26. Flight Data: Find the range, variance, and standard deviation. Express answers using appropriate units, such as "minutes."
Refer to Data Set 15 in Appendix B and use the times required to taxi out for takeoff.
Flight 1 1 1 1 1 1 1 1 1 1 1 1 3 3 3 3 3 3 3 3 3 3 3 3 19 19 19 19 19 19 19 19 19 19 19 19 21 21 21 21 21 21 21 21 21 21 21 21
Taxi Out 30 19 12 19 18 22 37 13 14 15 31 15 16 14 15 27 19 22 22 23 16 13 16 18 15 12 19 18 21 20 13 15 43 18 17 19 13 20 12 17 35 19 22 43 49 45 13 23
Taxi In 12 13 8 21 17 11 12 12 15 26 9 11 6 7 4 11 10 7 11 10 3 7 7 5 10 10 16 13 9 8 4 3 8 16 9 5 13 4 6 21 29 5 27 9 12 7 36 12
Range = highest value - lowest
Find the mean = 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.
The manager of a hotel has stated that the mean guest bill for a weekend is Birr 400 or
less. A member of the hotel’s accounting staff has noticed that the total charges for guest
bills have been increasing in recent months. The accountant will use a sample of weekend
guest bills to test the manager’s claim.
To find the range, variance, and standard deviation for the flight data (times required to taxi out for takeoff), follow these steps:
1. Calculate the range:
- The range is the difference between the maximum and minimum values in the data set.
- In this case, the maximum value is 49 minutes, and the minimum value is 3 minutes.
- So the range is 49 - 3 = 46 minutes.
2. Calculate the variance:
- Variance measures the spread of the data around the mean.
- To calculate the variance, you need to find the mean first.
Mean = (sum of all values) / (number of values)
= (30 + 19 + 12 + ... + 7 + 36 + 12) / 48
= 639 / 48
= 13.3125 minutes
- Subtract the mean from each value, square the result, and sum them all up.
- Divide the sum by the number of values.
Variance = ((30 - 13.3125)^2 + (19 - 13.3125)^2 + ... + (12 - 13.3125)^2 + (12 - 13.3125)^2) / 48
= (611.9241 + 8.3186 + ... + 2.4591 + 2.4591) / 48
= 2523.1646 / 48
= 52.5659 minutes^2
3. Calculate the standard deviation:
- The standard deviation is the square root of the variance.
Standard deviation = √(52.5659)
= 7.25 minutes
Therefore, the answers are:
- Range: 46 minutes
- Variance: 52.5659 minutes^2
- Standard deviation: 7.25 minutes
To find the range, variance, and standard deviation of the taxi out times for the given flight data, you can follow these steps:
Step 1: Organize the data
The first step is to organize the taxi out times. Looking at the given flight data, we have the following taxi out times:
Flight: 30 19 12 19 18 22 37 13 14 15 31 15 16 14 15 27 19 22 22 23 16 13 16 18 15 12 19 18 21 20 13 15 43 18 17 19 13 20 12 17 35 19 22 43 49 45 13 23
Step 2: Calculate the range
The range is the difference between the maximum and minimum values in the data set. To find the range, you need to find the maximum and minimum taxi out times:
Maximum taxi out time: 49 minutes
Minimum taxi out time: 12 minutes
Range = Maximum - Minimum = 49 - 12 = 37 minutes
So, the range of the taxi out times is 37 minutes.
Step 3: Calculate the variance
To find the variance, you can use the following formula:
Variance = Σ(x - μ)² / n
where Σ represents the sum, x represents the individual data points, μ represents the mean, and n represents the number of data points.
First, calculate the mean (average) of the taxi out times:
Sum of taxi out times: 854 minutes
Number of taxi out times: 50
Mean = Sum / n = 854 / 50 = 17.08 minutes
Next, calculate the squared differences from the mean for each data point:
(x - μ)² = (30 - 17.08)² + (19 - 17.08)² + (12 - 17.08)² + ... + (12 - 17.08)²
Calculate the sum of the squared differences:
Σ(x - μ)² = (30 - 17.08)² + (19 - 17.08)² + (12 - 17.08)² + ... + (12 - 17.08)² = 446.48
Finally, calculate the variance:
Variance = Σ(x - μ)² / n = 446.48 / 50 = 8.93 minutes²
So, the variance of the taxi out times is 8.93 minutes squared.
Step 4: Calculate the standard deviation
To find the standard deviation, you can take the square root of the variance. The standard deviation gives you a measure of the spread or dispersion of the data points.
Standard deviation = √(Variance) = √(8.93) = 2.99 minutes
So, the standard deviation of the taxi out times is 2.99 minutes.