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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

Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
18th Edition
ISBN:9780079039897
Author:Carter
Publisher:Carter
Chapter10: Statistics
Section: Chapter Questions
Problem 25SGR
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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
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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