https://file.aiquickdraw.com/mathbot/file/17286556301550ssxg8nt.jpg

We need to estimate how much the blood-pressure drug will lower a typical patient's systolic blood pressure, given a sample mean (x̄) of 31, a sample standard deviation (s) of 9, and a sample size (n) of 14 individuals. The confidence level is 90%, so we will calculate the margin of error and the confidence interval around the sample mean.

### Step-by-Step Solution:

1. **Sample Mean (x̄):** 31
2. **Sample Standard Deviation (s):** 9
3. **Sample Size (n):** 14
4. **Confidence Level:** 90%

For a 90% confidence level, we use the t-distribution because the sample size is small (n = 14 < 30), and we don't know the population standard deviation. We need the t-critical value (t\(_{\alpha/2}\)) for a 90% confidence interval and 13 degrees of freedom (n - 1).

#### Step 1: Find t-critical value
Using a t-table or calculator, for 13 degrees of freedom and a 90% confidence level, the t-critical value is approximately **1.771**.

#### Step 2: Calculate the margin of error (E)
The margin of error \(E\) is calculated using the formula:
\[
E = t_{\alpha/2} \times \frac{s}{\sqrt{n}}
\]
Substitute the known values:
\[
E = 1.771 \times \frac{9}{\sqrt{14}} \approx 1.771 \times 2.406 \approx 4.26
\]

#### Step 3: Calculate the confidence interval
The confidence interval is given by:
\[
\text{Point estimate} \pm \text{Margin of error}
\]
\[
31 \pm 4.26
\]

Thus, the estimate of how much the drug will lower the systolic blood pressure is:
\[
31.0 \pm 4.3
\]

### Final Answer:
The point estimate is **31.0** mmHg, and the margin of error is **4.3** mmHg.

Would you like further clarification or details on any step?

### Related Questions:
1. How do we interpret a 90% confidence interval in this context?
2. Why do we use the t-distribution instead of the normal distribution here?
3. What would happen to the margin of error if the sample size increased?
4. How is the confidence level related to the margin of error?
5. Can the standard deviation affect how wide the confidence interval is?

### Tip:
The higher the confidence level, the larger the t-critical value, which increases the margin of error, making the confidence interval wider.

The effectiveness of a blood-pressure drug is being investigated. An experimenter randomly selects 14 individuals and records their systolic blood pressure before the drug is given. He then checks their systolic blood pressure after administration of the drug several days later. The experimenter calculates the sample mean (x̄) and the sample standard deviation (s). He finds the reduction in systolic blood pressure has a mean of 31 and a standard deviation of 9. Estimate how much the drug will lower a typical patient's systolic blood pressure using a 90% confidence level. Give your answers to one decimal place and provide the point estimate with its margin of error.

Learn how to estimate the reduction in systolic blood pressure after administering a drug, using a 90% confidence interval. The solution includes key statistical concepts like margin of error and the t-distribution, with a step-by-step breakdown.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation