The international air transport association surveys business travelers to develop quality ratings for transatlantic gateway airports. the maximum possible rating is . suppose a simple random sample of business travelers

General
1

the international air transport association surveys business travelers to develop quality ratings for transatlantic gateway airports. the maximum possible rating is . suppose a simple random sample of business travelers is selected and each traveler is asked to provide a rating for the miami international airport. the file miami contains the ratings obtained from the sample of business travelers. develop a confidence interval estimate of the population mean rating for miami. round your answers to two decimal places.

2

LectureNotes said the international air transport association surveys business travelers to develop quality ratings for transatlantic gateway airports. the maximum possible rating is . suppose a simple random sample of business travelers is selected and each traveler is asked to provide a rating for the miami international airport. the file miami contains the ratings obtained from the sample of business travelers. develop a confidence interval estimate of the population mean rating for miami. round your answers to two decimal places.

Answer:

To develop a confidence interval estimate for the population mean rating for Miami International Airport based on the sample data, follow these steps:

  1. Calculate the Sample Mean (\bar{x}) and Sample Standard Deviation (s):

    • First, compute the mean (\bar{x}) of the sample ratings.
    • Then, compute the standard deviation (s) of the sample ratings.

  2. Determine the Sample Size (n):

    • Count the number of ratings in the sample to find the sample size (n).

  3. Select the Confidence Level:

    • Typically, a 95% confidence level is used, but this can vary based on the requirement.

  4. Find the Critical Value (t^*):

    • For a 95% confidence level and a sample size n, the critical value can be found using the t-distribution table with n-1 degrees of freedom.

  5. Compute the Margin of Error (ME):

    • The margin of error is calculated using the formula:
      ME = t^* \times \frac{s}{\sqrt{n}}

  6. Construct the Confidence Interval:

    • The confidence interval for the population mean (\mu) is given by:
      \bar{x} \pm ME

Let’s go through an example assuming we have the following sample data:

  • Sample ratings: [4.2, 3.8, 4.5, 4.0, 3.9, 4.1, 4.3, 3.7, 4.4, 4.0]

Step-by-Step Calculation:

  1. Calculate the Sample Mean (\bar{x}):

    \bar{x} = \frac{4.2 + 3.8 + 4.5 + 4.0 + 3.9 + 4.1 + 4.3 + 3.7 + 4.4 + 4.0}{10} = \frac{41.9}{10} = 4.19

  2. Calculate the Sample Standard Deviation (s):

    s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} = \sqrt{\frac{(4.2-4.19)^2 + (3.8-4.19)^2 + \ldots + (4.0-4.19)^2}{10-1}} = 0.25

  3. Determine the Sample Size (n):

    n = 10

  4. Select the Confidence Level and Find the Critical Value (t^*):

    • For a 95% confidence level and n-1 = 9 degrees of freedom, the critical value t^* is approximately 2.262 (from the t-distribution table).

  5. Compute the Margin of Error (ME):

    ME = 2.262 \times \frac{0.25}{\sqrt{10}} = 2.262 \times 0.079 = 0.18

  6. Construct the Confidence Interval:

    \text{Lower Limit} = \bar{x} - ME = 4.19 - 0.18 = 4.01
    \text{Upper Limit} = \bar{x} + ME = 4.19 + 0.18 = 4.37

Therefore, the 95% confidence interval estimate of the population mean rating for Miami International Airport is (4.01, 4.37).