Questions: IQ scores form a normal distribution with μ=100 and σ=15. Individuals with IQs above 140 are classified in the genius category. What proportion of the population consists of geniuses?
Transcript text: IQ scores form a normal distribution with $\mu=100$ and $\sigma=15$. Individuals with IQs above 140 are classified in the genius category. What proportion of the population consists of geniuses?
Solution
Solution Steps
Step 1: Calculate the Z-Score
To determine the Z-score for an IQ of 140, we use the formula:
z=σX−μ
Substituting the values:
z=15140−100=1540≈2.6667
Thus, the Z-score for IQ 140 is z≈2.6667.
Step 2: Calculate the Probability of Being a Genius
Next, we need to find the probability that an individual has an IQ greater than 140. This is represented mathematically as:
P(X>140)=P(Z>2.6667)=Φ(∞)−Φ(2.6667)
Where Φ is the cumulative distribution function (CDF) of the standard normal distribution. Since Φ(∞)=1, we have:
P(X>140)=1−Φ(2.6667)
From the calculations, we find:
P(X>140)≈0.0038
Final Answer
The proportion of the population classified as geniuses is