Assume the standard deviation of the heights of all five-year-old boys is 3.5 inches. How many five-year-old boys need to be sampled if we want to be 90% sure that the population mean height is estimated correctly to within 0.5 inches? 102 325 133 81 189
Assume the standard deviation of the heights of all five-year-old boys is 3.5 inches. How many five-year-old boys need to be sampled if we want to be 90% sure that the population mean height is estimated correctly to within 0.5 inches? 102 325 133 81 189
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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![**Statistical Sampling Example Problem**
**Problem Statement:**
Assume the standard deviation of the heights of all five-year-old boys is 3.5 inches. How many five-year-old boys need to be sampled if we want to be 90% sure that the population mean height is estimated correctly to within 0.5 inches?
**Options:**
1. 102
2. 325
3. 133
4. 81
5. 189
**Explanation:**
The problem involves determining the required sample size to estimate a population mean with a certain level of confidence and margin of error. This type of question is typical in inferential statistics where a sample size helps make estimates about a population parameter.
To calculate this, you can use the formula for the sample size for estimating a population mean:
\[ n = \left( \frac{Z \cdot \sigma}{E} \right)^2 \]
Where:
- \( n \) is the sample size
- \( Z \) is the z-value corresponding to the desired confidence level
- \( \sigma \) is the population standard deviation
- \( E \) is the margin of error
For a 90% confidence level, the z-value is approximately 1.645. The given standard deviation (\( \sigma \)) is 3.5 inches, and the desired margin of error (\( E \)) is 0.5 inches.
Plugging in these values:
\[ n = \left( \frac{1.645 \cdot 3.5}{0.5} \right)^2 \]
\[ n = \left( \frac{5.7575}{0.5} \right)^2 \]
\[ n = (11.515)^2 \]
\[ n = 132.551 \]
Since the sample size needs to be a whole number, you would round up to ensure the margin of error is met, so \( n = 133 \).
Therefore, the correct answer is:
- 133](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F49b6872f-8540-4d67-8f44-b13749eac041%2F7db8d32b-9780-4102-8c47-bf871e57eeed%2Fha5j3k4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Statistical Sampling Example Problem**
**Problem Statement:**
Assume the standard deviation of the heights of all five-year-old boys is 3.5 inches. How many five-year-old boys need to be sampled if we want to be 90% sure that the population mean height is estimated correctly to within 0.5 inches?
**Options:**
1. 102
2. 325
3. 133
4. 81
5. 189
**Explanation:**
The problem involves determining the required sample size to estimate a population mean with a certain level of confidence and margin of error. This type of question is typical in inferential statistics where a sample size helps make estimates about a population parameter.
To calculate this, you can use the formula for the sample size for estimating a population mean:
\[ n = \left( \frac{Z \cdot \sigma}{E} \right)^2 \]
Where:
- \( n \) is the sample size
- \( Z \) is the z-value corresponding to the desired confidence level
- \( \sigma \) is the population standard deviation
- \( E \) is the margin of error
For a 90% confidence level, the z-value is approximately 1.645. The given standard deviation (\( \sigma \)) is 3.5 inches, and the desired margin of error (\( E \)) is 0.5 inches.
Plugging in these values:
\[ n = \left( \frac{1.645 \cdot 3.5}{0.5} \right)^2 \]
\[ n = \left( \frac{5.7575}{0.5} \right)^2 \]
\[ n = (11.515)^2 \]
\[ n = 132.551 \]
Since the sample size needs to be a whole number, you would round up to ensure the margin of error is met, so \( n = 133 \).
Therefore, the correct answer is:
- 133
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