To construct the confidence interval for the sample mean, we'll use the t-distribution since the sample size is relatively small, and the population standard deviation is unknown. Here's how to construct the confidence interval:

Given Information

• Sample mean ($\bar{X}$): 25
• Sample standard deviation ($S_x$): 4.5
• Significance level ($\alpha$): 0.05
• Degrees of freedom ($df$): 20

Step 1: Find the Critical Value

For a confidence interval with $\alpha = 0.05$ and $df = 20$, we need to find the critical value ($t_{\alpha/2}$) from the t-distribution table. Since the confidence level is $1 - \alpha = 0.95$, the critical value for a two-tailed test at the 95% confidence level and 20 degrees of freedom is:

$t_{\alpha/2, df=20} \approx 2.086$

Step 2: Calculate the Margin of Error

The margin of error (ME) is calculated using the formula:

$\text{ME} = t_{\alpha/2} \cdot \frac{S_x}{\sqrt{n}}$

However, we don’t have the sample size ($n$) explicitly mentioned, so we assume that the degrees of freedom are $n - 1 = 20$, which implies $n = 21$. Substituting the values:

$\text{ME} = 2.086 \cdot \frac{4.5}{\sqrt{21}} \approx 2.046$

Step 3: Construct the Confidence Interval

The confidence interval for the mean is:

$\bar{X} \pm \text{ME}$

$25 \pm 2.046$

$(25 - 2.046, 25 + 2.046) \approx (22.954, 27.046)$

Final Answer

The 95% confidence interval for the sample mean is approximately:

$(22.95, 27.05)$

Would you like further details or have any questions about the process?

Related Questions

1. How would the confidence interval change if the sample size increased?
2. What happens to the confidence interval if we use a 99% confidence level instead of 95%?
3. Why do we use the t-distribution instead of the normal distribution here?
4. How can you interpret the meaning of this confidence interval in a practical context?
5. How would the interval change if the sample standard deviation ($S_x$) was higher?

Tip: Always check the degrees of freedom and the corresponding critical value from the t-distribution table for accurate results.