e of water by the residents of a small town in gallons per day. Th tudy found that for an average family the standard deviation is confidence, how large of a sample is required to estimate the m
e of water by the residents of a small town in gallons per day. Th tudy found that for an average family the standard deviation is confidence, how large of a sample is required to estimate the m
Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
18th Edition
ISBN:9780079039897
Author:Carter
Publisher:Carter
Chapter10: Statistics
Section: Chapter Questions
Problem 22SGR
Related questions
Question
![**Sample Size Estimation for Mean Household Water Usage**
**Problem Statement:**
The water works commission needs to determine the mean household water usage by the residents of a small town, measured in gallons per day. The goal is to ensure that the estimate has a maximum allowable error of 0.14 gallons. Based on previous research, it's known that the standard deviation of household water usage is 1.9 gallons, and the mean usage is 16.7 gallons per day. Given a 98% confidence level, the question is: what sample size is required to estimate the mean usage of water to satisfy these criteria? The answer should be rounded up to the next integer.
**Known Parameters:**
- Desired maximum error (E): 0.14 gallons
- Standard deviation (σ): 1.9 gallons
- Confidence level: 98%
**Explanation:**
The sample size (n) required can be calculated using the formula for the margin of error in the context of estimating a population mean with a known standard deviation:
\[ E = Z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}} \]
Where:
- \( E \) is the maximum error,
- \( Z_{\alpha/2} \) is the critical value from the standard normal distribution that corresponds to the desired confidence level,
- \( \sigma \) is the standard deviation,
- \( n \) is the sample size.
**Steps:**
1. Determine the critical value (\( Z_{\alpha/2} \)) for a 98% confidence level.
2. Rearrange the formula to solve for \( n \):
\[ n = \left( \frac{Z_{\alpha/2} \times \sigma}{E} \right)^2 \]
3. Plug in the known values and compute \( n \).
4. Round up to the nearest integer.
To achieve a clearer understanding, actual execution and calculation steps would typically be covered in educational materials.
**Conclusion:**
The calculation results in the necessary sample size required to estimate the mean house water usage with the specified confidence level and margin of error. Verifying these computations and concepts ensures proper resource planning and accurate data collection for the water works commission.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9d050742-a776-4688-997d-9b380febe959%2Fd274b68f-1529-4638-9f9b-3eadeec7f98a%2F3v0trgl_processed.png&w=3840&q=75)
Transcribed Image Text:**Sample Size Estimation for Mean Household Water Usage**
**Problem Statement:**
The water works commission needs to determine the mean household water usage by the residents of a small town, measured in gallons per day. The goal is to ensure that the estimate has a maximum allowable error of 0.14 gallons. Based on previous research, it's known that the standard deviation of household water usage is 1.9 gallons, and the mean usage is 16.7 gallons per day. Given a 98% confidence level, the question is: what sample size is required to estimate the mean usage of water to satisfy these criteria? The answer should be rounded up to the next integer.
**Known Parameters:**
- Desired maximum error (E): 0.14 gallons
- Standard deviation (σ): 1.9 gallons
- Confidence level: 98%
**Explanation:**
The sample size (n) required can be calculated using the formula for the margin of error in the context of estimating a population mean with a known standard deviation:
\[ E = Z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}} \]
Where:
- \( E \) is the maximum error,
- \( Z_{\alpha/2} \) is the critical value from the standard normal distribution that corresponds to the desired confidence level,
- \( \sigma \) is the standard deviation,
- \( n \) is the sample size.
**Steps:**
1. Determine the critical value (\( Z_{\alpha/2} \)) for a 98% confidence level.
2. Rearrange the formula to solve for \( n \):
\[ n = \left( \frac{Z_{\alpha/2} \times \sigma}{E} \right)^2 \]
3. Plug in the known values and compute \( n \).
4. Round up to the nearest integer.
To achieve a clearer understanding, actual execution and calculation steps would typically be covered in educational materials.
**Conclusion:**
The calculation results in the necessary sample size required to estimate the mean house water usage with the specified confidence level and margin of error. Verifying these computations and concepts ensures proper resource planning and accurate data collection for the water works commission.
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