Calculator Tip: The TI-83 and early versions of the TI-84 calculator have no function for invT
analogous to invNorm . The operating system of the TI-84 can be updated, but if you have a TI-83 you
are pretty well restricted to using Table B to find a value of t * . However, if you have a newer OS for
a TI-84, there is an invT function in the DISTR menu that makes it quite easy to find t * . It works
similarly to invNorm , only you need to indicate the number of degrees of freedom. The syntax is
invT(area of the left of t * , df) . In the first example above, then, t * = invT(0.95, 12)
= 1.782 and t * = invT(0.98, 12) = 2.303. In the second example, t * =
invT(0.975,1000) = 1.962 .General Form of a Confidence Interval
A confidence interval is composed of two parts: a point estimate of a population value and a margin of
error. We specify a level of confidence to communicate how certain we are that the interval contains the
true population parameter.
A level C confidence interval has the following form: (estimate) ± (margin of error). In turn, the
margin of error for a confidence interval is composed of two parts: the critical value of z or t (which
depends on the confidence level C ) and the standard error. Hence, all confidence intervals take the form:
(estimate) ± (margin of error) = (estimate) ± (critical value)(standard error).
A t confidence interval for μ would take the form:t * is dependent on C , the confidence level; s is the sample standard deviation; and n is the sample size.
The confidence level is often expressed as a percent: a 95% confidence interval means that C = 0.95,
or a 99% confidence interval means that C = 0.99. Although any value of C can be used as a confidence
level, typical levels are 0.90, 0.95, and 0.99.
Important: When we say that “We are 95% confident that the true population value lies in an
interval,” we mean that the process used to generate the interval will capture the true population value
95% of the time. We are not making any probability statement about the interval. Our “confidence” is in
the process that generated the interval. We do not know whether the interval we have constructed contains
the true population value or not—it either does or it doesn’t. All we know for sure is that, on average,
95% of the intervals so constructed will contain the true value.
Exam Tip: For the exam, be very, very clear on the discussion above. Many students seem to think that
we can attach a probability to our interpretation of a confidence interval. We cannot because probability
refers only to repeatable future random events. The interval has already been created, so talking about a
probability doesn’t make sense.example: Floyd told Betty that the probability was 0.95 that the 95% confidence interval he had
constructed contained the mean of the population. Betty corrected him by saying that his