Solve the problem.
Assume that math SAT scores are normally distributed with a mean of 500 and a standard deviation of 100. If you scored 560, what percentage of those faking the test scores below you?
To find the percentage of test-takers who scored below 560, we need to calculate the area under the normal distribution curve to the left of 560.
First, we need to standardize the score using the formula:
z = (x - μ) / σ
where:
x = the given score (560)
μ = the mean of the distribution (500)
σ = the standard deviation of the distribution (100)
z = (560 - 500) / 100
z = 60 / 100
z = 0.6
Next, we need to find the corresponding area under the standard normal distribution curve to the left of z = 0.6. This can be done by looking up the z-value in the standardized normal distribution table or by using a calculator.
Using a calculator or a standard normal distribution table, we find that the area to the left of z = 0.6 is approximately 0.7257.
Finally, we convert this proportion to a percentage:
0.7257 * 100 ≈ 72.57%
Therefore, approximately 72.57% of test-takers faked the score below 560.