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Part IV: Guesstimating and Hypothesizing with Confidence
The original data are in pairs, but you’re really interested only in the dif-
ference in reading scores (computer reading score minus phonics reading
score) for each pair, not the reading scores themselves. So the paired differ-
ences (the differences in the pairs of scores) are your new data set. See their
values in the last column of Table 15-1.
By examining the differences in the pairs of observations, you really only
have a single data set, and you only have a hypothesis test for one popula-
tion mean. In this case the null hypothesis is that the mean (of the paired dif-
ferences) is 0, and the alternative hypothesis is that the mean (of the paired
differences) is > 0.
If the two reading methods are the same, the average of the paired differ-
ences should be 0. If the computer method is better, the average of the
paired differences should be positive; the computer reading score is larger
than the phonics score.
The notation for the null hypothesis is H : μ = 0, where μ is the mean of the
o d d
paired differences for the population. (The d in the subscript just reminds you
that you’re working with the paired differences.)
The formula for the test statistic for paired differences is , where
is the average of all the paired differences found in the sample, and t is a
n–1
value on the t-distribution with n –1 degrees of freedom (see Chapter 10).
d
You use a t-distribution here because in most matched-pairs experiments the
sample size is small and/or the population standard deviation σ is unknown,
d
so it’s estimated by s . (See Chapter 10 for more on the t-distribution.)
d
To calculate the test statistic for paired differences, do the following:
1. For each pair of data, take the first value in the pair minus the second
value in the pair to find the paired difference.
Think of the differences as your new data set.
2. Calculate the mean, , and the standard deviation, s , of all the
d
differences.
3. Letting n represent the number of paired differences that you have,
d
calculate the standard error:
4. Divide by the standard error from Step 3.

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