so using the sample mean to estimate the population mean is, on average, correct. But since we don’t
know σ , we also have to use the sample standard deviation to estimate the population standard deviation.
And that is not, on average correct. s tends to underestimate σ . To compensate for this, we need to adjust
z , and the adjusted value is called t . The t statistic is given by
This statistic approximately follows a t distribution if the following are true.• The population from which the sample was drawn is approximately normal, or the sample is large
enough (rule of thumb: n ≥ 30).
• The sample is an SRS from the population.
There is a different t distribution for each n . The distribution is determined by the number of degrees
of freedom , df = n – 1. We will use the symbol t (k ) to identify the t distribution with k degrees of
freedom. The t -distribution is symmetric and mound-shaped, but it has heavier tails than a normal
distribution.
As n increases, the necessary adjustment gets smaller so the t distribution gets closer to the normal
distribution. In fact, if you look at the bottom row of Table B, you’ll see that the values of t for infinitely
many degrees of freedom are the same as those for the normal distribution. We can see this in the
following graphic:
The table used for t values is set up differently than the table for z . In Table A in the Appendices, the
marginal entries are z -scores, and the table entries are the corresponding areas under the normal curve to
the left of z . In the t table, Table B, the left-hand column is degrees of freedom, the top margin gives
upper tail probabilities, and the table entries are the corresponding critical values of t required to
achieve the probability. In this book, we will use t (or z ) to indicate critical values.
example: For 12 df, and an upper tail probability of 0.05, we see that the critical value of t is
1.782 (t * = 1.782). For an upper tail probability of 0.02, the corresponding critical value is
2.303 (t * = 2.303).
example: For 1000 df, the critical value of t for an upper tail probability of 0.025 is 1.962 (t * =
1.962). This is very close to the critical z -value for an upper tail probability of 0.025, which
is 1.96 (z * = 1.96).