The first quartile is. (Type an integer or decimal rounded to one decimal place as needed.)
The first quartile is. (Type an integer or decimal rounded to one decimal place as needed.)
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![### Problem Statement
Assume that adults have IQ scores that are normally distributed with a mean of 100.2 and a standard deviation of 15.2. Find the first quartile \( Q_1 \), which is the IQ score separating the bottom 25% from the top 75%. (Hint: Draw a graph.)
---
**The first quartile is** [ ]
(Type an integer or decimal rounded to one decimal place as needed.)
### Detailed Explanation (for Educational Context)
To determine the first quartile (\( Q_1 \)) for the given normally distributed IQ scores, follow these steps:
1. **Understanding the First Quartile \( Q_1 \)**:
- The first quartile (\( Q_1 \)) is the value below which 25% of the data falls.
- In a standard normal distribution, this corresponds to a z-score that separates the lower 25% of the distribution from the upper 75%.
2. **Standard Normal Distribution**:
- The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1.
- To find the z-score for the first quartile in a standard normal distribution, you can look up the value in a z-table or use a statistical tool.
3. **Converting to the Given Normal Distribution**:
- For the given normal distribution with mean (\( \mu \)) = 100.2 and standard deviation (\( \sigma \)) = 15.2, we need to find the corresponding IQ score for the z-score found in step 2.
4. **Calculation Steps**:
- Identify the z-score for \( Q_1 \) (25th percentile) using a z-table or statistical software. Typically, this z-score is approximately -0.674.
- Use the z-score formula for a normal distribution:
\[
Q_1 = \mu + (z \times \sigma)
\]
where:
- \(\mu = 100.2\)
- \(z = -0.674\)
- \(\sigma = 15.2\)
5. **Final Calculation**:
- Plug in the values:
\[
Q_1 = 100.2 + (-0.674 \times 15.2) \approx 100.2 + (-10.24) = 89.96](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fadca8183-f536-4768-a5dc-a2639f5a41ee%2F521840b0-127c-451a-a1ee-63957d72c59b%2Fm3ledj9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
Assume that adults have IQ scores that are normally distributed with a mean of 100.2 and a standard deviation of 15.2. Find the first quartile \( Q_1 \), which is the IQ score separating the bottom 25% from the top 75%. (Hint: Draw a graph.)
---
**The first quartile is** [ ]
(Type an integer or decimal rounded to one decimal place as needed.)
### Detailed Explanation (for Educational Context)
To determine the first quartile (\( Q_1 \)) for the given normally distributed IQ scores, follow these steps:
1. **Understanding the First Quartile \( Q_1 \)**:
- The first quartile (\( Q_1 \)) is the value below which 25% of the data falls.
- In a standard normal distribution, this corresponds to a z-score that separates the lower 25% of the distribution from the upper 75%.
2. **Standard Normal Distribution**:
- The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1.
- To find the z-score for the first quartile in a standard normal distribution, you can look up the value in a z-table or use a statistical tool.
3. **Converting to the Given Normal Distribution**:
- For the given normal distribution with mean (\( \mu \)) = 100.2 and standard deviation (\( \sigma \)) = 15.2, we need to find the corresponding IQ score for the z-score found in step 2.
4. **Calculation Steps**:
- Identify the z-score for \( Q_1 \) (25th percentile) using a z-table or statistical software. Typically, this z-score is approximately -0.674.
- Use the z-score formula for a normal distribution:
\[
Q_1 = \mu + (z \times \sigma)
\]
where:
- \(\mu = 100.2\)
- \(z = -0.674\)
- \(\sigma = 15.2\)
5. **Final Calculation**:
- Plug in the values:
\[
Q_1 = 100.2 + (-0.674 \times 15.2) \approx 100.2 + (-10.24) = 89.96
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