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 steps:
Step 1: Compute the sample mean (x̄) and the sample standard deviation (s) from the given data set.
- The sample mean, x̄, is the average rating of the sample.
- The sample standard deviation, s, measures the variability of the ratings within the sample.
Using the given data set, we can calculate the sample mean and the sample standard deviation as follows:
Sample mean (x̄) = (sum of all ratings) / (number of ratings)
= (6 + 4 + 6 + 8 + 7 + 7 + ... + 9 + 6) / 50
Sample standard deviation (s) = square root of [(sum of squared differences from the mean) / (number of ratings - 1)]
= square root of [(6 - x̄)² + (4 - x̄)² + (6 - x̄)² + ... + (6 - x̄)² + (9 - x̄)² + (6 - x̄)²] / (50 - 1)
Note: Calculating these values will give you specific numbers for x̄ and s, which are needed for the next steps.
Step 2: Determine the standard error (SE) of the sample mean.
- The standard error (SE) represents the variability of sample means that would be obtained if multiple samples were taken from the same population.
The formula for calculating the standard error (SE) is given by:
SE = s / sqrt(n)
where s is the sample standard deviation and n is the sample size.
Step 3: Calculate the margin of error (ME).
- The margin of error (ME) determines the range around the sample mean within which the population mean is likely to fall.
The formula for calculating the margin of error (ME) is given by:
ME = t * SE
where t is the critical value from the t-distribution corresponding to the desired confidence level and degrees of freedom (df).
For a 95% confidence level, the critical value (t) can be obtained from statistical tables or software. With a sample size of 50, the degrees of freedom is 50 - 1 = 49.
Step 4: Calculate the lower and upper bounds of the confidence interval.
- The lower bound is obtained by subtracting the margin of error (ME) from the sample mean (x̄).
- The upper bound is obtained by adding the margin of error (ME) to the sample mean (x̄).
The 95% confidence interval for the population mean rating for Miami International Airport is (lower bound, upper bound). This interval represents the range within which we can be 95% confident the true population mean rating lies.
To calculate the 95% confidence interval estimate of the population mean rating for Miami, we will use the formula:
Confidence Interval = sample mean ± (critical value) * (standard deviation / √sample size)
Step 1: Calculate the sample mean:
Add up all the ratings from the sample and divide by the sample size:
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 = 337
Sample mean = 337 / 50 = 6.74 (rounded to 2 decimal places)
Step 2: Calculate the standard deviation:
Find the squared difference between each rating and the sample mean. Add up these squared differences and divide by the sample size. Finally, take the square root of this value.
Step-by-step calculations are as follows:
(6-6.74)^2 + (4-6.74)^2 + (6-6.74)^2 + (8-6.74)^2 + ... + (8-6.74)^2 + (9-6.74)^2 + (6-6.74)^2 = 127.4
Sample standard deviation = √(127.4/49) = 1.43 (rounded to 2 decimal places)
Step 3: Calculate the critical value.
Since we want a 95% confidence interval, we need to find the critical value for a sample size of 50. Consult the Z-table or use a calculator to find the critical value corresponding to a 95% confidence level. The critical value for a 95% confidence level is approximately 1.96.
Step 4: Calculate the confidence interval.
Using the formula mentioned above:
Confidence Interval = 6.74 ± (1.96) * (1.43 / √50)
Calculations:
Confidence Interval = 6.74 ± (1.96) * (0.203)
Lower bound = 6.74 - (1.96 * 0.203) = 6.74 - 0.398 = 6.342 (rounded to 3 decimal places)
Upper bound = 6.74 + (1.96 * 0.203) = 6.74 + 0.398 = 7.138 (rounded to 3 decimal places)
Therefore, the 95% confidence interval estimate of the population mean rating for Miami is (6.342, 7.138) (rounded to 2 decimal places).
95% = mean ± 1.96 SEm
SEm = SD/√n
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
I'll let you do the calculations.