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.
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
a. Develop a 95% confidence interval estimate of the population mean rating for Miami (to 2 decimals).
95% Confidence:
Find the mean first = sum of scores/number of scores
Subtract each of the scores from the mean and square each difference. Find the sum of these squares. Divide that by the number of scores to get variance.
Standard deviation = square root of variance
SEm = SD/√n
P = .05 at ± 1.96 SEm
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability
I'll let you do the calculations.
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)
1. Calculate the sample mean: Add up all the ratings and divide by the sample size.
Sample Mean = (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) / 50
= 6.5 (rounded to 1 decimal place)
2. Find the critical value: Since the sample size is less than 30, we use a t-distribution and find the critical value with a confidence level of 95% and a degrees of freedom of n-1, which is 50-1 = 49.
Using a t-distribution table or a statistical calculator, the critical value for a 95% confidence level and 49 degrees of freedom is approximately 2.009.
3. Calculate the standard error: The standard error measures the variability of the sample mean.
Standard Error = Sample Standard Deviation / √Sample Size
First, calculate the sample standard deviation. To do this, subtract the sample mean from each rating, square the result, sum up all the squared differences, divide by the sample size minus 1, and finally, take the square root.
Sample Standard Deviation = √(((6-6.5)^2 + (4-6.5)^2 + ... + (6-6.5)^2) / (50-1))
= √(84.5 / 49)
= √1.724
= 1.31 (rounded to 2 decimal places)
Then, calculate the standard error:
Standard Error = 1.31 / √50
= 0.185 (rounded to 3 decimal places)
4. Calculate the confidence interval:
Confidence Interval = 6.5 ± (2.009 * 0.185)
= 6.5 ± 0.372 (rounded to 3 decimal places)
Therefore, the 95% confidence interval estimate of the population mean rating for Miami International Airport is approximately 6.128 to 6.872 (rounded to 3 decimal places).