Quandaries and Queries


Who is asking: Other
Level: Secondary


I WOULD appreciate it

Hooke's Law asserts that the magnitude of the force required to hold a spring is a linear function of the extension e of the spring. That is,
f = e0 + ke where k and e0 are constants depending only on the spring.
The following data was collected for a spring;

e: 9 , 11 , 12 , 16 , 19
f : 33 , 38 , 43 , 54 , 61

FIND the least square line f= B0 + B1x approximating this data and use it to approximate k.




There are a number of equivalent expressions for finding B0 and B1 in the least squares line y = B0 + B1x. In your case the independent variable is e rather than x, the dependent variable is f rather than y and the constants are e0 and k rather than B0 and B1.The formulas I use are


Let's look at your data, changing the variable names from e and f to x and y. You are asked to find k which is the coefficient of e in the equation. After our change of variables to x and y the coefficient of x is B1.

You have n = 5. To use the expressions above I need the sum of all the x's, the sum of the y's, the sum of the x2's and the sum of the x's times the y's. I am going to put your data in a table and calculate these sums.

x y x2 x y
9 33 92 = 81 9 33 = 297
11 48 121 528
12 43 144 516
16 54 256 864
19 61 361 1159
67 239 963 3364


The numbers in the bottom row of the table are the sums of the columns. Hence

x = 67, y = 239, x2 = 963 and xy = 3364

Now use the expression for B1 above to find the value of k.




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