Question
Assumethat adults have IQ scores that are normally distributed with a mean of 100 and a standard deviation of 20. Find the probability that a randomly selected adult has an IQ less than 128.
Answers
GPT-4o mini
To find the probability that a randomly selected adult has an IQ less than 128, we can use the properties of the normal distribution.
Given:
- Mean (\(\mu\)) = 100
- Standard deviation (\(\sigma\)) = 20
- IQ score = 128
We will first convert the IQ score to a z-score using the following formula:
\[
z = \frac{(X - \mu)}{\sigma}
\]
Where:
- \(X\) is the value we are interested in (128 in this case),
- \(\mu\) is the mean,
- \(\sigma\) is the standard deviation.
Substituting the values:
\[
z = \frac{(128 - 100)}{20} = \frac{28}{20} = 1.4
\]
Next, we will look up the z-score of 1.4 in the standard normal distribution table or use a calculator to find the corresponding probability.
Using a standard normal distribution table or calculator, we find that:
\[
P(Z < 1.4) \approx 0.9192
\]
Thus, the probability that a randomly selected adult has an IQ less than 128 is approximately **0.9192**, or **91.92%**.
Given:
- Mean (\(\mu\)) = 100
- Standard deviation (\(\sigma\)) = 20
- IQ score = 128
We will first convert the IQ score to a z-score using the following formula:
\[
z = \frac{(X - \mu)}{\sigma}
\]
Where:
- \(X\) is the value we are interested in (128 in this case),
- \(\mu\) is the mean,
- \(\sigma\) is the standard deviation.
Substituting the values:
\[
z = \frac{(128 - 100)}{20} = \frac{28}{20} = 1.4
\]
Next, we will look up the z-score of 1.4 in the standard normal distribution table or use a calculator to find the corresponding probability.
Using a standard normal distribution table or calculator, we find that:
\[
P(Z < 1.4) \approx 0.9192
\]
Thus, the probability that a randomly selected adult has an IQ less than 128 is approximately **0.9192**, or **91.92%**.