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The problem asks for an estimate of how much a drug will lower a typical patient's systolic blood pressure using a 90% confidence interval. 

From the image, the following values are provided:
- Sample mean ( \( \bar{x} \) ) = 31 (the average reduction in systolic blood pressure)
- Sample standard deviation ( \( s \) ) = 5
- Sample size ( \( n \) ) = 10 (as 10 individuals were selected)

### Step 1: Determine the critical value ( \( z^* \) ) for a 90% confidence level
For a 90% confidence interval, we look for the critical value that corresponds to the middle 90% of the standard normal distribution. This leaves 5% in each tail of the distribution. The \( z^* \) value is approximately 1.645 for a 90% confidence interval.

### Step 2: Calculate the margin of error ( \( E \) )
The formula for the margin of error is:

\[
E = z^* \times \frac{s}{\sqrt{n}}
\]

Substitute the given values:

\[
E = 1.645 \times \frac{5}{\sqrt{10}} = 1.645 \times \frac{5}{3.162} \approx 1.645 \times 1.581 \approx 2.6
\]

### Step 3: Calculate the confidence interval
Now, the confidence interval is:

\[
\text{Confidence Interval} = \bar{x} \pm E = 31 \pm 2.6
\]

Thus, the 90% confidence interval is approximately \( 31 \pm 2.6 \), meaning the drug is estimated to lower the patient's systolic blood pressure by between **28.4** and **33.6** units.

### Final answer:
The point estimate is **31** and the margin of error is **2.6**.

Do you want more details on any of the steps or have any questions?

Here are some related questions you might want to consider:
1. What if the confidence level was 95%? How would the critical value change?
2. How does increasing the sample size affect the margin of error?
3. What is the difference between using a z-distribution and a t-distribution for confidence intervals?
4. How would you interpret this confidence interval in a real-world medical context?
5. Why is it important to use a confidence interval instead of just reporting the sample mean?

**Tip**: Increasing your sample size generally decreases the margin of error, leading to a more precise estimate.

Estimate how much the drug will lower a typical patient's systolic blood pressure using a 90% confidence level. The sample mean is 31, the sample standard deviation is 5, and the sample size is 10. Provide the point estimate with its margin of error.

Learn how to estimate how much a drug will lower systolic blood pressure using a 90% confidence interval. This example involves a sample mean of 31, a standard deviation of 5, and a sample size of 10. The point estimate and margin of error are calculated step-by-step.

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