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 top 10% of students, B’s to the next 25%, and C’s to the next 42%.
a.) what is the bottom cutoff for a C grade?
b.) what is the bottom cutoff for a B grade?
c.) what is the bottom cutoff for a A grade?
what is the correct formula for this?
To find the cutoff scores for each grade, you can use the z-score formula. The z-score formula calculates how many standard deviations above or below the mean a given value falls.
The general formula for the z-score is:
z = (x - μ) / σ
Where:
z = z-score
x = given value
μ = mean
σ = standard deviation
To find the cutoff scores for each grade, you need to find the z-scores corresponding to the desired percentages. The inverse cumulative distribution function (also known as the percent-point function) can be used to determine the z-score for a given percentage.
a.) To find the bottom cutoff for a C grade (42%), you need to find the z-score that corresponds to the 58th percentile (100% - 42%). You can use a statistics table or a calculator with the inverse cumulative distribution function for the standard normal distribution to find the z-score.
b.) To find the bottom cutoff for a B grade (25%), you need to find the z-score that corresponds to the 75th percentile (100% - 25%).
c.) To find the bottom cutoff for an A grade (10%), you need to find the z-score that corresponds to the 90th percentile (100% - 10%).
Using these z-scores, you can calculate the corresponding test scores using the z-score formula.
Note: In some cases, you may also need to round the calculated test scores to the nearest whole number, as grades are commonly reported as whole numbers.
To find the cutoff scores for each grade, we can use the standard normal distribution table or a calculator.
The formula we'll use for this is:
Cutoff Score = Mean + (Z-score * Standard Deviation)
Here's how we can solve each part of the question:
a.) To find the bottom cutoff for a C grade (42% of students), we need to determine the Z-score at the 42nd percentile. To do this, we can use the Z-table or a calculator and find the Z-score that corresponds to the cumulative probability of 0.42. Once we have the Z-score, we can substitute it into the formula to get the C grade cutoff score.
b.) To find the bottom cutoff for a B grade (25% of students), we need to determine the Z-score at the 25th percentile. Again, we can use the Z-table or a calculator to find the Z-score that corresponds to the cumulative probability of 0.25, and then substitute it into the formula.
c.) To find the bottom cutoff for an A grade (10% of students), we need to determine the Z-score at the 10th percentile. Again, we can use the Z-table or a calculator to find the Z-score that corresponds to the cumulative probability of 0.10, and then substitute it into the formula.
Once we have the Z-scores, we can calculate the cutoff scores by multiplying each Z-score by the standard deviation and adding the result to the mean.
I'll now provide a step-by-step example solution for each part of the question.
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions/probabilities (.10, .35 & .77) and the corresponding Z scores. Insert into equation above.