In a large section of a statistics class, the points for the final exam are normally distributed, with a mean of 72 and a standard deviation of 9. Grades are to be assigned according to the following rule:
The top 10% receive A's.
The next 20% receive B's.
The middle 40% receive C's.
The next 20% received D's.
The bottom 10% receive F's
What is the least amount of points that a student must score, on the final exam, in order to earn a D? (Round your answer to two decimal places.)
67.32 is the answer
using same web site for normal dist
A = .1
M = 72
sd = 9
.1 area is anything below 60.46
If I try bottom .3, anything below a C, I get
A = .3 ---> below 67.28, that is the boundary for the HIGHEST D
To find the least amount of points a student must score to earn a D, we need to determine the threshold score that separates the bottom 60% (F, D) from the rest of the class (A, B, C).
First, let's find the z-score corresponding to the 60th percentile. The percentile can be converted to a z-score using the standard normal distribution table or a calculator.
The z-score formula is:
z = (x - mean) / standard deviation
Given:
Mean = 72
Standard deviation = 9
To find the 60th percentile, we need to find the z-score such that P(Z ≤ z) = 0.6.
Looking up the z-score in the standard normal distribution table, we find that the z-score corresponding to the 60th percentile is approximately 0.25.
Now, we can solve for the least amount of points a student must score to earn a D:
0.25 = (x - 72) / 9
Rearranging the equation to solve for x:
x - 72 = 0.25 * 9
x - 72 = 2.25
x = 2.25 + 72
x = 74.25
Therefore, the least amount of points a student must score on the final exam to earn a D is 74.25. Rounding to two decimal places, the answer is 74.25 or 74.26.
To find the least amount of points required to earn a D grade, we need to determine the cutoff score for the bottom 60% of students (D and F grades combined). Here's how you can calculate it:
1. Start by finding the z-score corresponding to the 20th percentile (to determine the cutoff for the bottom 20% receiving D grades).
Standardizing the value x using the formula:
z = (x - μ) / σ
where:
z is the z-score,
x is the raw score,
μ is the mean, and
σ is the standard deviation.
In this case, x is the unknown cutoff score for D grade, μ is the mean (72), and σ is the standard deviation (9).
Using the z-score table or a calculator, find the z-score corresponding to the 20th percentile (0.20). The z-score is approximately -0.84.
2. Substitute the value of the z-score (-0.84) into the formula:
-0.84 = (x - 72) / 9
3. Solve the equation for x (the unknown cutoff score):
-0.84 * 9 = x - 72
-7.56 = x - 72
x = 72 - 7.56
x = 64.44
Thus, the least amount of points a student must score on the final exam to earn a D grade is 64.44 (rounded to two decimal places).
Therefore, the answer provided (67.32) is incorrect. The correct answer should be 64.44.