Professor Smith gave a final exam in her history class. The scores on the final exam are
normally distributed with a mean of 75 and a standard deviation of 6. If Professor Smith
wishes to give A’s to the top 4%, what would be the cutoff score be for A’s?
a. 90.0
b. 87.3
c. 85.5
d. 84.0
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is it c?
To determine the cutoff score for A's, we need to find the score that separates the top 4% of the distribution. In other words, we need to find the score at which only 4% of the scores fall above.
To find this cutoff score, we can use the Z-score formula. The Z-score measures how many standard deviations a particular score is from the mean. We can then use a standard normal distribution table to find the proportion of scores above a given Z-score.
The formula to calculate the Z-score is:
Z = (X - μ) / σ
Where:
Z is the Z-score,
X is the raw score,
μ is the mean of the distribution, and
σ is the standard deviation of the distribution.
Given that the mean (μ) is 75 and the standard deviation (σ) is 6, let's calculate the Z-score for the top 4% (or 0.04) of students:
Z = (X - μ) / σ
0.04 = (X - 75) / 6
Now, we can solve for X (the cutoff score for A's) using algebra:
(X - 75) / 6 = 0.04
X - 75 = 0.04 * 6
X - 75 = 0.24
X = 75 + 0.24
X = 75.24
Therefore, the cutoff score for A's would be approximately 75.24.
Out of the given options, the closest value to 75.24 is 84.0 (option d). So, the correct answer is option d.