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To solve this problem, we need to construct an 80% confidence interval for the mean reduction in systolic blood pressure. Here are the steps we'll follow:

### Step 1: Gather information from the question
- Sample mean (\(\bar{x}\)): 58.5
- Sample size (\(n\)): 283
- Sample standard deviation (\(s\)): 14.5
- Confidence level: 80%

### Step 2: Calculate the Z-value for an 80% confidence level
For an 80% confidence interval, the confidence level leaves 20% in the tails of the normal distribution. Since we are constructing a two-tailed interval, each tail will have 10%. From the Z-table, the Z-value corresponding to 10% in one tail is approximately 1.28.

### Step 3: Calculate the standard error (SE)
The standard error of the mean is calculated as:

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

Let's calculate this next.The standard error (SE) is approximately \( 0.86 \).

### Step 4: Calculate the margin of error (ME)
The margin of error is calculated as:

\[
ME = Z \times SE = 1.28 \times 0.86
\]

Let's calculate the margin of error.The margin of error (ME) is approximately \( 1.10 \).

### Step 5: Construct the confidence interval
The confidence interval is calculated as:

\[
\text{Confidence Interval} = \bar{x} \pm ME = 58.5 \pm 1.10
\]

This gives us the lower and upper bounds of the confidence interval.

Let's calculate those bounds now.The 80% confidence interval for the mean reduction in systolic blood pressure is approximately:

\[
57.4 < \mu < 59.6
\]

This is the tri-linear inequality form requested by the problem.

Would you like any further explanation or details on this? Here are five related questions you could explore:

1. How would the confidence interval change with a 95% confidence level?
2. How do you interpret the confidence interval in terms of drug effectiveness?
3. What assumptions are made when constructing a confidence interval?
4. How does the sample size affect the margin of error?
5. How would you compute the confidence interval if the population standard deviation were known?

**Tip**: Remember, higher confidence levels result in wider intervals since they provide a higher degree of certainty in capturing the population mean.

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 size of 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). Enter your answer as a tri-linear inequality accurate to one decimal place.

Learn how to calculate the 80% confidence interval for the mean reduction in systolic blood pressure based on a sample of 283 patients. This problem involves key concepts in statistics, including confidence intervals and standard error. The margin of error and the Z-value for an 80% confidence level are also discussed.

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