On a recent English exam,if 20% of scores fell below 60 points and the standard deviation is 7.5,find the mean score (assuming the Normal model is appropriate).
A)12
B)1.50
C)67.5
D)-53.70
E)66.30

Question

Quizplus · Accepted Answer

Given that 20% of scores fell below 60 points, we can use the Z-score formula to find the mean. In a normal distribution, a Z-score of -0.84 corresponds to the 20th percentile. The Z-score formula is Z = (X - μ) / σ, where X is the score, μ is the mean, and σ is the standard deviation. Plugging in the values, we get -0.84 = (60 - μ) / 7.5. Solving for μ gives us μ = 60 + (-0.84)(7.5) = 60 - 6.3 = 53.7. However, this calculation was incorrect in determining the correct mean, which should be higher than 60 to reflect the distribution accurately. The correct approach involves recognizing the error in the calculation and understanding that the mean should be positioned above the 60-point mark to align with the given percentile and standard deviation. The correct mean, considering the normal distribution properties and the given data, aligns with option E, 66.30, after recalculating with the correct interpretation of the Z-score and its relation to the distribution's characteristics.

On a Recent English Exam,if 20% of Scores Fell Below