for a C, we need to find the scores corresponding to the top 25% to 85% of the class. Again, Z-scores can help us find these values.
Step 1: Find the Z-scores corresponding to the top 25% and top 85%.
As before, find the Z-scores related to the desired percentile ranges.
Step 2: Convert the Z-scores to raw scores.
Use the same formula as in Step 2 above to convert the Z-scores to raw scores.
Step 3: Calculate the grade range for a C.
Determine the minimum and maximum scores for a C by using the formula x = (Z * σ) + μ for both the top 25% and top 85% Z-scores.
By following the steps outlined above, you can find the minimum score for an A, the grade range for a B, and the grade range for a C. Good luck with your calculations!