368 Chapter 8 Confidence Intervals EXAMPLE 8.10 Finding a critical value A simple random sample of size 10 is drawn from a normal population. Find the critical value t��∕2 for a 95% confidence interval. Area = 0.025 Area = 0.025 Area = 0.95 0 −2.262 2.262 Figure 8.11 95% of the area under the Student’s t curve with 9 degrees of freedom is between t = −2.262 and t = 2.262. Solution The sample size is n = 10, so the number of degrees of freedom is n − 1 = 9. We consult Table A.3, looking in the row corresponding to 9 degrees of freedom, and in the column with confidence level 95% (the confidence levels are listed along the bottom of the table). The critical value is t��∕2 = 2.262. See Figure 8.11. ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 4 0.271 0.741 1.533 2.132 2.776 3.747 4.604 5.598 7.173 8.610 5 0.267 0.727 1.476 2.015 2.571 3.365 4.032 4.773 5.893 6.869 6 0.265 0.718 1.440 1.943 2.447 3.143 3.707 4.317 5.208 5.959 7 0.263 0.711 1.415 1.895 2.365 2.998 3.499 4.029 4.785 5.408 8 0.262 0.706 1.397 1.860 2.306 2.896 3.355 3.833 4.501 5.041 9 0.261 0.703 1.383 1.833 2.262 2.821 3.250 3.690 4.297 4.781 10 0.260 0.700 1.372 1.812 2.228 2.764 3.169 3.581 4.144 4.587 11 0.260 0.697 1.363 1.796 2.201 2.718 3.106 3.497 4.025 4.437 12 0.259 0.695 1.356 1.782 2.179 2.681 3.055 3.428 3.930 4.318 13 0.259 0.694 1.350 1.771 2.160 2.650 3.012 3.372 3.852 4.221 14 0.258 0.692 1.345 1.761 2.145 2.624 2.977 3.326 3.787 4.140 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 20% 50% 80% 90% 95% 98% 99% 99.5% 99.8% 99.9% Confidence Level What if the number of degrees of freedom isn’t in the table? The largest number of degrees of freedom shown in Table A.3 is 200. If the desired number of degrees of freedom is less than 200 and is not shown in Table A.3, use the next smaller number of degrees of freedom in the table. If the desired number of degrees of freedom is more than 200, use the z-value found in the last row of Table A.3, or use Table A.2. This problem will not arise if a calculator or computer is used to construct a confidence interval, because technology can compute critical values for any number of degrees of freedom. SUMMARY If the desired number of degrees of freedom isn’t listed in Table A.3, then ∙ If the desired number is less than 200, use the next smaller number that is in the table. ∙ If the desired number is greater than 200, use the z-value found in the last row of Table A.3, or use Table A.2. EXAMPLE 8.11 Finding a critical value Use Table A.3 to find the critical value for a 99% confidence interval for a sample of size 58. Solution Because the sample size is 58, there are 57 degrees of freedom. The number 57 doesn’t appear in the degrees of freedom column in Table A.3, so we use the next smaller number that does appear, which is 50. The value of t��∕2 corresponding to a confidence level of 99% is t��∕2 = 2.678. Explain It Again x̄ must be approximately normal: We need assumption 2 to be sure that the sampling distribution of x̄ is approximately normal. Following are the assumptions that are necessary to construct confidence intervals by using the Student’s t distribution. Assumptions for Constructing a Confidence Interval for �� When �� Is Unknown 1. We have a simple random sample. 2. Either the sample size is large (n > 30), or the population is approximately normal.
navidi_monk_elementary_statistics_2e_ch7-9
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