Statistics for the Behavioral Sciences by Frederick J. Gravetter, Larry B. Wallnau ISBN 10: 1305504917 ISBN 13: 9781305504912

SECTION 9.3 | Measuring Effect Size for the t Statistic 285
t distribution
df 5 8
Middle 80%
of t distribution
FIGURE 9.7
The distribution of t statistics
for df = 8. The t values pile
up around t = 0 and 80% of
all the possible values are
located between t = –1.397
and t = +1.397.
t 5 21.397
t 5 0
t 5 11.397
corresponds to a t value in this interval. Similarly, we can be 95% confident that the
mean for a sample of n = 9 scores corresponds to a t value between ±2.306. Notice that
we are able to estimate the value of t with a specific level of confidence. To construct a
confidence interval for μ, we plug the estimated t value into the t equation, and then we
can calculate the value of μ.
Before we demonstrate the process of constructing a confidence interval for an unknown
population mean, we simplify the calculations by regrouping the terms in the t equation.
Because the goal is to compute the value of μ, we use simple algebra to solve the equation
for μ. The result is μ = M – ts M
. However, we estimate that the t value is in an interval
around 0, with one end at +t and the other end at –t. The ±t can be incorporated in the
equation to produce
μ = M ± ts M
(9.6)
This is the basic equation for a confidence interval. Notice that the equation produces an
interval around the sample mean. One end of the interval is located at M + ts M
and the other
end is at M – ts M
. The process of using this equation to construct a confidence interval is
demonstrated in the following example.
EXAMPLE 9.5
Example 9.2 describes a study in which infants displayed a preference for the more attractive
face by looking at it, instead of the less attractive face, for the majority of a 20-second
viewing period. Specifically, a sample of n = 9 infants spent an average of M = 13 seconds,
out of a total 20-second period, looking at the more attractive face. The data produced
an estimated standard error of s M
= 1. We will use this sample to construct a confidence
interval to estimate the mean amount of time that the population of infants spends looking
at the more attractive face. That is, we will construct an interval of values that is likely to
contain the unknown population mean.
Again, the estimation formula is
μ = M ± t(s M
)
In the equation, the value of M = 13 and s M
= 1 are obtained from the sample data. The
next step is to select a level of confidence that will determine the value of t in the equation.
The most commonly used confidence level is probably 95% but values of 80%,