chapter7 - Chapter 7 Inference Concerning...

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Chapter 7 Inference Concerning Populations (Numeric Responses)
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Inference for Population Mean Practical Problem: Sample mean has sampling distribution that is Normal with mean μ and standard deviation σ / n (when the data are normal, and approximately so for large samples). σ is unknown . Have an estimate of σ , s obtained from sample data. Estimated standard error of the sample mean is: ) 1 ( ~ - - = = n t n s x t n s SE x μ When the sample is SRS from N ( μ , σ 29 then the t -statistic (same as z- with estimated standard deviation) is distributed t with n -1 degrees of freedom
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Family of t -distributions Symmetric, Mound-shaped, centered at 0 (like the standard normal ( z ) distribution Indexed by degrees of freedom ) , the number of independent observations (deviations) comprising the estimated standard deviation. For one sample problems ν = n -1 Have heavier tails (more probability over extreme ranges) than the z -distribution Converge to the z -distribution as ν gets large Tables of critical values for certain upper tail probabilities are available (inside back cover of text)
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df 0.25 0.2 0.15 0.1 0.05 0.025 0.02 0.01 0.005 0.0025 0.001 0.0005 1 1.000 1.376 1.963 3.078 6.314 12.71 15.89 31.82 63.66 127.3 318.3 636.6 2 0.816 1.061 1.386 1.886 2.920 4.303 4.849 6.965 9.925 14.09 22.33 31.60 3 0.765 0.978 1.250 1.638 2.353 3.182 3.482 4.541 5.841 7.453 10.21 12.92 4 0.741 0.941 1.190 1.533 2.132 2.776 2.999 3.747 4.604 5.598 7.173 8.610 5 0.727 0.920 1.156 1.476 2.015 2.571 2.757 3.365 4.032 4.773 5.894 6.869 6 0.718 0.906 1.134 1.440 1.943 2.447 2.612 3.143 3.707 4.317 5.208 5.959 7 0.711 0.896 1.119 1.415 1.895 2.365 2.517 2.998 3.499 4.029 4.785 5.408 8 0.706 0.889 1.108 1.397 1.860 2.306 2.449 2.896 3.355 3.833 4.501 5.041 9 0.703 0.883 1.100 1.383 1.833 2.262 2.398 2.821 3.250 3.690 4.297 4.781 10 0.700 0.879 1.093 1.372 1.812 2.228 2.359 2.764 3.169 3.581 4.144 4.587 11 0.697 0.876 1.088 1.363 1.796 2.201 2.328 2.718 3.106 3.497 4.025 4.437 12 0.695 0.873 1.083 1.356 1.782 2.179 2.303 2.681 3.055 3.428 3.930 4.318 13 0.694 0.870 1.079 1.350 1.771 2.160 2.282 2.650 3.012 3.372 3.852 4.221 14 0.692 0.868 1.076 1.345 1.761 2.145 2.264 2.624 2.977 3.326 3.787 4.140 15 0.691 0.866 1.074 1.341 1.753 2.131 2.249 2.602 2.947 3.286 3.733 4.073 16 0.690 0.865 1.071 1.337 1.746 2.120 2.235 2.583 2.921 3.252 3.686 4.015 17 0.689 0.863 1.069 1.333 1.740 2.110 2.224 2.567 2.898 3.222 3.646 3.965 18 0.688 0.862 1.067 1.330 1.734 2.101 2.214 2.552 2.878 3.197 3.610 3.922 19 0.688 0.861 1.066 1.328 1.729 2.093 2.205 2.539 2.861 3.174 3.579 3.883 20 0.687 0.860 1.064 1.325 1.725 2.086 2.197 2.528 2.845 3.153 3.552 3.850 21 0.686 0.859 1.063 1.323 1.721 2.080 2.189 2.518 2.831 3.135 3.527 3.819 22 0.686 0.858 1.061 1.321 1.717 2.074 2.183 2.508 2.819 3.119 3.505 3.792 23 0.685 0.858 1.060 1.319 1.714 2.069 2.177 2.500 2.807 3.104 3.485 3.768 24 0.685 0.857 1.059 1.318 1.711 2.064 2.172 2.492 2.797 3.091 3.467 3.745 25 0.684 0.856 1.058 1.316 1.708 2.060 2.167 2.485 2.787 3.078 3.450 3.725 26 0.684 0.856 1.058 1.315 1.706 2.056 2.162 2.479 2.779 3.067 3.435 3.707 27 0.684 0.855 1.057 1.314 1.703 2.052 2.158 2.473 2.771 3.057 3.421 3.689 28 0.683 0.855 1.056 1.313 1.701 2.048 2.154 2.467 2.763 3.047 3.408 3.674 29 0.683 0.854 1.055 1.311 1.699 2.045 2.150 2.462 2.756 3.038 3.396 3.660 30 0.683 0.854 1.055 1.310 1.697 2.042 2.147 2.457 2.750 3.030 3.385 3.646 40 0.681 0.851 1.050 1.303 1.684 2.021 2.123 2.423 2.704 2.971 3.307 3.551 50 0.679 0.849 1.047 1.299 1.676 2.009 2.109 2.403 2.678 2.937 3.261 3.496 60 0.679 0.848 1.045 1.296 1.671 2.000 2.099 2.390 2.660 2.915 3.232 3.460 80 0.678 0.846 1.043 1.292 1.664 1.990 2.088 2.374 2.639 2.887 3.195 3.416 100 0.677 0.845 1.042 1.290 1.660 1.984 2.081 2.364 2.626 2.871 3.174 3.390 1000 0.675 0.842 1.037 1.282 1.646 1.962 2.056 2.330 2.581 2.813 3.098 3.300 z* 0.674 0.842 1.036 1.282 1.645 1.960 2.054 2.326 2.576 2.807 3.090 3.290 Probability D e g r e e s o f F r e e d o m C r i t i c a l V a l u e s Critical Values
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t(5), t(15), t(25), z distributions -4 -3 -2 -1 0 1 2 3 4 Density t(5) t(15) t(25) z
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One-Sample Confidence Interval for μ SRS from a population with mean μ is obtained. Sample mean, sample standard deviation are obtained Degrees of freedom are ν = n -1, and confidence level C are selected Level C confidence interval of form: C t n t t P t t n s t x = - - ± ) ) 1 ( ( that so table - from selected * * * * Procedure is theoretically derived based on normally distributed data, but has been found to work well regardless for large n
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