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

4 7 6 5 8 8 4 8 9  8 8 4 10 10 1 1 1 9 1 9 1 1 8 8 1 8 8 9 8 7 7 1 8 8 10 1 8 7 9 7 8 9 7 7 7 8 1 6 10 7

Develop a 95% confidence interval estimate of the population mean rating for Miami. round your answers to two decimal places




To develop a 95% confidence interval for the population mean rating, we use the following formula:

\[
CI = \bar{x} \pm z \left(\frac{s}{\sqrt{n}}\right)
\]

Where:
- \(\bar{x}\) = sample mean
- \(z\) = z-score for 95% confidence level (1.96 for a two-tailed test)
- \(s\) = sample standard deviation
- \(n\) = sample size (which is 50)

### Step 1: Calculate the sample mean (\(\bar{x}\))

\[
\bar{x} = \frac{\sum x_i}{n}
\]

### Step 2: Calculate the sample standard deviation (\(s\))

\[
s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}
\]

### Step 3: Compute the margin of error

\[
\text{Margin of Error} = z \times \frac{s}{\sqrt{n}}
\]

### Step 4: Construct the confidence interval

\[
CI = \left( \bar{x} - \text{Margin of Error}, \bar{x} + \text{Margin of Error} \right)
\]

Let me calculate these values for you.The sample mean (\(\bar{x}\)) is 6.32, and the sample standard deviation (\(s\)) is 3.01.

Using a 95% confidence level, the confidence interval estimate for the population mean rating is:

\[
(5.48, 7.16)
\]

This means we are 95% confident that the true mean rating for Miami International Airport lies between 5.48 and 7.16.

Would you like more details on any step, or do you have any questions?

### Related questions:
1. What factors can influence the width of a confidence interval?
2. How does the sample size affect the confidence interval?
3. How is the z-score determined for different confidence levels?
4. What are the assumptions underlying the confidence interval calculation?
5. How would the confidence interval change if the sample had more variability?

### Tip:
A larger sample size reduces the margin of error, providing a more precise estimate of the population mean.

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: 4, 7, 6, 5, 8, 8, 4, 8, 9, 8, 8, 4, 10, 10, 1, 1, 1, 9, 1, 9, 1, 1, 8, 8, 1, 8, 8, 9, 8, 7, 7, 1, 8, 8, 10, 1, 8, 7, 9, 7, 8, 9, 7, 7, 7, 8, 1, 6, 10, 7. Develop a 95% confidence interval estimate of the population mean rating for Miami. Round your answers to two decimal places.

Calculate the 95% confidence interval for the population mean rating of Miami International Airport based on a sample of 50 business travelers. Learn how to apply key statistical concepts like the sample mean, standard deviation, z-scores, and confidence intervals.

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