In a large section of a Chemistry class, the points for the final exam are normally distributed, with a mean of 72, and a variance of 81. Grades are to be assigned according to the following rule: the top 10% receive As, the next 20% receive Bs, the middle 40% receive Cs, the next 20% receive Ds, and the bottom 10% receive Fs. Find the lowest score on the final exam that would qualify a student for an A, a B, a C, and a D.
P((x-72)/9)=90%
(x-72)/9=1.28 [normal distribution table]
x=83.52
83.52 at least for getting A
P((x-72)/81)=10%
(x-72)/81=-1.28 [normal distribution table]
x=60.48
60.48 maximum for getting D
for B, score is between 72 and 83.52
for C, score is between 60.48 and 72
To find the lowest score that would qualify a student for each grade, we need to determine the corresponding z-scores and then use the z-score formula to find the raw score.
First, let's find the z-scores corresponding to each grade cutoff using the mean and variance given. The z-score formula is:
z = (x - μ) / √σ
Where x is the raw score, μ is the mean, and σ is the standard deviation (which is the square root of the variance).
Given:
Mean (μ) = 72
Variance (σ^2) = 81
Standard Deviation (σ) = √81 = 9
Now, let's find the z-scores for each grade cutoff:
For an A grade (top 10%):
To find the z-score corresponding to the top 10%, we need to find the z-score that corresponds to a cumulative probability of 90% (since the top 10% corresponds to the remaining 90%). We can use a standard normal distribution table or a calculator to find this z-score.
Using a standard normal distribution table, we find that the z-score corresponding to a cumulative probability of 90% is approximately 1.28.
Using the z-score formula, we can calculate the raw score (x) for an A grade:
1.28 = (x - 72) / 9
Rearranging the formula:
x - 72 = 1.28 * 9
x - 72 = 11.52
x = 72 + 11.52
x ≈ 83.52
So, the lowest score that would qualify a student for an A grade is approximately 83.52.
For a B grade (next 20%):
To find the z-score corresponding to the next 20%, we need to find the z-score that corresponds to a cumulative probability of 70% (since the top 30% corresponds to the remaining 70%). Using a standard normal distribution table, we find that the z-score corresponding to a cumulative probability of 70% is approximately 0.52.
Using the z-score formula, we can calculate the raw score (x) for a B grade:
0.52 = (x - 72) / 9
Rearranging the formula:
x - 72 = 0.52 * 9
x - 72 = 4.68
x = 72 + 4.68
x ≈ 76.68
So, the lowest score that would qualify a student for a B grade is approximately 76.68.
For a C grade (middle 40%):
The middle 40% corresponds to the cumulative probability between 30% and 70%. Using a standard normal distribution table, we find the z-scores corresponding to these cumulative probabilities: -0.52 and 0.52.
Using the z-score formula, we can calculate the raw scores (x) for a C grade:
For the lower cutoff:
-0.52 = (x - 72) / 9
Rearranging the formula:
x - 72 = -0.52 * 9
x - 72 = -4.68
x = 72 - 4.68
x ≈ 67.32
For the upper cutoff:
0.52 = (x - 72) / 9
Rearranging the formula:
x - 72 = 0.52 * 9
x - 72 = 4.68
x = 72 + 4.68
x ≈ 76.68
So, the lowest score that would qualify a student for a C grade is approximately 67.32 and the upper cutoff is approximately 76.68.
For a D grade (next 20%):
To find the z-score corresponding to the next 20%, we need to find the z-score that corresponds to a cumulative probability of 10% (since the bottom 30% corresponds to the remaining 10%). Using a standard normal distribution table, we find that the z-score corresponding to a cumulative probability of 10% is approximately -1.28.
Using the z-score formula, we can calculate the raw score (x) for a D grade:
-1.28 = (x - 72) / 9
Rearranging the formula:
x - 72 = -1.28 * 9
x - 72 = -11.52
x = 72 - 11.52
x ≈ 60.48
So, the lowest score that would qualify a student for a D grade is approximately 60.48.
Therefore, the lowest score that would qualify a student for each grade is:
- A: approximately 83.52
- B: approximately 76.68
- C: approximately 67.32
- D: approximately 60.48
To find the lowest score on the final exam that would qualify a student for each grade, we need to determine the cutoff points for each grade category based on the given distribution.
Step 1: Calculate the standard deviation (σ) from the variance.
The standard deviation is the square root of the variance.
σ = √81 = 9
Step 2: Calculate the Z-score for each grade category.
The Z-score represents the number of standard deviations a given score is from the mean.
For A grade (top 10%):
To find the cutoff point for the top 10%, we need to find the Z-score that corresponds to the percentile of 90% (complement of 10%).
Using the Z-score table or a statistical calculator, we find that the Z-score corresponding to the percentile of 90% is approximately 1.28.
Z-score for A grade = 1.28
For B grade (next 20%):
To find the cutoff point for the next 20%, we need to find the Z-score that corresponds to the percentile of 80% (complement of 20%).
Using the Z-score table or a statistical calculator, we find that the Z-score corresponding to the percentile of 80% is approximately 0.84.
Z-score for B grade = 0.84
For C grade (middle 40%):
To find the cutoff point for the middle 40%, we need to find the Z-scores that correspond to the percentiles of 70% (half of 40%) and 30% (complement of 70%).
Using the Z-score table or a statistical calculator, we find that the Z-score corresponding to the percentile of 70% is approximately 0.52, and the Z-score corresponding to the percentile of 30% is approximately -0.52.
Z-score for C grade = 0.52 (upper cutoff)
Z-score for C grade = -0.52 (lower cutoff)
For D grade (next 20%):
To find the cutoff point for the next 20%, we need to find the Z-score that corresponds to the percentile of 10% (complement of 20%).
Using the Z-score table or a statistical calculator, we find that the Z-score corresponding to the percentile of 10% is approximately -1.28.
Z-score for D grade = -1.28
Step 3: Convert the Z-scores back to raw scores.
To convert the Z-scores back to raw scores, we use the formula:
Score = (Z-score * standard deviation) + mean
Raw score for A grade = (1.28 * 9) + 72 ≈ 83.52
Raw score for B grade = (0.84 * 9) + 72 ≈ 79.56
Raw score for C grade (upper cutoff) = (0.52 * 9) + 72 ≈ 76.68
Raw score for C grade (lower cutoff) = (-0.52 * 9) + 72 ≈ 67.32
Raw score for D grade = (-1.28 * 9) + 72 ≈ 60.88
Therefore, the lowest score on the final exam to qualify for each grade is approximately:
A grade: 83.52
B grade: 79.56
C grade (upper cutoff): 76.68
C grade (lower cutoff): 67.32
D grade: 60.88