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Sample Size and Power Calculations |

Sample size and power calculations are available for one-sample and
two-sample paired and independent designs, when the proposed
analysis is construction of confidence intervals of a mean
(one-sample) or difference of two means (two-sample), via the *t*-test.

To understand the power of a confidence interval, first define the
*precision* to be half the length of a two-sided confidence interval (or
the distance between the endpoint and the parameter estimate in a
one-sided interval). The power can then be considered to be the probability
that the desired precision is achieved, that is, the probability that
the length of the two-sided interval is no more than twice the desired
precision. Here, a slight modification of this concept is used. The
power is considered to be the conditional probability that the desired
precision is achieved, given that the interval includes the true value
of the parameter of interest. The reason for the modification is that
there is no reason for the interval to be particularly small if it
does not contain the true value of the parameter.

These computations assume equally sized groups.

Figure 12.6 displays the resulting dialog.

You specify whether you want to compute sample sizes or power, enter values for desired precision and standard deviation, enter the alpha levels, enter the sample sizes or power, and select if you want a power curve.

To request power for a paired confidence interval where the desired precision is 0.5 and the standard deviation is 2.462, follow these steps:

- Select
**Power**. - Enter 0.5 as the
**Desired precision:** - Enter 2.462 as the
**Std dev of diff:** - Enter 0.01, 0.025, 0.05, and 0.1 as values in
the field for
**Alpha:** - Enter 11 as the value for the
**From:**field in the line for**N:** - Enter 211 and 5 as the values under
**To:**and**By:**, respectively, in the line for**N:** - Select
**Power vs. N**. - Select
**2-sided**for**Interval**if it is not already selected.

Note that you can enter multiple values in these fields, for example,
for **Alpha:** and **Desired precision:**, separated by
commas or blanks, and the
analysis will be performed for all combinations of the input
values. Here, power will be computed for sample sizes ranging from
11 to 211 in multiples of 5.

Figure 12.7 contains the completed dialog. Note that, because multiple alpha values were entered, sets of results will be created for each one.

Figure 12.8 contains the power computations for the sample sizes ranging from 36 to 171.

The power analysis results in Figure 12.8 show that, for a two-sided paired confidence interval where the standard deviation of the differences is 2.462, the significance level is = 0.025, the sample size is 121, and the power is 0.415. That is, 0.415 represents the probability that a confidence interval containing the true parameter value has a length of no more than twice the desired precision of 0.5.

The requested plot is displayed in Figure 12.9 and includes one power curve for each specified alpha value.

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