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Use TI-84 plus calculator to show steps to find solution

To solve this problem using a **TI-84 Plus** calculator, we need to use the normal distribution function. Here's how to calculate the probability step by step:

### Given Data:
- Mean \( \mu = 101 \)
- Standard deviation \( \sigma = 15 \)
- We want to find the probability \( P(X > 120) \), where \( X \) represents IQ.

### Step-by-Step Process:

1. **Find the Z-score**:
   The Z-score for a given value is calculated using the formula:
   \[
   Z = \frac{X - \mu}{\sigma}
   \]
   Where:
   - \( X = 120 \) (the IQ value of interest),
   - \( \mu = 101 \) (the mean IQ),
   - \( \sigma = 15 \) (the standard deviation).

   Substituting the values:
   \[
   Z = \frac{120 - 101}{15} = \frac{19}{15} \approx 1.2667
   \]

2. **Use the NormalCDF function** on the TI-84 Plus:
   - Press the `2nd` button.
   - Press `VARS` to access the "DISTR" menu.
   - Scroll down to **normalcdf** and press `ENTER`.
   
   Since we are looking for \( P(X > 120) \), we need to compute the area to the right of this Z-score (1.2667). To do this:
   - Use `normalcdf(lower bound, upper bound, mean, standard deviation)`.
   
   For the lower bound, we use \( Z = 120 \), and for the upper bound, use a very large number (say \( 9999 \)) to represent infinity. So input:
   ```
   normalcdf(120, 9999, 101, 15)
   ```

   Press `ENTER`.

3. **Result**:
   The calculator will return a probability value.

   When calculated, the result is approximately:
   \[
   P(X > 120) \approx 0.1020
   \]

### Final Answer:
The probability that a randomly chosen person from this population has an IQ greater than 120 is **0.1020** (rounded to four decimal places).

Would you like any further clarifications or details? Here are a few related questions to explore this further:

1. What is the probability that an individual has an IQ less than 120?
2. How can you calculate the probability for a range of IQ scores, e.g., between 100 and 120?
3. How does changing the mean IQ affect the probability for IQs greater than 120?
4. What is the interpretation of a Z-score in normal distributions?
5. How do we compute the probability of an IQ less than 85 using the TI-84?

**Tip:** The **Z-table** can also be used to find probabilities if you don't have access to a calculator.

IQ is normally distributed with a mean of 101 and a standard deviation of 15. Suppose one individual at a school in New York City is randomly chosen. Find the probability that this person has an IQ greater than 120.

Learn how to calculate the probability of an IQ score greater than 120 using normal distribution. This problem involves calculating a Z-score and using the normalcdf function on a calculator. Suitable for high school students in Grades 9-12.

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