Basic Statistics

108 TESTS OF HYPOTHESES ON POPULATION MEANS

one-sided, upper tail test is appropriate. The null hypothesis is HO : pdso, so if it

is rejected we can say that pd > 0. In Table 8.3, the mean of the 10 differences is

calculated to be 2 = 1.32 months; the variance of the differences is sz = C(d -

d)2/(n- 1) = 22.488 and their standarddeviationis Sd = 4.74months. The variance

of the population of sample means is estimated by

22.488

s-=---^2 s2d - = 2.2488

dn 10

and by taking the square root of 2.2488 we have 1 SO months as an estimate of the

standard deviation s;i of the population. Then,

• d-0 1.32-0
  t=- - - = .880
  s;i 1.50

With Q = .05 and d.f. equal to n - 1 = 9, a t[.95] value of 1.833 would be necessary
to reject the null hypothesis (see Table A.3). With t = .880, far less than 1.833, we
cannot reject the null hypothesis; the mean age at first word may be the same for
cyanotic children as for their normal siblings. Alternatively, the P value for the test
can be obtained; fromTableA.3,75% of the area is below .703; 90% lies below 1.382.
The proportion oft values below 380 is between 75 and 90%, and P, the proportion
above .880, is between 10 and 25% or P < .25.
In making this test, we assumed that the d’s are a simple random sample from a
normal distribution. If we are unsure whether the differences are normally distributed,
then either a normal probability plot, a histogram, or a box plot should be graphed.
The differences should be at least approximately normally distributed.
When a computer program is used, the observations for the cyanotic children and
for their siblings are usually entered as separate variables. In some programs, the user
first has the program make a new variable, which is the difference between the ages
for the cyanotic children and their siblings, and then treats the problem as if it were
a single sample of observations and performs a t test for a single sample. In other
programs, the user tells the program which variable is the first variable and which is
the second, and the program gives the result of the paired t test. Usually, the P value
for a two-sided test is given.

8.4 CONCEPTS USED IN STATISTICAL TESTING

In this section we discuss the decision to accept or to reject a null hypotheses. The
two types of error are defined and explained.

8.4.1

After testing a null hypothesis, one of two decisions is made; either HO is accepted or
it is rejected. If we use a significance level Q = .05 and reject HO : p = 12.0, we feel
reasonably sure that p is not 12.0 because in the long run, in repeated experimentation,

Decision to Accept or Reject