31.8 EXERCISES
31.13 A similar technique to that employed in exercise 31.12 can be used to test
correlations between characteristics of sampled data. To illustrate this consider
the following problem.
During an investigation into possible links between mathematics and classical
music, pupils at a school were asked whether they had preferences (a) between
mathematics and english, and (b) between classical and pop music. The results
are given below.
Classical None Pop
Mathematics 23 13 14
None 17 17 36
English 30 10 40By computing tables of expected numbers, based on the assumption that no
correlations exist, and calculating the relevant values ofχ^2 , determine whether
there is any evidence for
(a) a link between academic and musical tastes, and
(b) a claim that pupils either had preferences in both areas or had no preference.
You will need to consider the appropriate value for the number of degrees of
freedom to use when applying theχ^2 test.
31.14 Three candidatesX,Y andZwere standing for election to a vacant seat on
their college’s Student Committee. The members of the electorate (current first-
year students, consisting of 150 men and 105 women) were each allowed to
cross out the name of the candidate they least wished to be elected, the other
two candidates then being credited with one vote each. The following data are
known.
(a) Xreceived 100 votes from men, whilstYreceived 65 votes from women.
(b)Zreceived five more votes from men thanXreceived from women.
(c) The total votes cast forXandYwere equal.
Analyse this data in such a way that aχ^2 test can be used todetermine whether
voting was other than random (i) amongst men and (ii) amongst women.
31.15 A particle detector consisting of a shielded scintillator is being tested by placing it
near a particle source whose intensity can be controlled by the use of absorbers.
It might register counts even in the absence of particles from the source because
of the cosmic ray background.
The number of countsnregistered in a fixed time interval as a function of the
source strengthsis given in as:
source strengths:0123456
countsn: 6 11 20 42 44 62 61At any given source strength, the number of counts is expected to be Poisson
distributed with mean
n=a+bs,
whereaandbare constants. Analyse the data for a fit to this relationship and
obtain the best values foraandbtogether with their standard errors.
(a) How well is the cosmic ray background determined?
(b) What is the value of the correlation coefficient betweenaandb?Isthis
consistent with what would happen if the cosmic ray background were
imagined to be negligible?
(c) Do the data fit the expected relationship well? Is there any evidence that the
reported data ‘are too good a fit’?