To construct an 80% confidence interval for the mean electricity usage per family, we will use the formula for the confidence interval for the population mean when the population standard deviation is known:

$\text{Confidence Interval} = \bar{x} \pm z \times \frac{\sigma}{\sqrt{n}}$

Where:

• $\bar{x}$ is the sample mean.
• $z$ is the z-score corresponding to the confidence level.
• $\sigma$ is the population standard deviation.
• $n$ is the sample size.

Given dataTo construct an 80% confidence interval for the mean electricity usage per family, we will use the formula for the confidence interval for the population mean when the population standard deviation is known:

$\text{Confidence Interval} = \bar{x} \pm z \times \frac{\sigma}{\sqrt{n}}$

Where:

• $\bar{x}$ is the sample mean.
• $z$ is the z-score corresponding to the confidence level.
• $\sigma$ is the population standard deviation.
• $n$ is the sample size.

Given

• Population standard deviation ($\sigma$) = 1.4 kWh.
• Sample mean ($\bar{x}$) = 15.5 kWh.
• Sample size ($n$) = 142.
• Confidence level = 80%, so the z-score for an 80% confidence interval (from the z-table) is approximately $z = 1.28$.

Step 1: Calculate the standard error of the mean (SE)

$\text{SE} = \frac{\sigma}{\sqrt{n}} = \frac{1.4}{\sqrt{142}} \approx \frac{1.4}{11.916} \approx 0.1175$

Step 2: Calculate the margin of error (ME)

$\text{ME} = z \times \text{SE} = 1.28 \times 0.1175 \approx 0.1504$

Step 3: Construct the confidence interval

$\text{Confidence Interval} = \bar{x} \pm \text{ME} = 15.5 \pm 0.1504$

Lower limit = $15.5 - 0.1504 \approx 15.3$

Upper limit = $15.5 + 0.1504 \approx 15.7$

Final Answer:

The 80% confidence interval for the mean usage of electricity is approximately 15.3 kWh to 15.7 kWh per day.

---

Would you like more details or explanations on any part?

Here are 5 related questions you could explore:

1. How is the z-score for different confidence levels determined?
2. What happens if the population standard deviation is unknown?
3. How would the confidence interval change with a larger sample size?
4. Why is the z-score used instead of the t-score in this case?
5. How does increasing the confidence level affect the width of the confidence interval?

Tip: As you increase the confidence level (e.g., from 80% to 95%), the confidence interval becomes wider because you are more certain that the true population mean lies within the interval.