A random sample is selected of 25 patients who have recovered from a specific surgical procedure. The mean recovery time, from surgery to discharge, is 2.27 days, and the sample standard deviation is 0.93 days.

Compute a 95% confidence interval for the mean recovery time from the surgery.

Give your final answer with at least 33 dp.

Lower bound of the 95% CI (SMALLER NUMBER HERE):

Answer 1 Question 2

Upper bound of the 95% CI (BIGGER NUMBER HERE):

Answer 2 Question 2


To calculate the 95% confidence interval (CI) for the mean recovery time, we use the formula for a confidence interval for the mean:

\[
\text{CI} = \bar{x} \pm t_{\alpha/2} \times \frac{s}{\sqrt{n}}
\]

Where:
- \(\bar{x}\) is the sample mean.
- \(t_{\alpha/2}\) is the t-score corresponding to the desired confidence level and degrees of freedom (\(df = n - 1\)).
- \(s\) is the sample standard deviation.
- \(n\) is the sample size.

### Step 1: Calculate the degrees of freedom
\[
df = n - 1 = 25 - 1 = 24
\]

### Step 2: Find the t-score for a 95% confidence level with 24 degrees of freedom

### Step 3: Plug the values into the formula and calculate the confidence interval

Let's compute these values.The 95% confidence interval for the mean recovery time from the surgery is:

- **Lower bound** (smaller number): 1.886
- **Upper bound** (bigger number): 2.654

Would you like further details or have any questions?

Here are 5 related questions to consider:
1. How would the confidence interval change if the sample size were larger?
2. What assumptions are made when using the t-distribution for this calculation?
3. How does the variability in the data (standard deviation) affect the confidence interval?
4. What is the impact of choosing a different confidence level, such as 90% or 99%?
5. How might outliers in the data affect the mean recovery time and its confidence interval?

**Tip:** The t-distribution is used instead of the z-distribution when the sample size is small (typically \(n < 30\)) and the population standard deviation is unknown.

Learn how to calculate the 95% confidence interval for the mean recovery time from a surgical procedure using sample mean, sample standard deviation, and the t-distribution. Detailed step-by-step solution provided.

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