The 78th percentile i (Round to two decimal places 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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![**Understanding Percentiles in a Normal Distribution**
**Problem Statement:**
Assume the random variable X is normally distributed with a mean (μ) of 50 and a standard deviation (σ) of 7. Find the 78th percentile.
**Instructions:**
Click the icon for a table of areas under the normal curve.
**Solution:**
The 78th percentile is represented by the z-score where the cumulative area to the left under the standard normal curve is 0.78. Use the standard normal distribution table to find this z-score.
**Graph Explanation:**
On the right side, there's a diagram of a normal distribution curve that shows a shaded area representing the cumulative area up to z.
**Table – Standard Normal Distribution (Excerpt):**
| z | .00 | .01 | .02 | .03 | .04 | .05 | .06 | .07 | .08 | .09 |
|-------|-------|-------|-------|-------|-------|-------|-------|-------|-------|-------|
| 0.7 | 0.7580| 0.7608| 0.7636| 0.7664| 0.7692| 0.7719| 0.7745| 0.7771| 0.7794| 0.7823|
To find the 78th percentile:
1. Locate the closest value to 0.78 in the table. The value 0.7794 at z = 0.77 approximates this.
2. Convert the z-score:
\[ X = μ + z \times σ = 50 + 0.77 \times 7 \approx 55.39 \]
**Conclusion:**
The 78th percentile of X is approximately 55.39. Round to two decimal places as needed.
Explore further by adjusting the parameters or checking other percentile values using the normal distribution table.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe55ca600-6ecd-431e-a2f3-7959f6a21e5c%2F1b1745fe-46ca-48e8-8939-522823af99c3%2Fac7atxb_processed.png&w=3840&q=75)
Transcribed Image Text:**Understanding Percentiles in a Normal Distribution**
**Problem Statement:**
Assume the random variable X is normally distributed with a mean (μ) of 50 and a standard deviation (σ) of 7. Find the 78th percentile.
**Instructions:**
Click the icon for a table of areas under the normal curve.
**Solution:**
The 78th percentile is represented by the z-score where the cumulative area to the left under the standard normal curve is 0.78. Use the standard normal distribution table to find this z-score.
**Graph Explanation:**
On the right side, there's a diagram of a normal distribution curve that shows a shaded area representing the cumulative area up to z.
**Table – Standard Normal Distribution (Excerpt):**
| z | .00 | .01 | .02 | .03 | .04 | .05 | .06 | .07 | .08 | .09 |
|-------|-------|-------|-------|-------|-------|-------|-------|-------|-------|-------|
| 0.7 | 0.7580| 0.7608| 0.7636| 0.7664| 0.7692| 0.7719| 0.7745| 0.7771| 0.7794| 0.7823|
To find the 78th percentile:
1. Locate the closest value to 0.78 in the table. The value 0.7794 at z = 0.77 approximates this.
2. Convert the z-score:
\[ X = μ + z \times σ = 50 + 0.77 \times 7 \approx 55.39 \]
**Conclusion:**
The 78th percentile of X is approximately 55.39. Round to two decimal places as needed.
Explore further by adjusting the parameters or checking other percentile values using the normal distribution table.
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