To determine the lower boundary of the 95% confidence interval for the mean number of vacation days spent in Colorado, follow these steps:
1. Identify the given information:
- Mean ([tex]\(\mu\)[/tex]) = 4.8 days
- Standard Error (SE) = 0.125 days
- Z-score for a 95% confidence interval = 1.96
2. Calculate the margin of error (ME):
The margin of error is calculated using the formula:
[tex]\[
\text{Margin of Error} = Z \times \text{Standard Error}
\][/tex]
Substituting the given values:
[tex]\[
\text{Margin of Error} = 1.96 \times 0.125 = 0.245
\][/tex]
3. Determine the lower boundary of the confidence interval:
The lower boundary is found by subtracting the margin of error from the mean:
[tex]\[
\text{Lower Boundary} = \mu - \text{Margin of Error}
\][/tex]
Substituting the given values:
[tex]\[
\text{Lower Boundary} = 4.8 - 0.245 = 4.555
\][/tex]
Therefore, the lower boundary of the interval within which we are 95% confident that the true mean number of vacation days lies is 4.555 days.
