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The problem you’ve provided involves constructing a confidence interval for the mean reduction in systolic blood pressure. Here's a step-by-step guide on how to approach this:

### Given Data:
- Sample mean (\(\bar{x}\)) = 58.5
- Sample size (\(n\)) = 283
- Standard deviation (\(s\)) = 14.5
- Confidence level = 80%

### Steps to Solve:
1. **Find the Z-score for the 80% confidence interval:**
   The confidence interval formula for the mean is:

   \[
   \bar{x} \pm Z_{\alpha/2} \cdot \frac{s}{\sqrt{n}}
   \]

   For an 80% confidence level, the corresponding Z-score (\(Z_{\alpha/2}\)) is approximately 1.28.

2. **Calculate the standard error (SE):**

   \[
   SE = \frac{s}{\sqrt{n}} = \frac{14.5}{\sqrt{283}} 
   \]

3. **Compute the margin of error (MOE):**

   \[
   MOE = Z_{\alpha/2} \cdot SE = 1.28 \cdot SE
   \]

4. **Construct the confidence interval:**

   \[
   \text{Lower limit} = \bar{x} - MOE
   \]
   \[
   \text{Upper limit} = \bar{x} + MOE
   \]

I'll now calculate the values for the interval.

The effectiveness of a blood-pressure drug is being investigated. An experimenter finds that, on average, the reduction in systolic blood pressure is 58.5 for a sample of size 283 and standard deviation 14.5. Estimate how much the drug will lower a typical patient's systolic blood pressure (using a 80% confidence level).

Estimate the reduction in systolic blood pressure caused by a drug using a confidence interval. Given a sample mean of 58.5, sample size of 283, standard deviation of 14.5, and an 80% confidence level, calculate the confidence interval with step-by-step guidance.

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