interval either does contain the value (P = 1) or it doesn’t (P = 0). This interval could be one
of the 95 out of every 100 on average that does contain the population mean, or it might be one
out of the 5 out of every 100 that does not. Remember that probability values apply to the
expected relative frequency of future events, not events that have already occurred.
example: Find the critical value of t required to construct a 99% confidence interval for a
population mean based on a sample size of 15.
solution: To use the t distribution table (Table B in the Appendixes), we need to know the upper-
tail probability.
Looking in the row for df = 15 – 1 = 14, and the column for 99% confidence we find t * =
2.977. Note that the table is set up so that if you look at the bottom of the table and find 99%,
you are in the correct column.
Because C = 0.99, and confidence intervals are two sided, the upper-tail probability isUsing the newer version of the TI-84, the solution is given by invT(0.995,14) = 2.97 7 .
example: Find the critical value of z required to construct a 95% confidence interval for a
population proportion.
solution: We are reading from Table A, the table of Standard Normal Probabilities. Remember
that table entries are areas to the left of a given z -score. With C = 0.95, we wantin each tail, or 0.975 to the left if z *. Finding 0.975 in the table, we have z * = 1.96. On the TI-
83/84, the solution is given by invNorm(0.975)=1.960.Confidence Intervals for Means and Proportions
In the previous section we discussed the concept of a confidence interval. In this section, we get more
specific by actually constructing confidence intervals for each of the parameters under consideration. The
chart below lists each parameter for which we will construct confidence intervals, the conditions under
which we are justified in constructing the interval, and the formula for actually constructing the interval.