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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.
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.
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