Following are airborne times (in minutes) for 10 randomly selected flights from San Francisco to Washington Dulles:

\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|}
\hline
270 & 256 & 267 & 286 & 274 & 275 & 266 & 258 & 271 & 281 \\
\hline
\end{tabular}

1. Calculate a [tex]$90 \%$[/tex] confidence interval for the mean airborne time for flights from San Francisco to Washington Dulles. (Round your answers to three decimal places.)
[tex]\[
\boxed{\text{Lower Bound}} \quad \boxed{\text{Upper Bound}}
\][/tex]

2. Interpret a 90\% confidence interval for the mean airborne time for flights from San Francisco to Washington Dulles.

A. There is a 90\% chance that the true difference in the mean airborne time for flights from San Francisco to Washington Dulles is directly in the middle of these two values.

B. We are 90\% confident that the true mean airborne time for flights from San Francisco to Washington Dulles is directly in the middle of these two values.

C. There is a 90\% chance that the true mean airborne time for flights from Washington Dulles to San Francisco is directly in the middle of these two values.

D. We are 90\% confident that the true mean airborne time for flights from Washington Dulles to San Francisco is between these two values.

E. We are 90\% confident that the mean airborne time for flights from San Francisco to Washington Dulles is between these two values.



Answer :

To calculate the 90% confidence interval for the mean airborne time for flights from San Francisco to Washington Dulles based on the provided data set of airborne times, we will follow several steps:

1. List the Data: The given airborne times (in minutes) for 10 flights are:
270, 256, 267, 286, 274, 275, 266, 258, 271, 281.

2. Calculate the Sample Mean ([tex]\( \bar{X} \)[/tex]):
[tex]\[ \bar{X} = \frac{270 + 256 + 267 + 286 + 274 + 275 + 266 + 258 + 271 + 281}{10} = 270.4 \text{ minutes} \][/tex]

3. Calculate the Sample Standard Deviation (s):
[tex]\[ s = 9.324 \text{ minutes} \quad \text{(to three decimal places)} \][/tex]

4. Determine the Sample Size (n):
[tex]\[ n = 10 \][/tex]

5. Calculate the Standard Error of the Mean ([tex]\( SE \)[/tex]):
[tex]\[ SE = \frac{s}{\sqrt{n}} = \frac{9.324}{\sqrt{10}} = 2.948 \text{ minutes} \quad \text{(to three decimal places)} \][/tex]

6. Determine the t-multiplier for a 90% Confidence Level:
For a 90% confidence level and degrees of freedom [tex]\( df = n - 1 = 9 \)[/tex]:
[tex]\[ t_{\alpha/2} = 1.833 \quad \text{(from t-distribution table)} \][/tex]

7. Calculate the Margin of Error (ME):
[tex]\[ ME = t_{\alpha/2} \times SE = 1.833 \times 2.948 = 5.405 \text{ minutes} \quad \text{(to three decimal places)} \][/tex]

8. Compute the Confidence Interval:
[tex]\[ \text{Lower Bound} = \bar{X} - ME = 270.4 - 5.405 = 264.995 \text{ minutes} \quad \text{(to three decimal places)} \][/tex]
[tex]\[ \text{Upper Bound} = \bar{X} + ME = 270.4 + 5.405 = 275.805 \text{ minutes} \quad \text{(to three decimal places)} \][/tex]

So, the 90% confidence interval for the mean airborne time for flights from San Francisco to Washington Dulles is:
[tex]\[ \boxed{264.995} \quad \text{to} \quad \boxed{275.805} \][/tex]

Interpretation of the 90% Confidence Interval:
We are 90% confident that the mean airborne time for flights from San Francisco to Washington Dulles is between these two values.