Finding error in following difference table?

Question

I've been trying to solve a question on my book on finding error in the following difference table:

My teacher told me to expand the difference table until I find the proper Binomial coefficients, hence I've expanded the difference table to y5.

According to my solution, the error is originating from 6th entry of the table. In the y5 column, I am using Binomial factors to find the error.

Error = Largest value in a column / Corresponding coefficient of ε in that column

Error = $.095/4 = 0.02375$

Since my teacher told me that the error will always be subtracted from orginal entry

Corrected Value = $ 0.589 - 0.02375 = 0.56525 $

but in my book, the answer is: $0.598$

Where am I missing?

Answer 1

In y^5, the entry with the highest deviation is -0.093. In the corresponding binomial table, the highest entry in y^5 is 10. Therefore, the answer is [{0.589-(-0.093/10)}=0.5983]. 0.5983 can be rounded down to 0.598

Answer 2

The greatest numerical value in the fourth column is -0.057. We know that the greatest element at the fourth column is 6 X the error (6e). Therefore, equate 6e to -0.057 and solve for e. Notice that e = -0.0095. Now the actual value is the wrong value - the error. Which is 0.589 - (-0.0095)

The correct value is now 0.5985.

Hope this has helped

Answer 3

The first column with a binomially scaled sub-sequence over a clearly random-noise background is $y_2$. Then for the error $e$ that has to be corrected one has $[-0.01,0.019,-0.009]\sim e·[1,-2,1]$. By the least-square formula one gets $$ e=-\frac{0.01+0.038+0.009}{1+4+1}=-\frac{0.058}{6}=-0.00967 $$ This corrects the value of $y(6)$ to $0.589+0.00967=0.59867$ which rounds to $0.599$. This is also more in line with a gradual shortening of the distance between the values.

Fitting the valid values to a polynomial of degree 3 gives the missing value as $y(6)=0.598692$, which rounds to the same.