Mathematical Methods for Physics and Engineering : A Comprehensive Guide

STATISTICS

number of degrees of freedom isn=N−M= 8. Settingn=8andα=0.05 in (31.127)
we find from table 31.2 thatk=15.51. Hence our rejection region is

χ^2 (m,ˆˆc)> 15. 51.

Since above we foundχ^2 (m,ˆˆc)=11.5, we cannot reject the null hypothesis that the
underlying model for the data is a straight liney=mx+c.

As mentioned above, our analysis is only valid if the functionf(x;a) is linear

in the parametersa. Nevertheless, it is so convenient that it is sometimes applied

in non-linear cases, provided the non-linearity is not too severe.

31.8 Exercises

31.1 A group of students uses a pendulum experiment to measureg, the acceleration
of 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 and
standard error forgas 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 is
a 50% chance that the correct value lies?
31.3 The following are the values obtained by a class of 14 students when measuring
a physical quantityx: 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 the
best value forx.
(b) Without calculation, estimate how much reliance could be placed upon your
answer to (a).
(c) Databooks give the value ofxas 53.6 with negligible error. Are the data
obtained by the students in conflict with this?

31.4 Two physical quantitiesxandyare connected by the equation

y^1 /^2 =

x

ax^1 /^2 +b

,

and measured pairs of values forxandyare as follows:

x:10121620

y: 409 196 114 94

Determine the best values foraandbby graphical means, and (either by hand
or by using a built-in calculator routine) by a least-squares fit to an appropriate
straight line.
31.5 Measured quantitiesxandyare known to be connected by the formula

y=

ax

x^2 +b

,

whereaandbare constants. Pairs of values obtained experimentally are

x: 2.0 3.0 4.0 5.0 6.0

y: 0.32 0.29 0.25 0.21 0.18

Use these data to make best estimates of the values ofythat would be obtained

for (a)x=7.0, and (b)x=− 3 .5. As measured by fractional error, which estimate

is likely to be the more accurate?