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Copyright © 2006 by Stephen L. Morgan and Stanley N. Deming. All rights reserved
The p-value (0.000175207) is the probability that a value of t greater than or equal to the
calculated value (t
calc
= 8.22) could have occurred by chance if there were no difference in the
means. The p-value is twice (two-sided) the fractional area of the right tail of the t-distribution
above the calculated value of t
calc
. The risk of getting the value |t
calc
| = 8.22 (ignoring sign) by
chance is less than Alpha = 0.05. Because this value is larger than that which could be expected
to occur by chance if the null hypothesis is true, the null hypothesis is rejected.
The label for the p-value listed in the table [“P(T<=t) two-tail”] is difficult to interpret and can be
misleading. First, “t Stat” was used to label that calculated value of t, not “T”. Second, it is
unusual to see this label written “backward” as it is in the Excel table –writing “P(t >= T)” would
have been better.
ONE-SIDED TWO-SAMPLE t-TEST: The one-sided t-test is appropriate if it is desired wanted
to know, before looking at the data, if one population mean (say,
μ
2
) is greater than the other
(
μ
1
). In the one-sided test, the test statistic for judging the significance of the difference in the
two population means is the ratio of the difference in the means to the standard error of the
difference in the means, a calculated value of t:
()
21
12
11
−
=
⎛⎞
+
⎜⎟
⎝⎠
calc
p
xx
t
s
nn
Note the absence of an absolute value sign in the numerator for the one-sided test: the sign of the
difference is of concern. The null and alternative hypotheses for the one-sided two-sample t-test
are:
null hypothesis, H
0
:
21
μ
≤
alternative hypothesis, H
a
:
21
μ
>
For a one-sided test, it is important to define the difference between the means to be placed in the
numerator of the calculated t-statistic to be positive if the alternative hypothesis, H
a
, is true. Then
all one-sided tests will be ‘greater than” tests and the “less than” possibility doesn't have to be
handled separately. This is accomplished by always selecting the data which is claimed to larger
by the alternative hypothesis (
μ
1
above) to be entered in the Variable 2 Range.
OUTPUT AND INTERPRETATION OF THE ONE-SIDED TWO-SAMPLE t-TEST
ASSUMING EQUAL VARIANCES:
The decision to reject the null hypothesis and to accept the
alternative hypothesis is made by comparing t
calc
to the critical or tabular value of Student’s t
(t
crit
) at a level of risk α for the number of degrees of freedom associated with the standard error
of the difference in means (n
1
+ n
2
– 2). The statistical decision is made using the following
logic: (a) If t
calc
> t
crit
, then reject the null hypothesis and accept the alternative hypothesis; or (b)
If t
calc
≤ t
crit
, then do not reject the null hypothesis. Note the absence of absolute value signs in
these comparisons (unlike with the two-sided t-test). Note also that a negative value of t
calc
could
(i.e., the sample mean for the group entered in the Variable 2
Range is actually less than the
sample mean of the data entered in the Variable 1
Range. In this case, the sign of the difference