Suppose that SAT scores among U.S. college students are normally distributed with a mean of 450 and a standard deviation of 150. What is the probability that a randomly selected individual from this population has an SAT score at or below 600?
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion for the Z score calculated.
To find the probability that a randomly selected individual from this population has an SAT score at or below 600, we need to use the normal distribution and calculate the z-score.
The z-score is a measure of how many standard deviations an individual value is from the mean. It is calculated using the formula:
z = (x - μ) / σ
where
x = the value we want to calculate the probability for (600 in this case),
μ = the mean of the distribution (450 in this case), and
σ = the standard deviation of the distribution (150 in this case).
Substituting the values into the formula:
z = (600 - 450) / 150
z = 150 / 150
z = 1
Once we have the z-score, we can look up the corresponding probability from a standard normal distribution table or use a calculator.
The probability that a randomly selected individual from this population has an SAT score at or below 600 is the probability associated with the z-score of 1.
From a standard normal distribution table, a z-score of 1 corresponds to a probability of approximately 0.8413. This means that approximately 84.13% of individuals will have a SAT score at or below 600.
So, the probability is 0.8413 or 84.13%.