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 calculate the 95% confidence interval estimate for the population mean rating for Miami, we can use the formula:
Confidence Interval = sample mean ± (critical value * standard deviation / square root of sample size)
First, let's calculate the sample mean:
Sum all the ratings: 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 = 365
Sample mean = 365 / 50 = 7.3
Next, let's calculate the standard deviation:
Step 1: Compute the differences between each rating and the sample mean, squared them, and sum them up.
(6-7.3)^2 + (4-7.3)^2 + (6-7.3)^2 + (8-7.3)^2 + (7-7.3)^2 + (7-7.3)^2 + (6-7.3)^2 + (3-7.3)^2 + (3-7.3)^2 + (8-7.3)^2 + (10-7.3)^2 + (4-7.3)^2 + (8-7.3)^2 + (7-7.3)^2 + (8-7.3)^2 + (7-7.3)^2 + (5-7.3)^2 + (9-7.3)^2 + (5-7.3)^2 + (8-7.3)^2 + (4-7.3)^2 + (3-7.3)^2 + (8-7.3)^2 + (5-7.3)^2 + (5-7.3)^2 + (4-7.3)^2 + (4-7.3)^2 + (8-7.3)^2 + (4-7.3)^2 + (5-7.3)^2 + (6-7.3)^2 + (2-7.3)^2 + (5-7.3)^2 + (9-7.3)^2 + (9-7.3)^2 + (8-7.3)^2 + (4-7.3)^2 + (8-7.3)^2 + (9-7.3)^2 + (9-7.3)^2 + (5-7.3)^2 + (9-7.3)^2 + (7-7.3)^2 + (8-7.3)^2 + (3-7.3)^2 + (10-7.3)^2 + (8-7.3)^2 + (9-7.3)^2 + (6-7.3)^2 = 358.2
Step 2: Divide the sum by (n-1) where n is the sample size.
Standard deviation = sqrt(358.2 / (50-1)) = sqrt(358.2 / 49) = sqrt(7.3102) ≈ 2.70
Now, we need to find the critical value for a 95% confidence level using a t-distribution with (n-1) degrees of freedom. Since the sample size is 50, the degrees of freedom will be 49.
Using a t-table or software, the critical value for a 95% confidence level with 49 degrees of freedom is approximately 2.01.
Finally, let's calculate the confidence interval:
Confidence Interval = 7.3 ± (2.01 * 2.70 / sqrt(50))
Confidence Interval = 7.3 ± (2.01 * 2.70 / 7.07)
Confidence Interval = 7.3 ± (5.447 / 7.07)
Confidence Interval = 7.3 ± 0.771
Confidence Interval ≈ (6.53, 7.97)
Therefore, the 95% confidence interval estimate of the population mean rating for Miami is approximately (6.53, 7.97).
To develop a confidence interval estimate of the population mean rating for Miami International Airport, we need to calculate the sample mean and the margin of error.
Here are the steps to calculate the 95% confidence interval:
1. Find the sample mean:
Add up all the ratings in the sample and divide by the number of ratings (50 in this case).
In this example, the sample mean is (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 = 6.54
2. Calculate the standard deviation (SD) of the sample:
Subtract the sample mean from each rating, square each difference, sum them up, divide by n-1 (where n is the sample size), and take the square root.
In this example, the formula for standard deviation is sqrt(((6-6.54)^2 + (4-6.54)^2 + (6-6.54)^2 + ... + (6-6.54)^2) / (50-1)).
3. Determine the standard error (SE):
Divide the standard deviation by the square root of the sample size.
In this example, the standard error is SD / sqrt(sample size) = SD / sqrt(50).
4. Find the margin of error:
Multiply the standard error by the critical value, which is obtained from the t-distribution for the desired confidence level.
For a 95% confidence level, the critical value is approximately 1.96.
5. Calculate the confidence interval:
Subtract the margin of error from the sample mean to get the lower bound, and add the margin of error to the sample mean to get the upper bound.
The confidence interval is (sample mean - margin of error, sample mean + margin of error).
Now let's plug in the values and calculate the confidence interval:
Sample mean = 6.54
SD = √(((6-6.54)^2 + (4-6.54)^2 + (6-6.54)^2 + ... + (6-6.54)^2) / (50-1))
Standard error (SE) = SD / sqrt(50)
Margin of error = 1.96 * SE
Lower bound = Sample mean - margin of error
Upper bound = Sample mean + margin of error
Once you calculate these values, you'll have the 95% confidence interval estimate of the population mean rating for Miami International Airport.
(5.70, 6.94)
t table
(5.73, 6.95)