Math Problem Statement

The water works commission needs to know the mean household usage of water by the residents of a small town in gallons per day. They would like the estimate to have a maximum error of 0.14 gallons. A previous study found that for an average family the standard deviation is 2 gallons and the mean is 16 gallons per day. If they are using a 95% level of confidence, how large of a sample is required to estimate the mean usage of water? Round your answer up to the next integer.

Solution

To determine the required sample size for estimating the mean household water usage, we use the formula for the sample size of a mean when the standard deviation is known:

n=(ZσE)2n = \left( \frac{Z \cdot \sigma}{E} \right)^2

Where:

  • nn is the required sample size,
  • ZZ is the z-score corresponding to the desired confidence level (95% confidence level corresponds to Z=1.96Z = 1.96),
  • σ\sigma is the standard deviation of the population (σ=2\sigma = 2 gallons),
  • EE is the maximum error of the estimate (E=0.14E = 0.14 gallons).

Step-by-Step Calculation:

  1. Plug the values into the formula: n=(1.9620.14)2n = \left( \frac{1.96 \cdot 2}{0.14} \right)^2

  2. Calculate the numerator: 1.962=3.921.96 \cdot 2 = 3.92

  3. Divide by the maximum error: 3.920.1428\frac{3.92}{0.14} \approx 28

  4. Square the result: 282=78428^2 = 784

  5. Round up to the next whole number: n=784n = 784

Final Answer:

The required sample size is 784 households.


Do you want further clarification on any of the steps or related concepts?

Related Questions:

  1. What is the significance of using the z-score in confidence interval calculations?
  2. How would the sample size change if the maximum error was reduced to 0.1 gallons?
  3. Why is it important to round up to the nearest whole number for the sample size?
  4. How does the standard deviation affect the required sample size?
  5. What is the relationship between confidence level and sample size?

Tip:

When designing surveys or experiments, always ensure that the desired level of precision is realistic given the resources and time available for data collection.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Statistics
Confidence Intervals
Sample Size Calculation

Formulas

n = (Z * σ / E)^2

Theorems

Central Limit Theorem
Properties of the Normal Distribution

Suitable Grade Level

Grades 10-12