SECTION 6.2 Confidence Intervals for the Mean (s Unknown) 311 Table 5 in Appendix B lists critical values of t for selected confidence intervals and degrees of freedom. Finding Critical Values of t Find the critical value tc for a 95% confidence level when the sample size is 15. SOLUTION Because n = 15, the degrees of freedom are d.f. = n - 1 = 15 - 1 = 14. A portion of Table 5 is shown. Using d.f. = 14 and c = 0.95, you can find the critical value tc, as shown by the highlighted areas in the table. Level of confidence, c 0.80 0.90 0.98 One tail, 0.10 0.05 0.01 d.f. Two tails, 0.20 0.10 0.02 1 3.078 6.314 31.821 3 1.638 2.353 4.541 13 1.350 1.771 2.650 15 1.341 1.753 2.131 2.602 1.886 2.920 6.965 2 0.95 0.025 0.05 12.706 3.182 4.303 12 1.356 1.782 2.179 2.681 14 1.345 1.761 2.145 2.624 16 1.337 1.746 2.120 2.583 2.160 a a 0.99 0.005 0.01 63.657 9.925 5.841 3.055 3.012 2.977 2.947 2.921 From the table, you can see that tc = 2.145. The figure shows the t@distribution for 14 degrees of freedom, c = 0.95, and tc = 2.145. c = 0.95 0.025 0.025 tc = 2.145 −tc = −2.145 t You can use technology to find tc. To use a TI-84 Plus, you need to know the area under the curve to the left of tc, which is 0.95 + 0.025 = 0.975. Area to the left of tc From the TI-84 Plus display at the left, tc ≈ 2.145. Interpretation So, for a t@distribution curve with 14 degrees of freedom, 95% of the area under the curve lies between t = {2.145. TRY IT YOURSELF 1 Find the critical value tc for a 90% confidence level when the sample size is 22. Answer: Page A40 When the number of degrees of freedom you need is not in the table, use the closest number in the table that is less than the value you need (or use technology, as shown in Example 1). For instance, for d.f. = 57, use 50 degrees of freedom. This conservative approach will yield a larger confidence interval with a slightly higher level of confidence c. EXAMPLE 1 Study Tip Critical values in the t@distribution table for a specific confidence interval can be found in the column headed by c in the appropriate d.f. row. (The symbol a will be explained in Chapter 7.) TI-84 PLUS invT(.975,14) 2.144786681
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