In a large section of a statistics class, the points for the final exam are normally distributed, with a mean of 70 and a standard deviation of 8 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% receive D's,

the bottom 10% receive F's

Find the lowest score on the final exam that would qualify a student for an A, a B: a C, and a D.

You can play around with Z table stuff at

davidmlane.com/hyperstat/z_table.html

To find the lowest score on the final exam that would qualify a student for each grade, we need to use the concept of Z-scores and the standard normal distribution.

Step 1: Convert the given mean and standard deviation to Z-scores.
The formula for calculating the Z-score is: Z = (X - μ) / σ
Where:
- Z is the Z-score
- X is the individual value
- μ is the mean of the distribution
- σ is the standard deviation of the distribution

For our case:
- Z_A: Z-score for an A grade (top 10%)
- Z_B: Z-score for a B grade (next 20%)
- Z_C: Z-score for a C grade (middle 40%)
- Z_D: Z-score for a D grade (next 20%)

Step 2: Calculate the Z-scores using the formula.
For an A grade:
Z_A = (X - μ) / σ
Z_A = (X - 70) / 8

For a B grade:
Z_B = (X - μ) / σ
Z_B = (X - 70) / 8

For a C grade:
Z_C = (X - μ) / σ
Z_C = (X - 70) / 8

For a D grade:
Z_D = (X - μ) / σ
Z_D = (X - 70) / 8

Step 3: Find the corresponding Z-scores for the given percentile values.

For an A grade: Since the top 10% receive A's, we need to find the Z-score corresponding to the 90th percentile. We can use a Z-table or a calculator to find the Z-score associated with a percentile of 0.90. This Z-score represents the lowest score for an A grade.

For a B grade: Since the next 20% receive B's, we need to find the Z-score corresponding to the 70th percentile. We can use a Z-table or a calculator to find the Z-score associated with a percentile of 0.70. This Z-score represents the lowest score for a B grade.

For a C grade: Since the middle 40% receive C's, we need to find the Z-score corresponding to the 30th percentile. We can use a Z-table or a calculator to find the Z-score associated with a percentile of 0.30. This Z-score represents the lowest score for a C grade.

For a D grade: Since the next 20% receive D's, we need to find the Z-score corresponding to the 10th percentile. We can use a Z-table or a calculator to find the Z-score associated with a percentile of 0.10. This Z-score represents the lowest score for a D grade.

Step 4: Convert the Z-scores back to individual scores (X) using the formula.
X = Z * σ + μ

Substitute the respective Z-score values into the formula to find the lowest score for each grade.

To find the lowest score on the final exam that would qualify a student for each letter grade, we can use the standard normal distribution table (also known as the z-score table).

Let's first find the z-scores corresponding to each grade cutoff.

1. A grade (top 10%):
The top 10% of scores correspond to z-scores that fall in the upper 10% of the distribution. Using the standard normal distribution table, we can find the z-score corresponding to the 90th percentile (100% - 10% = 90%).
From the table, the z-score at the 90th percentile is approximately 1.28. So, the z-score cutoff for an A grade is 1.28.

2. B grade (next 20%):
The next 20% of scores correspond to z-scores that fall between the 90th and 70th percentiles. We need to find the z-score at the 70th percentile (100% - (10% + 20%) = 70%).
From the table, the z-score at the 70th percentile is approximately 0.52. So, the z-score cutoff for a B grade is 0.52.

3. C grade (middle 40%):
The middle 40% of scores correspond to z-scores that fall between the 70th and 30th percentiles. We need to find the z-score at the 30th percentile (100% - (10% + 20% + 40%) = 30%).
From the table, the z-score at the 30th percentile is approximately -0.52 (since it's the negative of the z-score for the 70th percentile). So, the z-score cutoff for a C grade is -0.52.

4. D grade (next 20%):
The next 20% of scores correspond to z-scores that fall between the 30th and 10th percentiles. We need to find the z-score at the 10th percentile (100% - (10% + 20% + 40% + 20%) = 10%).
From the table, the z-score at the 10th percentile is approximately -1.28 (since it's the negative of the z-score for the 90th percentile). So, the z-score cutoff for a D grade is -1.28.

Now, we can convert these z-scores to actual scores by using the formula: score = (z-score * standard deviation) + mean.

1. Lowest score for an A grade:
A score that qualifies for an A grade corresponds to a z-score of 1.28.
Using the formula, the lowest score for an A grade is (1.28 * 8) + 70 = 79.04 (rounded to 79).

2. Lowest score for a B grade:
A score that qualifies for a B grade corresponds to a z-score of 0.52.
Using the formula, the lowest score for a B grade is (0.52 * 8) + 70 = 74.16 (rounded to 74).

3. Lowest score for a C grade:
A score that qualifies for a C grade corresponds to a z-score of -0.52.
Using the formula, the lowest score for a C grade is (-0.52 * 8) + 70 = 65.36 (rounded to 65).

4. Lowest score for a D grade:
A score that qualifies for a D grade corresponds to a z-score of -1.28.
Using the formula, the lowest score for a D grade is (-1.28 * 8) + 70 = 59.76 (rounded to 60).

Therefore, the lowest scores needed for each grade are as follows:
A grade: 79
B grade: 74
C grade: 65
D grade: 60