In the movie *Forrest Gump*, the public school required an IQ of at least 80 for admittance.

**(a)** If IQ test scores are normally distributed with mean 100 and standard deviation 16, what percentage of people would qualify for admittance to the school? **(Round your answer to 2 decimal places.)**

**(b)** If the public school wishes 95 percent of all children to qualify for admittance, what minimum IQ test score should be required for admittance? **(Round the z value to 3 decimal places. Round your answer to 2 decimal places.)**

We are working with a normally distributed population where IQ scores have a mean (\(\mu\)) of 100 and a standard deviation (\(\sigma\)) of 16. Let's break down each part of the problem.

### **(a)** What percentage of people would qualify for admittance (IQ \(\geq 80\))?

We need to find the percentage of people whose IQ is at least 80. This is a question of finding the area under the normal curve to the right of \(x = 80\).

To solve this:

1. **Standardize the score** by converting the raw score of 80 into a z-score:
   \[
   z = \frac{x - \mu}{\sigma} = \frac{80 - 100}{16} = \frac{-20}{16} = -1.25
   \]

2. **Find the cumulative probability** corresponding to this z-score from standard normal distribution tables (or using a calculator). The cumulative probability for \(z = -1.25\) represents the area to the left of \(z = -1.25\), i.e., the percentage of people with an IQ less than 80.

   From standard normal tables, the cumulative probability for \(z = -1.25\) is approximately 0.1056 (or 10.56%).

3. **Calculate the percentage of people who qualify for admittance**:
   \[
   P(X \geq 80) = 1 - P(X < 80) = 1 - 0.1056 = 0.8944
   \]
   Therefore, approximately 89.44% of people would qualify for admittance to the school.

### **Answer for (a)**: 89.44%

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### **(b)** What minimum IQ score should be required if the school wants 95% of all children to qualify for admittance?

Here, we are given that 95% of children should qualify, meaning we need to find the IQ score such that the cumulative probability to the left is 5% (or 0.05).

1. **Find the z-score corresponding to the lower 5th percentile**:
   The z-score for the 5th percentile (the point where only 5% of the distribution is below) is approximately \(z = -1.645\) (from standard normal distribution tables).

2. **Convert the z-score back to an IQ score** using the formula:
   \[
   x = \mu + z \cdot \sigma = 100 + (-1.645) \cdot 16 = 100 - 26.32 = 73.68
   \]

### **Answer for (b)**: The minimum IQ score required should be approximately 73.68.

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### Summary:
- **(a)** Approximately **89.44%** of people would qualify for admittance.
- **(b)** The minimum IQ score for 95% of children to qualify should be **73.68**.

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Would you like any more details on the solution or have further questions?

Here are 5 related questions to consider:
1. How do changes in the mean IQ affect the percentage of people who qualify for admittance?
2. What happens if the standard deviation of IQ scores increases? How does that affect admittance percentages?
3. How can z-scores be used in other types of normally distributed data?
4. What would happen if the school raised the required IQ threshold to 90?
5. How can cumulative distribution functions (CDF) help in solving real-world problems like this?

**Tip**: Understanding how to convert between z-scores and raw scores in a normal distribution is key for many statistical analyses.

In the movie *Forrest Gump*, the public school required an IQ of at least 80 for admittance. (a) If IQ test scores are normally distributed with mean 100 and standard deviation 16, what percentage of people would qualify for admittance to the school? (b) If the public school wishes 95 percent of all children to qualify for admittance, what minimum IQ test score should be required for admittance?

Solve a normal distribution problem where IQ test scores are normally distributed with a mean of 100 and a standard deviation of 16. Find what percentage of people qualify for school admittance and determine the minimum IQ required for 95% qualification. Learn key concepts like Z-scores and percentiles. Suitable for high school students in Grades 10-12.

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