https://file.aiquickdraw.com/mathbot/file/1729995631025vmve3qvd.jpg

To calculate a 99% confidence interval for the true mean diastolic blood pressure, we can use the formula:

\[
\text{Confidence Interval} = \bar{x} \pm z \cdot \frac{\sigma}{\sqrt{n}}
\]

where:
- \(\bar{x} = 91\) (the sample mean),
- \(\sigma = 10\) (the population standard deviation),
- \(n = 50\) (the sample size),
- \(z\) is the z-score for a 99% confidence level.

For a 99% confidence level, the z-score is approximately \(2.576\).

### Step-by-Step Solution
1. **Calculate the Standard Error (SE):**
   \[
   SE = \frac{\sigma}{\sqrt{n}} = \frac{10}{\sqrt{50}} \approx 1.414
   \]

2. **Calculate the Margin of Error (ME):**
   \[
   ME = z \cdot SE = 2.576 \cdot 1.414 \approx 3.644
   \]

3. **Determine the Confidence Interval:**
   \[
   \text{Lower limit} = \bar{x} - ME = 91 - 3.644 \approx 87.4
   \]
   \[
   \text{Upper limit} = \bar{x} + ME = 91 + 3.644 \approx 94.6
   \]

### Final Answer
- **Lower limit:** 87.4
- **Upper limit:** 94.6

Would you like a more detailed explanation of each step?

---

Here are some related questions to expand your understanding:

1. How would the confidence interval change if we used a 95% confidence level?
2. What effect does increasing the sample size have on the width of the confidence interval?
3. Why do we use the z-score instead of the t-score in this calculation?
4. How would we calculate the confidence interval if the population standard deviation was unknown?
5. What does the confidence interval imply about the true mean diastolic blood pressure?

**Tip:** Confidence intervals provide a range within which we expect the population mean to lie, based on our sample data and chosen confidence level.

The mean diastolic blood pressure for a random sample of 50 people was 91 millimeters of mercury. If the standard deviation of individual blood pressure readings is known to be 10 millimeters of mercury, find a 99% confidence interval for the true mean diastolic blood pressure of all people. Then give its lower limit and upper limit.

Calculate the 99% confidence interval for the mean diastolic blood pressure of a sample population with a known mean of 91 and a standard deviation of 10. This detailed solution includes key statistical concepts like confidence intervals, standard error, and z-scores.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation