Appendix 3 APPENDIX 5. GUIDANCE MATERIAL ON MYHODS TO CALCULATE
CONFTDENCE INTERVALS IVTRODUCHTON the use of APD maps requires confidence intervals to be determined using a more general formulation than used for a cluster of data points, For this more general case, confidence intervals should be calculated about a regression curve for flight test data and for static test data These confidence intervals should be used in calculating the confidence intervals for the pooled data sets that constitute NPD maps.
2. CONFIDENCE INTERVAL FOR THE MEAN OF CLUSTERED DATA Let I = data (EPNL) N = mumber of data points K: order of regression curve (K = O for clustered data) v = degrees of freedom = N - K - 1 t = Student's t distribution for 90 percent confidence interval and v degrees of freedom for example:
t = 2.015 for N = 6 (v = 5) t = 1.645 for N = infinity N 1 L (mean) = N = 1L, - [12 12 - 141 S (standard deviation) = N- 1 N (H-1) S CI (confidence interval) = † t Par 1
AC 36~4B 3/23/88 Appendix 5
3. CONFIDENCE INTERVAL FOR A MEAN REGRESSION CURVE.
The general expression for a regression curve may be written as:
EPNL = ao + ajx + azx? +ayxk
where EPNL = noise level X = parameter such as EPR, RPM, altitude, etc.
a = coefficients for the regression curve K = polynomial order of regression curve define: AEPN = difference between regression curve and individual data point (X constant) N = mumber of data points v = degrees of freedom = N - K -1 t = students t distribution for 90 percent ENLO, Xo = a particular point on the regression curve CIo = confidence interval for a particular point For a linear (k = 1) regression, the equation is as follows:
2 N 2 (Xo - x) 2 CIo = I t N-K-1 N N where X = N Note: K is normally limited to K = 2. K = 3 may be approved when there is a physical basis for that polynomial order of curve fit.
2 Par 3
Appendix 5 For a higher order regression curve, a particular PPNLO is:
ENLO = ao + a1xo + a2x2 + .. + ays K For higher order regression curves, the regression coefficients are usually calculated using computer-matrix solutions, since there are many terms.
Defining the vectors: x' = [1 x x* • a row vector Ко = Г 1 1 Хо !
a colum vector and &A-1 : the varience-covarience matrix for the ax values Then the 90 percent confidence interval for a particular noise level, LENo, on a regression curve of EPNL vs. X is;
2 · (AEL) 2 2 Xo A-1 Xo CIo = I t N-K-1 Par 3
Appendix 5 Linear Example 100 8 8 8 5 8 8 82 50 23 (Thousands) regression Quadratic Example 110 105 105 104 102 00 EPNL 21 203 24 Par 3
Appendix 5
CONFIDENCE INTERVAL FOR POOLED DATA SENS The confidence interval for pooled data sets may be calculated as follows. For example, for three data sets (flight test data and two different static tests) - each with its own confidence interval:
let IZ1 = (CI1/t1)2 (square of the confidence interval without the Student's t term for each data set) then for three data sets, the confidence interval for the pooled data 1s:
CIo= +/-+' (221+222 +223) 1/2 241t1 + 222t2 + 2zztz where t'= 2L1 + 272 + 273
5. REFERENCES
(a) Kendall, M G. and Stuart, A, The Advanced Theory of Statistics, Volumes 1 and 2, Hafner, New York, 1973.
(b) Yule, G.U. and Kendall, M. G., An Introduction to the Theory of Statistics, 14th ed, Griffin, New York, 1950.
(c) Walpole, R.E. and Myers, R. H, Probability and Statistics for Engineers and Scientists, MacMillan, New Yoek, 1972.
Cochran, W.G., Approximate Significance Levels of the Behrens-Fisher Test, Biometrics, 20, 191-195, 1964.
(e) Shedecor, G.W. and Cochran, W. G., Statistical Methods, 6th ed, Iowa State University Press, Ames Iowa, 1968, Par 4 5
Appendix 5 Table 5-1 STUDENT'S t DISTRIBUTION FOR 90 PERCENT CONFIDENCE INTERVAL Degree of freedom (V) t
6.314
2.920
2.353 4 2.132 5 2.015
1.943
1.895
1.860
1.833
1.812
1.782
1.761
1.746
1.734
1.725
1.711
1.697
1.671
1.645 6 Par 5 *U.S. GOVERNMENT PRINTING OFFICE: 1992 - 312 - 68 75 5 5 96