The scores on an economics examination are normally distributed with a mean of 66 and a standard deviation of 14. If the instructor wishes to assign a grade of A to the top 10% of the class, what is the lowest score a student may have and still obtain an A?
To find the lowest score a student may have and still obtain an A, we need to determine the cutoff score that separates the top 10% of the class.
Since the scores on the economics examination are normally distributed, we can use the standard normal distribution table to find the cutoff z-score corresponding to the top 10%.
Step 1: Find the z-score corresponding to the top 10%
The top 10% corresponds to the area under the curve to the right of the cutoff score. Since the z-table provides the area to the left of the cutoff score, we subtract the top 10% from 100% to find the area to the left.
Area to the left = 100% - 10% = 90%
Step 2: Find the z-score from the z-table
Using the z-table or a calculator, we can find the z-score corresponding to an area of 90%. The closest value we can find is 1.28.
Step 3: Calculate the lowest score
Now that we have the z-score, we can use the formula for z-scores to calculate the lowest score:
z = (x - μ) / σ
where:
z = z-score
x = score
μ = mean
σ = standard deviation
Rearranging the formula, we can solve for the lowest score (x):
x = (z * σ) + μ
Plugging in the values, we have:
x = (1.28 * 14) + 66
x = 18.22 + 66
x ≈ 84.22
Therefore, the lowest score a student may have and still obtain an A is approximately 84.22.
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion (.10) related to a Z score. Then insert the values to solve for the score.