To determine the probability that an individual's IQ is between 80 and 110, we use the properties of the normal distribution. Specifically, we calculate the z-scores for 80 and 110, then find the probabilities corresponding to these z-scores using the standard normal distribution table or software.

Step 1: Calculate the z-scores

The formula for the z-score is:

$z = \frac{x - \mu}{\sigma}$

Where:

• $x$ is the value
• $\mu$ is the mean ($100$)
• $\sigma$ is the standard deviation ($10$)

For $x = 80$: $z = \frac{80 - 100}{10} = -2$

For $x = 110$: $z = \frac{110 - 100}{10} = 1$

Step 2: Find the probabilities for these z-scores

Using the standard normal distribution:

• The cumulative probability for $z = -2$ is approximately $P(Z \leq -2) = 0.0228$.
• The cumulative probability for $z = 1$ is approximately $P(Z \leq 1) = 0.8413$.

Step 3: Calculate the probability between 80 and 110

The probability that an individual’s IQ is between 80 and 110 is:

$P(80 \leq X \leq 110) = P(Z \leq 1) - P(Z \leq -2)$

Substitute the values: $P(80 \leq X \leq 110) = 0.8413 - 0.0228 = 0.8185$

Final Answer:

The probability that an individual’s IQ is between 80 and 110 is 0.8185, or approximately 81.85%.

---

Would you like a detailed breakdown or explanation of the normal distribution? Here are some follow-up questions to explore this topic:

1. How do we interpret z-scores in the context of probabilities?
2. What tools can be used to calculate probabilities for normal distributions?
3. What is the significance of the area under the curve in a normal distribution?
4. How would the probability change if the standard deviation were larger or smaller?
5. Can you explain the empirical rule (68-95-99.7 rule) in relation to this example?

Tip: For quick calculations, keep a standard normal distribution table or a scientific calculator with statistical functions handy!