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UNIT IV : INTTERPOLATION AND EXTRAPOLATION, INTERPOLATION :, , Interpolation is a technique of estimating the value of dependent variable (Y) for any, intermediate value of the independent variable (X)., , EXTRAPOLATION, , Extrapolation is a technique of estimating the value of dependent variable (Y) for any yaluAgf, the independent variable (X) which is outside the range of the given series., , ASSUMPTION OF INTERPOLATION AND EXTRAPOLATIO QO, , 1. There are no sudden jumps in the values of dependent variable Her 10d to, another., 2. There is a sort of uniformity in the rise or fall of the values,of t jendent variable., 3. There will be no consecutive missing values in the wah, METHODS OF INTERPOLATION AND er, , 1, Binomial expansion method, 2. Newton’s advancing difference metho, , IN, , BINOMIAL EXPANSION METH), This method is simplest ig algebraic method, which can be used both for, , interpolation and e:, ASSUMPTION OF BN AANSION METHOD, endenvariable should have a common difference. That is the values, , 1, The values of in, ithmeti) progression., , should be in ai, , , , , 2. The value Nforewhich the value of Y is to be interpolated must be one of the values, of X,, ©, THE BI EXPANSION METHOD OF ESTIMATING TWO MISSING, VAL OLVES THE FOLLOWING STEPS., , &y Find the number of known values (n) of the dependent variable Y., Then find Ao"= 0 = (y-1)"=0., , 3. Expand (y-1)"=0 according to Binomial expansion method. Substitute the values and, obtain one missing value., To get another missing value use the same equation by doing a small alteration that is, raising y suffix by 1., 5. Substitute the values and obtain the second missing value, , +, , Anjum sultana 8884206666 [email protected], , Scanned with CamScanner

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PROBLEMS, , 1. Estimate the production for the year 2001 and 2005 from the following data, Year 1995 1997 1999 2001 2003 2005, Production | 20 40 70 ? 130 ?, , , , , , , , , , , , , , , , , , Solution : let X be year and Y be the production, x 1995 1997 1999 2001 2003 2005, Y¥ 20(yo) 40(y1) 70(y2) Tys) 130(ya) 3, The number of known values are n=4,, , s Ao" = 0 > (y-1)"=0. ©, (y-Dt=0. OC, (y-1)" = ya — 4y3 + 6y2 — dy + yo=0. °, , = 130 —4ys + 6(70) - 4(40) + 20 =0 NS, , = 130 — dys + 420 — 160 + 20=0 ey, =410—4y3=0, , =-dyi=-410, , =y3= 102.5, , To find another value use, , (y-1) = ya—4y3 + 6y2 — 4y1 + yo= 0., , Raise the suffixes by 1, ys — 4ya + 6y3 — 4y2 + yi = 0., , Y5 — (130) + 6(102.5)— 4 Rs =0, Y5— 520 + 615 — 280 + 40k 0}, , Ys— 145=0, , , , , , , , , , , , , , , , , , , , , , , , Ys= 145, NN, PRACTICAL PROBLE:, , 1. Working class co! iting indices of a certain place for some years are given below., Interpolate t issim index number for 1995 and 1999., , , , , , , , 1994 1995 1996 1997 1998 1999, , , , , , 300 ? 280 278 250 2, , , , , , infateyhe number of persons for the ages 24 and 30 years with the help of the, , , , , , a, i wing data., “AS, , , , , , , , , , , , , , , , , , , , , , , , , , , , ge 18 20 22 24 26 28 30, No of 20 22 26 ? 35 39 ?, persons, 3. Find out the missing values in the data., x 5 10 15 20 25 30, Y 5 8 ? 13 15 2, , , , , , , , , , , , , , , , , , 4. From the following data estimate probable life expectancy of an average Indian for the, years 1980 and 2010., , Anjum sultana 8884206666 [email protected], , Scanned with CamScanner

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Birth year_| 1950 1960 1970 1980 1990 _| 2000 2010, Life 68.7 69.7 70.8 & 74 71 ?, expectancy, , 5. For the following data relating to the production. Estimate the production for the year, , 2006 and 2010, Year 2005 2006 2007 2008 2009 2010, Production | 5 2? 10 15 26 2, , 6. Estimate sales of a business concern for the year 2007 and 2011., Year 2006 2007 2008 2009 2010, Sales 13 ? 25 38 65, i), , 7. Interpolate the business in 1982 and extrapolate for the year 1984 from! the fi Towing data, Year 1978 1979 1980 1981 1982° , 7 | 1984, Business _| 80 150 235 365 2 fe ?, , 8. Estimate the production for the years 2000 and 2010 with the hayes he following table., Year 1980 1985 1990 1995 =| 2005 2010, Production | 5 12 19 40 2, , 9. Estimate the production for the years 2000 and the following data, Year 1975 1980 | { 2000 | 2005, Production | 10 ll 2 33, , 10. Interpolate the production for the years 1989 with the help of the following, , table., Year 1986 1987 _.€ JLB 1989 1990 1991, Production | 120 122, 36 ? 135 ?, , 11. Estimate the production for thy j 982 and 1985 from the following data, Years 1979 1981 1982 1983 1984, Production | 100 150 2? 525 780, , , , , , , , , , , , , , , , , , , , NEWTON’ NCING DIFFERENCE METHOD, NEWTON: 2 ANCING DIFFERENCE METHOD OF, INTER ON, , This x is Applicable in those cases where the independent variable X increases by, Is., , NY, IN’S FORMULA OF INTERPOLATION IS, , , , x(X-1 x(x-1)(x-2 X(X-1)(x-2)(x-3, R= yo +xAo! + BED gt + AEBED Agh 4 HE DORDIEND Agt + cesteserseeees SOON, Where, xX = The value of X to be interpolated-th_value of x at the origin, The dif ference bwteen the adjoining values of X, FOR TIME SERIES DATA, x = The year which to be interpolated-th_year at the origin, The dif ference bwteen the adjoining years, Anjum sultana 8884206666 [email protected], , Scanned with CamScanner

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The value of X to be interpolated is 25, The value of x at the origin is 20, Common difference in x series is 10, , X=05., Yo= 53, , Ao! = -4, Ao? = -9, a 9, , The above data is extracted (, Newton’s general formula is “, , = =1(x- =1)(x-2)(x- ®, Yx = yo + xAq! + 2 Day (fee, ve 2) Ag+ pl ee Bae, For the present problem we can take formula till Ao’, Yu = yo + xAo! + 22 Dag? +2 VERD a3 &, , 3!, Substitute the values neatly in the above fogmul, Yu =53 = 0.5(-4) +5050 1 (9) + 0508 ), , Remove the brackets by multiplying vy,, = 532 4 2505) at 0.5) (9) + 0.5(-0.8)(Now carefully ‘multiply signs any, , =53- 2+ 228 4 3375, , =53-—2:+ L125 +: can), =52.6875 = 53 persong al ige 25,, PRACTIC. Ne, , 1. From the hg data interpolate the value for the year 1998., , 1996 | 2000 | 2004 | 2008 | 2012, is) 147 | 84 105 [111 | 114, , , , , , , , , , Ive the factorials, , , , , , , , , , , , , , , , , , , , , , , , 300 | 500 | 700 | 900 | 1100, umber of employees | 36 |31 |24 |22 | 18, , OS Using newton’s forward difference method find the value of y when x = 25., , X {15 | 19 | 23 | 27 | 31, ¥ [17 | 18 | 22 [28 | 35, 4. The table shows the expectation at the life at different ages. Find the expectation, of life at the age 26., , Age in years 15 | 20 | 25 | 30 | 35, Expectation of life | 30 | 29 | 27 | 22 | 20, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , Anjum sultana 8884206666 [email protected], , Scanned with CamScanner