An exam is standardized to a Normal model, with a mean of 105 and a standard deviation of 18. Based on the Normal model N(105, 18) What exam score represents the 96th percentile? What percentile of an exam score of 100 represent?
To find the exam score that represents the 96th percentile, you need to use the z-table or a statistical calculator. The 96th percentile corresponds to a z-score of approximately 1.75.
To calculate the z-score, you can use the formula:
z = (X - μ) / σ
Where:
X is the exam score,
μ is the mean (105 in this case),
and σ is the standard deviation (18 in this case).
Rearranging the formula to solve for X, we get:
X = z * σ + μ
Plugging in the values:
X = 1.75 * 18 + 105
X ≈ 138.5
So, the exam score that represents the 96th percentile is approximately 138.5.
To find the percentile of an exam score of 100, you need to calculate the z-score for 100 using the formula mentioned earlier.
z = (X - μ) / σ
Plugging in the values:
z = (100 - 105) / 18
z ≈ -0.2778
To convert the z-score into a percentile, you can use the z-table or a statistical calculator. The z-table provides the area under the curve to the left of the z-score.
The area to the left of a z-score of -0.28 is approximately 0.3907. This means that an exam score of 100 would fall in the 39th percentile.
Therefore, an exam score of 100 represents approximately the 39th percentile.