402 CHAPTER 6 Inferential Statistics
H 0 : p=. 55 , Ha:p <. 55.The test statistic would then be
Z =
P̂−p
√̂
P(1−P̂)/n,
which isnis large enough is approximately normally distributed. There-
fore testing the hypothesis at the (1−α)% level of significance can be
handled in the usual fashion.
6.5.4 Matched pairs
One of the most frequent uses of statistics comes in evaluating the effect
of one or moretreatmentson a set of subjects. For example, people
often consider the effects of listening to Mozart while performing an
intellectual task (such as a test). In the same vein, one may wish to
compare the effects of two insect repellants.
To effect such comparisons, there are two basic—but rather different—
experimental designs which could be employed. The first would be to
divide a group of subjects into two distinct groups and apply the dif-
ferent “treatments” to the two groups. For example, we may divide
the group of students taking a given test into groupsAandB, where
groupAlistens to Mozart while taking the test, whereas those in group
Bdo not. Another approach to comparing treatments is to successively
apply the treatments to the same members of a given group; this design
is often called amatched-pairs design. In comparing the effects of
listening of Mozart, we could take the same group of students and allow
them to listen to Mozart while taking one test and then at another time
have them take a similar test without listening to Mozart.
In such situations, we would find ourselves comparingμ 1 versusμ 2 ,
whereμ 1 , μ 2 represent means associated with treatments 1 and 2. The
sensible null hypothesis would be expressed asH 0 : μ 1 =μ 2 and the
alternative will be either one or two sided, depending on the situation.
Without delving into the pros and cons of the above two designs,
suffice it to say that the statistics used are different. The first design