Mathematical Methods for Physics and Engineering - Matematica.NET

STATISTICSnumber of degrees of freedom is n = N − M = 8. Setting n =8andα =0.05 in (31.127)we find from table 31.2 that k =15.51. Hence our rejection region isχ 2 ( ˆm, ĉ) > 15.51.Since above we found χ 2 ( ˆm, ĉ) = 11.5, we cannot reject the null hypothesis that theunderlying model for the data is a straight line y = mx + c. ◭As mentioned above, our analysis is only valid if the function f(x; a) is linearin the parameters a. Nevertheless, it is so convenient that it is sometimes appliedin non-linear cases, provided the non-linearity is not too severe.31.8 Exercises31.1 A group of students uses a pendulum experiment to measure g, the accelerationof free fall, and obtains the following values (in m s −2 ): 9.80, 9.84, 9.72, 9.74,9.87, 9.77, 9.28, 9.86, 9.81, 9.79, 9.82. What would you give as the best value andstandard error for g as measured by the group?31.2 Measurements of a certain quantity gave the following values: 296, 316, 307, 278,312, 317, 314, 307, 313, 306, 320, 309. Within what limits would you say there isa 50% chance that the correct value lies?31.3 The following are the values obtained by a class of 14 students when measuringa physical quantity x: 53.8, 53.1, 56.9, 54.7, 58.2, 54.1, 56.4, 54.8, 57.3, 51.0, 55.1,55.0, 54.2, 56.6.(a) Display these results as a histogram and state what you would give as thebest value for x.(b) Without calculation, estimate how much reliance could be placed upon youranswer to (a).(c) Databooks give the value of x as 53.6 with negligible error. Are the dataobtained by the students in conflict with this?31.4 Two physical quantities x and y are connected by the equationy 1/2 x=ax 1/2 + b ,and measured pairs of values for x and y are as follows:x: 10 12 16 20y: 409 196 114 94Determine the best values for a and b by graphical means, and (either by handor by using a built-in calculator routine) by a least-squares fit to an appropriatestraight line.31.5 Measured quantities x and y are known to be connected by the formulay =axx 2 + b ,where a and b are constants. Pairs of values obtained experimentally arex: 2.0 3.0 4.0 5.0 6.0y: 0.32 0.29 0.25 0.21 0.18Use these data to make best estimates of the values of y that would be obtainedfor (a) x =7.0, and (b) x = −3.5. As measured by fractional error, which estimateis likely to be the more accurate?1298