Solve the problem
Scores on a test are approximately normally distributed with a mean of 70 and a standard deviation of 9. The teacher wants to give A’s to the too 10% of students, B’s to the next 25%,, C’s to the next 40%, Do’s to the next 16% and F’s to the bottom 9%. What is the bottom cutoff for a grade D? Round your answer to the nearest whole number
To find the bottom cutoff for a grade D, we need to find the score below which the next 16% of students fall.
Since the scores are approximately normally distributed, we can use the z-score formula to determine the cutoff point.
The z-score formula is: z = (x - μ) / σ
Where:
- z is the z-score
- x is the raw score
- μ is the mean
- σ is the standard deviation
In this case, the mean is 70 and the standard deviation is 9. We need to find the score below which the next 16% of students fall, which means we want the z-score corresponding to the 84th percentile (100% - 16%).
Using a standard normal distribution table or a calculator, we can find that the z-score corresponding to the 84th percentile is approximately 1.405.
Now we can rearrange the z-score formula to solve for x:
x = z * σ + μ
x = 1.405 * 9 + 70
x = 12.645 + 70
x ≈ 82.645
Rounding to the nearest whole number, the bottom cutoff for a grade D is 83.