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B-1 A P P E N D I X B Example of Estimating Operations from Sample Counts Using Forms from FAA-APO-85-7, Statistical Sampling of Aircraft Operations at Non-Towered Airports
B-2 Using Form 1 as a worksheet, the total daily operaons are tallied. For a sample that includes 2 weeks per season, 8 Form 1s would be needed. The total daily operaons will be transferred to Form 2, Weekly Data, Column B. Using Form 2 as a worksheet, the number of weekly total operaons is calculated. Again, there would be Form 2 for each Form 1 completed (8 total). The data is then transferred to Form 3. Sll using Form 2 as a worksheet, the average daily operaons for each week are calculated. The data from the two Form 2s for each season are transferred to Form 3 resulng in 4 Form 3s. In this step, Form 3 is used to find average daily operaons for each season and to calculate addional numbers that will be used on Form 4. Sll using Form 3, the number of total operaons for each season is calculated and transferred to Form 4. Using Form 4, the sample data is expanded to esmate total annual operaons and to calculate addional measures. Sll on Form 4, addional measures can be calculated, such as precision of the esmate of annual operaons, confidence interval and percent sample error. STEP 1 Daily Operaons STEP 2 Weekly Operaons STEP 3 Average Daily Operaons for Each Week STEP 4 Average Daily Operaons for Each Season STEP 5 Total Operaons for Each Season STEP 6 Total Annual Operaons Esmate STEP 7 Confidence Interval and Sampling Error 8 Form 1s Transferred to 8 Form 2s Transferred to 4 Form 3s Transferred to 1 Form 4 Using the attached forms, follow these steps to estimate annual operations from sample counts. These forms are reproduced from FAA-APO-85-7, Statistical Sampling of Aircraft Operations at Non-Towered Airports. The following example is also taken directly from the FAA-APO-85-7. While the forms may appear intimidating, all the mathematical functions can be completed on a simple calculator and include calculations no harder than addition, subtraction, multiplication, division, and square root.
B-3
B-4 Example FORM 1 Airport Name Tri-City State Season 1 Week 1 Day Day 1 Day 2 Day 3 Day 4 Day 5 Day 6 Day 7 Time of Day 7/21/2014 7/22/2014 7/23/2014 7/24/2014 7/25/2014 7/26/2014 7/27/2014 Midnight 1:00 AM 2:00 AM 3:00 AM 4:00 AM 5:00 AM 6:00 AM 7:00 AM 2 8:00 AM 4 9:00 AM 10:00 AM 2 2 4 11:00 AM 2 2 12:00 PM 6 2 2 2 2 2 1:00 PM 2 2 2 2:00 PM 10 2 2 2 3:00 PM 2 2 2 4:00 PM 6 4 5:00 PM 2 2 6:00 PM 7:00 PM 8:00 PM 9:00 PM 10:00 PM 11:00 PM TOTALS 28 6 8 6 10 4 14 FORM 1 Airport Name Tri-City State Season 1 Week 2 Day Day 1 Day 2 Day 3 Day 4 Day 5 Day 6 Day 7 Time of Day 9/16/2014 9/17/2014 9/18/2014 9/19/2014 9/20/2014 9/21/2014 9/22/2014 Midnight 1:00 AM 2:00 AM 3:00 AM 4:00 AM 5:00 AM 6:00 AM 7:00 AM 8:00 AM 9:00 AM 2 10:00 AM 2 11:00 AM 12:00 PM 4 10 1:00 PM 2 2 6 2:00 PM 3:00 PM 4 4:00 PM 5:00 PM 2 6:00 PM 7:00 PM 2 8:00 PM 9:00 PM 10:00 PM 11:00 PM TOTALS 8 2 2 12 0 6 6
B-5 FORM 1 Airport Name Season Week Day Day 1 Day 2 Day 3 Day 4 Day 5 Day 6 Day 7 Time of Day Midnight 1:00 AM 2:00 AM 3:00 AM 4:00 AM 5:00 AM 6:00 AM 7:00 AM 8:00 AM 9:00 AM 10:00 AM 11:00 AM 12:00 PM 1:00 PM 2:00 PM 3:00 PM 4:00 PM 5:00 PM 6:00 PM 7:00 PM 8:00 PM 9:00 PM 10:00 PM 11:00 PM TOTALS
B-6 INSTRUCTIONS FOR FORM 2 (Reproduced from FAA-APO-85-7) WEEKLY DATA Column A Fill in the date of each day sampled on the lines in Column A. Column B Fill in the number of operations counted on each date on the lines in Column B. Operations for each date are obtained from the bottom row of Form 1 if daily observations are taken. Box C Add the values in Column B. Enter the answer, total weekly operations, in Box C. Box D Divide the value in Box C by the number of days counted in the week (normally 7). Enter the answer, average daily operations, in Box D, This number will be used in Column E and transferred to Form 3, Column A. Column E Subtract the value in Box D from the value on line 'in Column B. Enter the answer on line 1 of Column E. Repeat this step, subtracting D from each value in Column B. Column F Square each value in Column E and enter the answers in Column F. (All squared numbers will be positive.) Box G Add the values in Column F. Enter the answer in Box G, This number will be transferred to Form 3, Column K. In the boxes at the top of Form 2 fill in the name of the airport sampled and the season number and week number from which the data were collected. If you have two weeks of sampling, then you will need two of these forms. The purpose of Form 2 is to calculate total operations and average daily operations for each week, and other numbers that will be used on Form 3. There will be one Form 2 produced for each Form 1 completed.
B-7 Example FORM 2 Airport Name Tri-City State Season 1 Week 1 A B E F Day DATE Daily Operaons Column B Minus Box D Column E Squared 1 7/21/2014 28 17.14 293.78 2 7/22/2014 6 -4.86 23.62 3 7/23/2014 8 -2.86 8.18 4 7/24/2014 6 -4.86 23.62 5 7/25/2014 10 -0.86 0.74 6 7/26/2014 4 -6.86 47.06 7 7/27/2014 14 3.14 9.86 C = TOTAL WEEKLY OPERATIONS 76 G = Sum of Column F 406.86 D = AVERAGE DAILY OPERATIONS 10.86 Columns Box C = The sum of Colum B D = C ÷ by number of days counted Mulply the number directly to the le in column E by itself Box G Box D
B-8 FORM 2 Airport Name Season Week A B E F Day DATE Daily Operaons Column B minus Box D Column E Squared 1 2 3 4 5 6 7 C = TOTAL WEEKLY OPERATIONS G = Sum of Column F D = AVERAGE DAILY OPERATIONS
B-9 INSTRUCTIONS FOR FORM 3 (Reproduced from FAA-APO-85-7) SEASONAL DATA Column A Fill in the average daily operations per week on the lines in Column A. Average daily operations per week are transferred from Form 2. Box D. Column A may have 2 or more lines depending on the number of weeks sampled. Box B Add the values in Column A and enter the answer in Box B. Box C Divide the value in Box B by n (find n in box at top of Form 3). Enter the answer in Box C. This number will be used in Column D and transferred to Form 4, Column H. Column D Subtract the value in Box C from the value on line 1 in Column A. Enter the answer on line 1 in Column D. Repeat this step for each value in Column A. Column E Square each value in Column D and enter the answer (always a positive number) in Column E. Box F Add the values in Column E and enter the answer in Box F. Box G Subtract 1 from n (find n in box at top of Form 3) and enter the answer in Box G. Box H Divide the value in Box F by the value in Box G and enter the answer in Box H. Box l Subtract 1 from N (find N in box at top of Form 3) and enter the answer in Box I. Box J Multiply the value in Box H by the value in Box I. Enter the answer in Box J. Column K Enter the value from Form 2, Box G for each week on the appropriate line in Column K. Box L Add the values in Column K and enter the answer in Box L. Box P Subtract 1 from M (find M in box at top of Form 3). Multiply the value (M-1) by n and enter the answer in Box P. Box Q Divide the value in Box L by the value in Box P and enter the answer in Box Q. Box R Subtract 1 from M. Multiply the value (M-I) by N and enter the answer in Box R. Box S Multiply the value in Box Q by the value in Box R and enter the answer in Box S. Box T Add the values in Box J and Box S and enter the answer in Box T. Box U Multiply N times M and then subtract 1. Enter the answer in Box U. Box V Divide the value in Box T by the value in Box U and enter the answer in Box V. This value is also transferred to Form 4, Column K. The purpose of Form 3 is to find average daily operations for each season and to calculate numbers that will be used on Form 4. Two week samples in each season will produce four Form 3sâone for each season. In the boxes at the top of the Form, fill in the name of the airport sampled. Fill in the season from which the sample data were collected. Also fill in the season size (N = total number of weeks in the season), the sample size (n = number of weeks sampled in the season), and the week size (M = number of days [normally 7]). (Note: Form 3 assumes a cluster of 7 days (one week) is sampled, and therefore N, n, and M are based on weeks. If a cluster of days less than one week or greater than one week is sampled, refer to FAA-APO-8507.)
B-10 FORM 3 Airport Name Tri-City State Season 1 TOTAL WEEKS IN SEASON = N 13 WEEKS SAMPLED IN SEASON = n 2 DAYS IN WEEK = M 7 A D E K Week Average Daily OperaÂons (Transfer Box D from Form 2) Column A minus Box C Column D Squared 1 10.86 2.86 8.18 2 5.14 -2.86 8.18 3 4 B = Sum of Column A 16.00 F= Sum Col. E 16.36 L = Sum of Column K 509.72 G = n - 1 1 P = n x (M - 1) 12 H = F ÷ G 16.36 Q = L ÷ P 42.48 C = Average Daily OperaÂons for Season 8.00 I = N - 1 12 R = N x (M-1) 78 J = H x I 196.31 S = Q x R 3313.18 T = J + S 3509.49 U = (N x M) - 1 90 V = T ÷ U 38.99 Transfer Box G from Form 2 (Sum of column F on Form 2) 406.86 102.86 Columns Refer to Form 2 for this informaÂon Box F, Box G, Box H, etc.
B-11 FORM 3 Airport Name Season TOTAL WEEKS IN SEASON = N WEEKS SAMPLED IN SEASON = n DAYS IN WEEK = M A D E K Week Average Daily OperaÂons (Transfer Box D from Form 2) Column A minus Box C Column D Squared 1 2 3 4 B = Sum of Column A F= Sum Col. E L = Sum of Column K G = n - 1 P = n x (M - 1) H = F ÷ G Q = L ÷ P C = Average Daily OperaÂons for Season I = N - 1 R = N x (M-1) J = H x I S = Q x R T = J + S U = (N x M) - 1 V = T ÷ U Transfer Box G from Form 2 (Sum of Column F on Form 2)
B-12 INSTRUCTIONS FOR FORM 4 (Reproduced from FAA-APO-85-7) ANNUAL ESTIMATES In the boxes at the top of Form 4 fill in the name of the airport sampled and the period of the sample. Column A Fill in the total number of days in each season D, on the lines in Column A. Use the actual number of calendar days, not number of weeks, N, times 7 days. (If data is not stratified by season, only one line will be used.) Column B Square each value of D and enter the answer in Column B. Column C Fill in the number of days sampled, d, in each season on the lines in Column C. (When sampling 2 weeks in each season, D = 14.) Column D For each season divide D2 in Column B by d in Column C and enter the answer in Column D. Column E For each season divide d in Column C by D in Column A and enter the answer in Column E. Column F Subtract d÷D in Column E from 1 and enter the answer in Column F. Transfer the value of D for each season from Column A to the appropriate line in Column G. Column H Fill in the average daily operations for each season on the lines in Column H. Average daily operations are obtained from Form 3, Box C. Column l For each season, multiply D in Column G by average daily operations in Column H. Enter the answer on the appropriate line in Column I. Box J Add the values in Column I and enter the answer, annual operations, in Box J. Column K Fill in the value from Form 3, Box V for each season on the appropriate line in Column K. Column L For each season, transfer the values for D2÷ d from Column D to Column L. Column M For each season multiply the value in Column K by D2÷ d in Column L, Enter the answer in Column M. Column N For each season transfer the value 1 - (d ÷ D) from Column F to Column N. Column O For each season multiply the value in Column M by 1 - (d ÷ D) in Column N and enter the answer in Column O. This value is the variance of operations in each season. Box P Add the values in Column O. Enter the answer, the variance of estimated annual operations, in Box P. Box Q Take the square root of the value in Box P and enter the answer in Box Q. (The symbol on a calculator for square root is Box R Find the appropriate t-value from the chart, to the right based on the value (d-1). (Find d from Column C, then subtract 1.) Use the lowest value of d in Column C. Enter the t-value in Box . (Example t- value if d-1=13, then t=2.1 ) Box S Multiply the value in Box Q by the t-value in Box . Enter the answer in Box S. This is the percent confidence interval or range of the estimated operations. It is a measure of the precision of the annual operations estimate. Box T Divide the value in Box S by the value in Box J and multiply by 1 . Enter the answer in Box T. This is the percent sampling error, which is also a measure of the precision of the annual operations estimate. The purpose of Form 4 is to expand the sample data to estimate total annual operations and to calculate measures of precision of the estimate of annual operations, including the confidence interval and the percent sampling error.
B-13 Example FORM 4 Airport Name Tri-City State A B C D E F Season D = Total Days in Season D2 d = number of days sampled D 2 ÷ d d ÷ D 1 - (d ÷ D) 1 92 8464 14 604.57 0.15 0.85 2 92 8464 14 604.57 0.15 0.85 3 91 8281 14 591.5 0.15 0.85 4 91 8281 14 591.5 0.15 0.85 G H I Season D (Transferred from Column A above) Average (Transfer Box C from Form 3) Total (Column G x Column H) 1 92 8 736 2 92 4.57 420.44 3 91 6.76 615.16 4 91 5.52 502.32 J = Annual OperaÂons 2274 K L M N O Season Transfer Box V from Form 3 D2 ÷ d (Transfer from Column D above) Column K x Column L 1 - (d ÷ D) (Transfer from Column F above) Column M x Column N 1 39 604.57 23578.23 0.85 20041.5 2 18.85 604.57 11396.15 0.85 9686.73 3 99.17 591.5 58659.06 0.85 49860.2 4 21.68 591.5 12823.72 0.85 10900.16 P = Sum of Column O (This is the Variance of OperaÂons) 40488.59 Q = square root of P 300.81 R = t-value for d-1 (see t-value chart to right) 2.16 Use the lowest value for d found in Column C. Enter corresponding amount for t S = Q x R (This is the 95% Confidence Interval or range of the esÂmated operaÂons. 650 It is a measure of the precision of the annual operaÂons esÂmate.) T = (S ÷ J) x 100 (Percent Sampling error) 28.6 Columns Columns If d - 1 = then t = 3 3.182 4 2.776 5 2.571 6 2.447 7 2.365 8 2.306 9 2.262 10 2.228 11 2.201 12 2.179 13 2.160 14 2.145 15 2.131 16 2.120 17 2.110 18 2.101 19 2.093 20 2.066 30 2.042 40 2.021 50 2.009 60 2.000 80 1.990 t-value Chart
B-14 FORM 4 Airport Name A B C D E F Season D = Total Days in Season D2 d = number of days sampled D 2 ÷ d d ÷ D 1 - (d ÷ D) 1 2 3 4 G H I Season D (Transfer from Column A above) Average (Transfer Box C from Form 3) Total (Column G x Column H) 1 2 3 4 J = Annual Operations K L M N O Season Transfer Box V from Form 3 D2 ÷ d (Transfer from Column D above) Column K x Column L 1 - (d ÷ D) (Transfer from Column F above) Column M x Column N 1 2 3 4 P = Sum of Column O (This is the Variance of OperaÂons) Q = square root of P R = t-value for d-1 (see t-value chart to right) Use the lowest value for d found in Column C. Enter corresponding amount for t S = Q x R (This is the 95% Confidence Interval or range of the esÂmated operaÂons. It is a measure of the precision of the annual operaÂons esÂmate.) T = (S ÷ J) x 100 (Percent Sampling error) If d - 1 = then t = 3 3.182 4 2.776 5 2.571 6 2.447 7 2.365 8 2.306 9 2.262 10 2.228 11 2.201 12 2.179 13 2.160 14 2.145 15 2.131 16 2.120 17 2.110 18 2.101 19 2.093 20 2.066 30 2.042 40 2.021 50 2.009 60 2.000 80 1.990 t-value Chart