QuestionFind the percentage of people with an IQ over 130, given a normal distribution with mean 100 and standard deviation 15.
Studdy Solution
STEP 1
Assumptions1. The distribution of IQ is approximately normal.
. The mean (average) IQ is100.
3. The standard deviation of IQ is15.
4. We are looking for the percentage of people with an IQ over130.
5. We are using the standard deviation rule, which states that for a normal distribution, about68% of values lie within1 standard deviation of the mean, about95% lie within standard deviations, and about99.7% lie within3 standard deviations.
STEP 2
First, we need to find how many standard deviations above the mean an IQ of130 is. We can do this by subtracting the mean from the given IQ and dividing by the standard deviation.
where- Z is the number of standard deviations from the mean (Z-score),
- X is the value we are interested in (in this case,130),
- μ is the mean (in this case,100), and- σ is the standard deviation (in this case,15).
STEP 3
Now, plug in the given values for X, μ, and σ to calculate the Z-score.
STEP 4
Calculate the Z-score.
STEP 5
Now that we have the Z-score, we can use the standard deviation rule to estimate the percentage of people with an IQ over130. According to the rule, about95% of values lie within2 standard deviations of the mean. This means that about5% of values lie outside of2 standard deviations from the mean. Since the distribution is symmetric, we can divide this by2 to find the percentage of values that are more than2 standard deviations above the mean.
STEP 6
Calculate the percentage.
So, according to the standard deviation rule, about2.5% of people have an IQ over130.
Was this helpful?
