**Ten Mile Run** ~ A random sample of 50 runners from all the runners who finished the Cherry Blossom Ten Mile Run in 2009 had a mean finishing time 94.5 minutes and a standard deviation of 18.5 minutes.

What is the lower bound for a 95% confidence interval for the actual mean finishing time? Give your answer to 4 decimal places.

To find the lower bound for a 95% confidence interval for the actual mean finishing time, we use the formula for the confidence interval for a mean when the sample standard deviation is known and the sample size is relatively large (so the \( t \)-distribution approximates the normal distribution):

\[ \text{CI} = \bar{x} \pm t^* \left( \frac{s}{\sqrt{n}} \right) \]

Where:
- \(\bar{x}\) is the sample mean
- \(s\) is the sample standard deviation
- \(n\) is the sample size
- \(t^*\) is the critical value from the \( t \)-distribution for a 95% confidence level with \( df = n - 1 \)

Given:
- \(\bar{x} = 94.5\) minutes
- \(s = 18.5\) minutes
- \(n = 50\)
- \(df = 50 - 1 = 49\)

For a 95% confidence interval, \( \alpha = 0.05 \) and \( \alpha/2 = 0.025 \). We look up the \( t \)-value for \( df = 49 \) and \( \alpha/2 = 0.025 \) in the \( t \)-distribution table or use a calculator to find \( t^* \approx 2.0096 \).

Next, we calculate the standard error of the mean (SEM):

\[ \text{SEM} = \frac{s}{\sqrt{n}} = \frac{18.5}{\sqrt{50}} \approx \frac{18.5}{7.0711} \approx 2.6151 \]

Then, we calculate the margin of error (ME):

\[ \text{ME} = t^* \times \text{SEM} = 2.0096 \times 2.6151 \approx 5.2552 \]

Finally, the lower bound of the confidence interval is:

\[ \text{Lower Bound} = \bar{x} - \text{ME} = 94.5 - 5.2552 \approx 89.2448 \]

Thus, the lower bound for the 95% confidence interval for the actual mean finishing time is approximately \( 89.2448 \) minutes.

Would you like more details or have any questions?

Here are 8 related questions to expand your understanding:

1. How is the critical value \( t^* \) determined for different confidence levels?
2. What are the assumptions required for constructing a confidence interval for the mean?
3. How does increasing the sample size affect the confidence interval width?
4. What is the difference between a one-sided and a two-sided confidence interval?
5. How do you interpret a 95% confidence interval in practical terms?
6. What happens to the confidence interval if the standard deviation increases?
7. Why is the \( t \)-distribution used when the sample size is relatively small?
8. How do you calculate a confidence interval if the population standard deviation is known?

**Tip:** Always double-check the degrees of freedom and corresponding \( t \)-value when constructing a confidence interval, as mistakes in these values can lead to incorrect results.

Learn how to calculate the lower bound of a 95% confidence interval for the mean finishing time based on a sample of runners. Includes step-by-step solution using sample mean, sample standard deviation, and critical values from the t-distribution. Suitable for undergraduate level statistics.

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