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This question is asking for the 80% confidence interval to estimate the population mean of systolic blood pressure reduction, based on the provided data. Let’s go step by step:

### Given Data:
- Sample mean (\(\bar{x}\)) = 24.4 mmHg
- Sample size (\(n\)) = 304
- Sample standard deviation (\(s\)) = 12.7 mmHg
- Confidence level = 80%

### Step 1: Find the Z-score for an 80% confidence level
For an 80% confidence interval, the critical value \(z_{\alpha/2}\) (for a two-tailed test) corresponds to the central 80% of the standard normal distribution, leaving 10% in each tail. The corresponding Z-score for 80% confidence is approximately **1.28**.

### Step 2: Calculate the standard error (SE)
The standard error of the mean is calculated as:

\[
SE = \frac{s}{\sqrt{n}} = \frac{12.7}{\sqrt{304}} 
\]

### Step 3: Compute the margin of error (ME)
The margin of error is:

\[
ME = z_{\alpha/2} \times SE
\]

### Step 4: Construct the confidence interval
The confidence interval for the population mean is given by:

\[
\bar{x} - ME < \mu < \bar{x} + ME
\]

Let me calculate that for you.The 80% confidence interval for the population mean reduction in systolic blood pressure is approximately:

\[
23.5 \, \text{mmHg} < \mu < 25.3 \, \text{mmHg}
\]

This means we are 80% confident that the true mean reduction in systolic blood pressure falls between 23.5 mmHg and 25.3 mmHg.

Do you have any questions or would you like more details?

Here are 5 related questions you might find useful:
1. How does the confidence level affect the width of the confidence interval?
2. What is the formula for calculating a 95% confidence interval, and how would it differ from the 80% interval?
3. What is the importance of sample size in estimating population parameters?
4. How does the sample standard deviation impact the margin of error?
5. Can we use the Z-distribution for smaller sample sizes, or is a t-distribution more appropriate?

**Tip:** Increasing the confidence level (e.g., from 80% to 95%) will increase the width of the confidence interval, as we require a higher degree of certainty.

The effectiveness of a blood-pressure drug is being investigated. An experimenter finds that, on average, the reduction in systolic blood pressure is 24.4 mmHg (millimeters of mercury) for a sample of size 304 and a sample standard deviation of 12.7 mmHg. How much of mmHg will lower for a typical patient's systolic blood pressure after taking the drug? Estimate with an 80% confidence.

Learn how to estimate the reduction in systolic blood pressure for a population after taking a drug, using an 80% confidence interval. This example calculates the confidence interval based on a sample of size 304, with a mean reduction of 24.4 mmHg and a standard deviation of 12.7 mmHg.

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