Page 177 - Statistics for Dummies

Chapter 10: The t-Distribution
you look at the row for df = 9. The 95th percentile is the number where 95%
of the values lie below it and 5% lie above it, so you want the right-tail area to
be 0.05. Move across the row, find the column for 0.05, and you get t = 1.833.
9
This is the 95th percentile of the t-distribution with 9 degrees of freedom.
Now, if you increase the sample size to n = 20, the value of the 95th percen-
tile decreases; look at the row for 20 – 1 = 19 degrees of freedom, and in the
column for 0.05 (a right-tail probability of 0.05) you find t = 1.729. Notice
19
that the 95th percentile for the t distribution is less than the 95th percen-
19
tile for the t distribution (1.833). This is because larger degrees of freedom
9
indicate a smaller standard deviation and the t-values are more concentrated
about the mean, so you reach the 95th percentile with a smaller value of t.
(See the section “Discovering the effect of variability on t-distributions,”
earlier in this chapter.)
Picking out t*-values for confidence intervals
Confidence intervals estimate population parameters, such as the population 161
mean, by using a statistic (for example, the sample mean) plus or minus a
margin of error. (See Chapter 13 for all the information you need on confi-
dence intervals and more.) To compute the margin of error for a confidence
interval, you need a critical value (the number of standard errors you add
and subtract to get the margin of error you want; see Chapter 13). When the
sample size is large (at least 30), you use critical values on the Z-distribution
(shown in Chapter 13) to build the margin of error. When the sample size is
small (less than 30) and/or the population standard deviation is unknown, you
use the t-distribution to find critical values.
To help you find critical values for the t-distribution, you can use the last
row of the t-table, which lists common confidence levels, such as 80%, 90%,
and 95%. To find a critical value, look up your confidence level in the bottom
row of the table; this tells you which column of the t-table you need. Intersect
this column with the row for your df (see Chapter 13 for degrees of freedom
formulas). The number you see is the critical value (or the t*-value) for your
confidence interval. For example, if you want a t*-value for a 90% confidence
interval when you have 9 degrees of freedom, go to the bottom of the table,
find the column for 90%, and intersect it with the row for df = 9. This gives
you a t*-value of 1.833 (rounded).
Across the top row of the t-table, you see right-tail probabilities for the
t-distribution. But confidence intervals involve both left- and right-tail proba-
bilities (because you add and subtract the margin of error). So half of the
probability left from the confidence interval goes into each tail. You need to
take that into account. For example, a t*-value for a 90% confidence interval
has 5% for its greater-than probability and 5% for its less-than probability
(taking 100% minus 90% and dividing by 2). Using the top row of the t-table,

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