professor has found that the grades on a Statistics Exam are normally distributed with mean 72 and a standard deviation of 12. If only the best 15 % of the grades in the class will get an A, what grade must a student get in order to get an A?
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability (.15) and its Z score. Insert Z value into above equation and solve for score.
answer:
x = 72 + (12*1.036) = 84.43
To find the grade required to get an A, we need to determine the corresponding z-score that represents the top 15% of the grades in a standard normal distribution.
The z-score is calculated using the formula:
z = (x - mean) / standard deviation
To find the z score for the top 15%, we can use the inverse normal distribution table or a calculator.
Looking up the z-score for 15% in the table, we find that the z-value is approximately 1.04.
Next, we can use the formula to find the corresponding grade (x):
x = z * standard deviation + mean
Plugging in the values:
x = 1.04 * 12 + 72
Calculating:
x ≈ 12.48 + 72
x ≈ 84.48
Therefore, a student needs to get a grade of approximately 84.48 in order to get an A in the class.
To determine the grade a student must get in order to get an A, we need to find the cutoff point for the top 15% of grades.
Step 1: Convert the percentage to a z-score.
Since the grades are normally distributed, we can use the standard normal distribution table. The top 15% corresponds to the area under the curve to the left of the cutoff point, which is the complement of 0.15 (1 - 0.15 = 0.85).
Step 2: Find the z-score using the standard normal distribution table.
Looking up the z-score corresponding to 0.85 in the standard normal distribution table, we find it to be approximately 1.04.
Step 3: Use the z-score formula to find the grade.
The z-score formula is: z = (x - mean) / standard deviation
Rearranging the formula to solve for x (grade):
x = (z * standard deviation) + mean
Plugging in the known values:
x = (1.04 * 12) + 72 = 84.48
Therefore, a student must get a grade of approximately 84.48 or higher to get an A.