I'm stumped on this question.
A teacher informs her history class that a test is very difficult, but the grades would be curved. Scores for the test are normally distributed with a mean of 25 and a standard deviation of 5. If the grades are curved according to the following scheme, find the numerical limits for each grade.
A: Top 10%
B: Scores above the bottom 70% and below the top 10%
C: Scores above the bottom 30% and below the top 30%
D: Scores above the bottom 10% and below the top 70%.
F: Bottom 10%
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions and their Z scores. Insert the values into equation above to find the scores.
To find the numerical limits for each grade, you need to calculate the corresponding z-scores. A z-score measures the number of standard deviations a data point is from the mean in a normal distribution.
Here's how you can find the numerical limits for each grade:
1. Grade A (top 10%):
To find the z-score corresponding to the top 10%, you need to find the z-score that corresponds to the cumulative probability of 0.9. Using a standard normal distribution table or a statistical calculator, you can find that the z-score for a cumulative probability of 0.9 is approximately 1.28.
Now, you can find the raw score corresponding to this z-score using the formula:
z = (x - mean) / standard deviation
Rearranging the formula, you get:
x = z * standard deviation + mean
Substituting the values, you get:
x = 1.28 * 5 + 25 = 31.4
Therefore, the numerical limit for an A grade is above 31.4.
2. Grade B (scores above the bottom 70% and below the top 10%):
To find the z-score corresponding to the bottom 70%, you need to find the z-score that corresponds to a cumulative probability of 0.3 (1 - 0.7). Using a standard normal distribution table or a statistical calculator, you can find that the z-score for a cumulative probability of 0.3 is approximately -0.52.
Similarly, to find the z-score corresponding to the top 10%, you already know it is approximately 1.28.
Using the formulas mentioned above, you can find the raw scores for the bottom and top z-scores:
For the bottom z-score:
x = -0.52 * 5 + 25 = 22.4
For the top z-score:
x = 1.28 * 5 + 25 = 31.4
Therefore, the numerical limits for a B grade are between 22.4 and 31.4.
3. Grade C (scores above the bottom 30% and below the top 30%):
Following the same logic as above, you can find the z-scores for the bottom 30% (cumulative probability of 0.3) and the top 30% (cumulative probability of 0.7) using a standard normal distribution table or a statistical calculator. The z-scores are approximately -0.52 and 0.52, respectively.
Calculating the raw scores:
For the bottom z-score:
x = -0.52 * 5 + 25 = 22.4
For the top z-score:
x = 0.52 * 5 + 25 = 27.6
Therefore, the numerical limits for a C grade are between 22.4 and 27.6.
4. Grade D (scores above the bottom 10% and below the top 70%):
The z-score for the bottom 10% (cumulative probability of 0.1) is approximately -1.28, and the z-score for the top 70% (cumulative probability of 0.3) is approximately 0.52.
Calculating the raw scores:
For the bottom z-score:
x = -1.28 * 5 + 25 = 18.6
For the top z-score:
x = 0.52 * 5 + 25 = 27.6
Therefore, the numerical limits for a D grade are between 18.6 and 27.6.
5. Grade F (bottom 10%):
The z-score for the bottom 10% (cumulative probability of 0.1) is approximately -1.28.
Calculating the raw score:
For the bottom z-score:
x = -1.28 * 5 + 25 = 18.6
Therefore, the numerical limit for an F grade is below 18.6.
To summarize:
- A grade: Above 31.4
- B grade: Between 22.4 and 31.4
- C grade: Between 22.4 and 27.6
- D grade: Between 18.6 and 27.6
- F grade: Below 18.6