The scores on a standardized exam are normally distributed with mean= 50 and variance=16. (a) what percentage of the scores exceed 58? (b) what percentage of the scores lie between 38 and 54? (c) what is the 84th percentile of the distribution?
To solve these questions, we need to use the standard normal distribution. First, we need to convert the given normal distribution to the standard normal distribution by using the formula:
Z = (X - μ) / σ
where:
Z = standard score
X = raw score
μ = mean of the distribution
σ = standard deviation of the distribution
(a) To find the percentage of scores that exceed 58, we need to find the area under the curve to the right of 58.
Step 1: Calculate the Z-value for X = 58.
Z = (58 - 50) / √16 = 8 / 4 = 2
Step 2: Use a Z-table or a calculator to find the area to the right of this Z-value. The area to the left of Z = 2 is approximately 0.9772.
Step 3: Subtract this value from 1 to find the area to the right of 58.
Area to the right of 58 = 1 - 0.9772 = 0.0228 or 2.28%.
Therefore, approximately 2.28% of the scores exceed 58.
(b) To find the percentage of scores that lie between 38 and 54, we need to find the area under the curve between these two Z-values.
Step 1: Calculate the Z-values for X = 38 and X = 54.
Z1 = (38 - 50) / √16 = -12 / 4 = -3
Z2 = (54 - 50) / √16 = 4 / 4 = 1
Step 2: Use a Z-table or a calculator to find the area to the right of Z1 and Z2.
The area to the left of Z = -3 is approximately 0.0013.
The area to the left of Z = 1 is approximately 0.8413.
Step 3: Subtract the smaller area from the larger area to find the area between Z1 and Z2.
Area between -3 and 1 = 0.8413 - 0.0013 = 0.84
Therefore, approximately 84% of the scores lie between 38 and 54.
(c) To find the 84th percentile of the distribution, we need to find the Z-value that corresponds to an area of 0.84 to the left.
Step 1: Use a Z-table or a calculator to find the Z-value that corresponds to an area of 0.84 to the left.
The Z-value corresponding to an area of 0.84 is approximately 0.97.
Step 2: Calculate the raw score (X) using the Z-value and the formula:
X = Z * σ + μ
X = 0.97 * √16 + 50 = 0.97 * 4 + 50 = 3.88 + 50 = 53.88
Therefore, the 84th percentile of the distribution is approximately 53.88.