The International Air Transport Association surveys business travelers to develop quality ratings for transatlantic gateway airports. The maximum possible rating is 10. Suppose a simple random sample of 50 business travelers is selected and each traveler is asked to provide a rating for the Miami International Airport. The ratings obtained from the sample of 50 business travelers follow.
Excel or Minitab users: The data set is available in the file named "Miami." All data sets can be found in your eBook or on your Student CD.
6 4 6 8 7 7 6 3 3 8 10 4 8
7 8 7 5 9 5 8 4 3 8 5 5 4
4 4 8 4 5 6 2 5 9 9 8 4 8
9 9 5 9 7 8 3 10 8 9 6
Develop a 95% confidence interval estimate of the population mean rating for Miami (to 2 decimals).
95% Confidence:
( , )
To develop a 95% confidence interval estimate of the population mean rating for Miami International Airport, we can use the following formula:
Confidence Interval = sample mean ± (critical value * standard error)
First, let's calculate the sample mean:
Average Rating = (6 + 4 + 6 + 8 + 7 + 7 + 6 + 3 + 3 + 8 + 10 + 4 + 8 + 7 + 8 + 7 + 5 + 9 + 5 + 8 + 4 + 3 + 8 + 5 + 5 + 4 + 4 + 8 + 4 + 5 + 6 + 2 + 5 + 9 + 9 + 8 + 4 + 8 + 9 + 9 + 5 + 9 + 7 + 8 + 3 + 10 + 8 + 9 + 6) / 50
Average Rating ≈ 6.66 (rounded to two decimal places)
Next, let's calculate the standard error:
Standard Error = (standard deviation / √n)
To calculate the standard deviation, we need to find the variance first:
Variance = [(6 - 6.66)^2 + (4 - 6.66)^2 + (6 - 6.66)^2 + (8 - 6.66)^2 + ... + (8 - 6.66)^2 + (9 - 6.66)^2 + (6 - 6.66)^2] / 49
Variance ≈ 3.368 (rounded to three decimal places)
Standard Deviation = √(Variance) ≈ √3.368 ≈ 1.835 (rounded to three decimal places)
Now, we can calculate the standard error:
Standard Error = 1.835 / √50 ≈ 0.259 (rounded to three decimal places)
The critical value for a 95% confidence interval can be obtained from the t-distribution table or using a statistical software. For a sample size of 50 (n=50), the critical value is approximately 2.009 (rounded to three decimal places).
Finally, we can plug these values into the formula to calculate the confidence interval:
Confidence Interval = 6.66 ± (2.009 * 0.259)
Confidence Interval ≈ (6.66 ± 0.521)
Therefore, the 95% confidence interval estimate for the population mean rating for Miami International Airport is approximately (6.14, 7.18) (rounded to two decimal places).