To calculate the margin of error and the confidence interval, we need to use the following formulas:

1. Margin of Error (ME): $ME = z \cdot \left(\frac{s}{\sqrt{n}}\right)$ where $z$ is the z-score corresponding to the desired confidence level, $s$ is the sample standard deviation, and $n$ is the sample size.

2. Confidence Interval (CI): $CI = \left(\bar{x} - ME, \bar{x} + ME\right)$ where $\bar{x}$ is the sample mean.

Let's first find the z-scores for the 80% and 93% confidence levels:

• For 80% confidence level, the z-score is approximately 1.282.
• For 93% confidence level, the z-score is approximately 1.812.

Now we can calculate the margin of error and the confidence intervals for both confidence levels.

Given:

• Sample mean ($\bar{x}$) = 1.85
• Sample standard deviation ($s$) = 0.67
• Sample size ($n$) = 21

Let's calculate these values.Here are the results:

• 80% Confidence Level:

  • Margin of Error: $\approx 0.19$
  • Confidence Interval: $(1.66, 2.04)$

• 93% Confidence Level:

  • Margin of Error: $\approx 0.26$
  • Confidence Interval: $(1.59, 2.11)$

Would you like more details or have any questions?

Here are five questions you can ask next:

1. How do you interpret a confidence interval in a practical context?
2. Can you show how to calculate the z-scores manually?
3. What is the difference between confidence level and confidence interval?
4. How would these calculations change for a different sample size?
5. How can we use confidence intervals in hypothesis testing?

Tip: Always ensure your sample is randomly selected to make your confidence intervals reliable and representative of the population.