Assume that adults have IQ scores that are normally distributed with a mean of \mu=105μ=105and a standard deviation \sigma=20σ=20. Find the probability that a randomly selected adult has an IQ less than 145.
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to the Z score.
To find the probability that a randomly selected adult has an IQ score less than 145, we need to calculate the area under the normal distribution curve to the left of 145.
Here's how you can do it step by step:
Step 1: Standardize the value
To work with the normal distribution, we need to standardize the value of 145. We can do this by subtracting the mean (105) from 145 and then dividing the result by the standard deviation (20):
Z = (145 - 105) / 20
= 40 / 20
= 2
Step 2: Look up the standardized value
Using a standard normal distribution table or a calculator, we can find the cumulative probability for the standardized value of Z = 2. In this case, the cumulative probability is the area under the curve to the left of Z = 2.
The cumulative probability for Z = 2 is approximately 0.9772.
Step 3: Find the probability of a randomly selected adult having an IQ less than 145
Since the cumulative probability of Z = 2 represents the proportion of the population with IQ scores less than 145, we can use it to find the probability:
P(X < 145) = 0.9772
Therefore, the probability that a randomly selected adult has an IQ less than 145 is approximately 0.9772, or 97.72%.
Note: Keep in mind that this calculation assumes a normal distribution and that IQ scores follow this specific normal distribution with a mean of 105 and a standard deviation of 20.