The electric cooperative needs to know the mean household usage of electricity by its non-commercial customers in kWh per day.  They would like the estimate to have a maximum error of 0.15
 kWh.  A previous study found that for an average family the standard deviation is 1.9
 kWh and the mean is 16.7
 kWh per day.  If they are using a 98%
 level of confidence, how large of a sample is required to estimate the mean usage of electricity? Round your answer up to the next integer.

To calculate the required sample size for estimating the mean with a specified maximum error, we use the following formula for sample size \( n \):

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

Where:
- \( Z \) is the z-score corresponding to the desired confidence level,
- \( \sigma \) is the population standard deviation,
- \( E \) is the maximum error (or margin of error) allowed.

### Step-by-Step Calculation:

1. **Identify the values:**
   - Confidence level = 98% → corresponding z-score (\( Z \)) = 2.33
   - Standard deviation (\( \sigma \)) = 1.9 kWh
   - Maximum error (\( E \)) = 0.15 kWh

2. **Substitute the values into the formula:**

\[
n = \left( \frac{2.33 \cdot 1.9}{0.15} \right)^2
\]

3. **Simplify the expression:**

\[
n = \left( \frac{4.427}{0.15} \right)^2 = \left( 29.5133 \right)^2
\]

4. **Square the result:**

\[
n = 870.90
\]

Since the sample size must be a whole number, and we round up, the required sample size is:

\[
n = 871
\]

### Final Answer:
The required sample size is **871**.

Would you like more details or have any other questions?  

Here are 5 related questions to consider:
1. How does changing the confidence level affect the required sample size?
2. What happens if the standard deviation is different in future studies?
3. How is the z-score determined for different confidence levels?
4. What if a smaller margin of error is desired?
5. How can the cooperative ensure the sample is representative of the population?

**Tip**: When estimating means, a higher confidence level or a smaller margin of error will always increase the required sample size.

The electric cooperative needs to know the mean household usage of electricity by its non-commercial customers in kWh per day. They would like the estimate to have a maximum error of 0.15 kWh. A previous study found that for an average family the standard deviation is 1.9 kWh and the mean is 16.7 kWh per day. If they are using a 98% level of confidence, how large of a sample is required to estimate the mean usage of electricity? Round your answer up to the next integer.

Learn how to calculate the required sample size for estimating mean household electricity usage with a maximum error of 0.15 kWh. The problem uses a 98% confidence level and a population standard deviation of 1.9 kWh. Follow the step-by-step solution to determine the sample size, including the use of z-scores and confidence intervals. Suitable for college-level statistics students.

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