IQ is normally distributed with a mean of 100 and a standard deviation of 15. Suppose one individual is randomly chosen. Let X = IQ of an individual. Find the probability that the... IQ is normally distributed with a mean of 100 and a standard deviation of 15. Suppose one individual is randomly chosen. Let X = IQ of an individual. Find the probability that the person has an IQ greater than 120.
Understand the Problem
The question is asking us to find the probability that a randomly chosen individual has an IQ greater than 120 given that IQ follows a normal distribution with a mean of 100 and a standard deviation of 15. To solve this, we will need to calculate the z-score for an IQ of 120 and then use the standard normal distribution to find the corresponding probability.
Answer
The probability is approximately $0.0918$.
Answer for screen readers
The probability that a randomly chosen individual has an IQ greater than 120 is approximately $0.0918$.
Steps to Solve
- Calculate the Z-score for IQ of 120
To find the Z-score, we use the formula:
$$ Z = \frac{(X - \mu)}{\sigma} $$
where $X$ is the value we are interested in (120), $\mu$ is the mean (100), and $\sigma$ is the standard deviation (15).
Substituting the values, we get:
$$ Z = \frac{(120 - 100)}{15} = \frac{20}{15} \approx 1.33 $$
- Look up the Z-score in the standard normal distribution table
Using the Z-score of approximately 1.33, we find the corresponding cumulative probability from the standard normal distribution table.
The cumulative probability for $Z = 1.33$ is approximately $0.9082$. This value represents the probability of an individual having an IQ less than 120.
- Calculate the probability of IQ greater than 120
To find the probability of an IQ greater than 120, we subtract the cumulative probability from 1:
$$ P(X > 120) = 1 - P(X < 120) = 1 - 0.9082 = 0.0918 $$
Thus, the probability that a randomly chosen individual has an IQ greater than 120 is approximately $0.0918$.
The probability that a randomly chosen individual has an IQ greater than 120 is approximately $0.0918$.
More Information
This means about 9.18% of individuals in this normal distribution have an IQ higher than 120. In a practical context, this is useful for understanding how people are distributed based on IQ scores in relation to benchmarks.
Tips
- Confusing the cumulative probability with the probability of greater values. Remember to subtract from 1 if you need the probability of exceeding a specific value.
- Miscalculating the Z-score, especially when using the wrong mean or standard deviation.
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