The mean diastolic blood pressure for a random sample of 60 people was 80 millimeters of mercury. If the standard deviation of individual blood pressure readings is known to be 11 millimeters of mercury, find a 95% confidence interval for the true mean diastolic blood pressure of all people. Then give its lower limit and upper limit.
Carry your intermediate computations to at least three decimal places. Round your answers to one decimal place. (If necessary, consult a list of formulas.)

To find the 95% confidence interval for the true mean diastolic blood pressure, we can use the formula for the confidence interval of the population mean when the population standard deviation is known:

\[
\text{Confidence Interval} = \bar{x} \pm z_{\alpha/2} \times \left(\frac{\sigma}{\sqrt{n}}\right)
\]

Where:
- \(\bar{x} = 80\) (sample mean)
- \(\sigma = 11\) (population standard deviation)
- \(n = 60\) (sample size)
- \(z_{\alpha/2}\) is the critical value for a 95% confidence level (which corresponds to 1.96 for a normal distribution)

### Step-by-step calculation:

1. **Compute the standard error (SE):**

\[
\text{SE} = \frac{\sigma}{\sqrt{n}} = \frac{11}{\sqrt{60}} \approx 1.420
\]

2. **Compute the margin of error (MOE):**

\[
\text{MOE} = z_{\alpha/2} \times \text{SE} = 1.96 \times 1.420 \approx 2.783
\]

3. **Compute the confidence interval:**

The confidence interval is:

\[
80 \pm 2.783
\]

This gives:
- **Lower limit**: \( 80 - 2.783 \approx 77.2 \)
- **Upper limit**: \( 80 + 2.783 \approx 82.8 \)

### Final result:

- The 95% confidence interval for the true mean diastolic blood pressure is \( (77.2, 82.8) \).
- **Lower limit**: 77.2
- **Upper limit**: 82.8

Let me know if you'd like more details on this, or if you have any questions!

Here are 5 related questions for further exploration:
1. How would the confidence interval change if the sample size was larger, say 100 people?
2. What happens to the confidence interval if we increase the confidence level to 99%?
3. How would the calculation differ if the population standard deviation was unknown?
4. What role does the normal distribution play in confidence interval estimation?
5. How would we interpret the confidence interval in a real-world health study?

**Tip**: Increasing the sample size reduces the standard error, leading to a narrower confidence interval.

The mean diastolic blood pressure for a random sample of 60 people was 80 millimeters of mercury. If the standard deviation of individual blood pressure readings is known to be 11 millimeters of mercury, find a 95% confidence interval for the true mean diastolic blood pressure of all people. Then give its lower limit and upper limit. Carry your intermediate computations to at least three decimal places. Round your answers to one decimal place. (If necessary, consult a list of formulas.)

Find the 95% confidence interval for the true mean diastolic blood pressure with a sample of 60 people, a mean of 80 mmHg, and a standard deviation of 11 mmHg. Learn how to calculate standard error, margin of error, and the confidence interval step-by-step. Suitable for high school and college students.

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