CHI-SQUARE AND DISTRIBUTION-FREE TESTS 615J
Problem 5. A manager of a manufacturer is
concerned about suspected slow progress in
dealing with orders. He wants at least half of the
orders received to be processed within a work-
ing day (i.e. 7 hours). A little later he decides to
time 17 orders selected at random, to check if
his request had been met. The times spent by the
17 orders being processed were as follows:
434 h9^34 h15^12 h11h 8^14 h6^12 h9h 834 h10^34 h3^12 h8^12 h9^12 h1514 h 13h 8h 734 h6^34 hUse the sign test at a significance level of 5%
to check if the managers request for quicker
processing is being met.Using the above procedure:
(i) The hypotheses areH 0 :t=7handH:t>7h,
wheretis time.
(ii) SinceH 1 ist>7 h, a one-tail test is assumed,
i.e.α 1 =5%.
(iii) In the sign test each value of data is assigned
a+or−sign. For the above data let us assign
a+for times greater than 7 hours and a−for
less than 7 hours. This gives the following
pattern:
−++++−+++
−++++++−
(iv) The test statistic,S, in this case is the number of
minus signs (−ifH 0 were true there would be
an equal number of+and−signs). Table 63.3
gives critical values for the sign test and is given
in terms of small values; hence in this caseSis
the number of−signs, i.e.S= 4.
(v) From Table 63.3, with a sample sizen=17, for
a significance level ofα 1 =5%,S≤ 4.
SinceS=4 in our data, the resultis significant
atα 1 =5%, i.e.the alternative hypothesis
is accepted—it appears that the managers
request for quicker processing of orders is
not being met.Problem 6. The following data represents the
number of hours that a portable car vacuum
cleaner operates before recharging is required.Operating
time (h) 1.4 2.3 0.8 1.4 1.8 1.5
1.9 1.4 2.1 1.1 1.6Use the sign test to test the hypothesis, at a 5%
level of significance, that this particular vacuum
cleaner operates, on average, 1.7 hours before
needing a recharge.Using the procedure:
(i) Null hypothesisH 0 :t= 1 .7h
Alternative hypothesisH 1 :t= 1 .7h.
(ii) Significance level, α 2 =5%(since this is a
two-tailed test).
(iii) Assuming a+sign for times>1.7 and a−sign
for times<1.7 gives:−+−−+−+−+−−(iv) There are 4 plus signs and 7 minus signs; taking
the smallest number,S= 4.
(v) From Table 63.3, wheren=11 andα 2 =5%,
S≤ 1.
SinceS=4 falls in the acceptance region (i.e. in
this case in greater than 1),the null hypothesis
is accepted, i.e. the average operating time is
not significantly different from 1.7 h.Problem 7. An engineer is investigating two
different types of metering devices,AandB, for
an electronic fuel injection system to determine
if they differ in their fuel mileage performance.
The system is installed on 12 different cars, and
a test is run with each metering system in turn
on each car. The observed fuel mileage data (in
miles/gallon) is shown below:A 18.7 20.3 20.8 18.3 16.4 16.8
B 17.6 21.2 19.1 17.5 16.9 16.4A 17.2 19.1 17.9 19.8 18.2 19.1
B 17.7 19.2 17.5 21.4 17.6 18.8Use the sign test at a level of significance of
5% to determine whether there is any difference
between the two systems.Using the procedure:
(i)H 0 :FA=FBandH 1 :FA=FBwhereFAand
FBare the fuels in miles/gallon for systemsA
andBrespectively.
(ii)α 2 =5%(since it is a two-tailed test).
(iii) The difference between the observations is
determined and a+or a−sign assigned to