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Lecture 20

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Department
Statistics
Course
STAB22H3
Professor
Michael Krashinsky
Semester
Fall

Description
Lecture Twenty ▯ Many times scientists and statisticians are interested in estimating the mean of a distribution. ▯ The sample mean is a good point estimate of the distribution mean; if you are going to propose one number for ▯, the sample mean is good but ... { The sample mean is itself a random variable and subject to sampling vari- ability. { Proposing the sample mean alone is inadequate without also describing the variability in the sample mean (to measure the con▯dence we have in the estimate.) { The variability in the sample mean is proportional to the variability in the distribution sampled from and inversely proportional to the size of the sample. ▯ More often we propose a range of values for ▯ ▯ A con▯dence interval is a range of values proposed for the distribution mean ▯ that comes with a measure of our con▯dence in the procedure used to generate that range of values. ▯ Patiently and with an eye toward that goal we ... ▯ Recall, suppose that 1 ;2 ;:::;Xnis a sample from a distribution with mean ▯ and standard deviation ▯ then ... Distribution Sampled From N(▯;▯) ▯, ▯ (Not Necessarily Normal) n small n small ▯ p ▯ X ▯ N(▯;▯= n) Distribution of X depends on distribution of the X’s n large n large (let’s say bigger than 30) X ▯ N(▯;▯= n)p X▯ a▯ N(▯;▯= n) p 1 ▯ Recall, we had the hugely important fact that a random variable X ▯ N(▯;▯) X▯▯ if and only if Z ▯ ▯ N(0;1). ▯ We used this fact every time we standardize to take advantage of table A. p ▯ So, whenever X ▯ N(▯;▯= n), it is the case that X ▯ ▯ Z = p ▯ N(0;1) ▯= n p ▯ And whenever X ap▯ N(▯;▯= n), it is the case that X ▯ ▯ approx Z = p ▯ N(0;1) ▯= n ▯ So we can improve our table. ▯ If 1 ;X2;:::;Xnis a sample from a distribution with mean ▯ and standard deviation ▯ then ... Distribution Sampled From N(▯;▯) ▯, ▯ (Not Necessarily Normal) n small n small p X ▯ N(▯;▯= n) Distribution of X depends ▯ on distribution of the X’s X▯p ▯ N(0;1) ▯= n n large n large (let’s say bigger than 30) p p X ▯ N(▯;▯= n) X▯ a▯ N(▯;▯= n) X▯p ▯ N(0;1) Xp▯ a▯ N(0;1) ▯= n ▯= n ▯ There is another probability distribution that shows up a lot in statistics, in particular when we are sampling from the normal distribution. 2 ▯ De▯nition: If X1;X2;:::;Xnis a sample from the normal distribution with mean ▯ and standard deviation ▯, X is the sample mean and S is the sample standard deviation, then the t distribution with n▯1 degrees of freedom is the probability distribution of the random variable X ▯ ▯ T = S= n ▯ The sample standard deviation should be pretty close to the distribution stan- dard deviation, ie S ▯ ▯, (especially if the sample size is large) so the t distribution is pretty close to the standard normal distribution. ▯ In fact if n is large, say bigger than 30, the t distribution with n ▯ 1 degrees of freedom is approximately equal to the standard normal distribution. ▯ So we can add to our table. ▯ If X1;X2;:::;Xnis a sample from a distribution with mean ▯ and standard deviation ▯ then ... Distribution Sampled From N(▯;▯) ▯, ▯ (Not Necessarily Normal) n small n small p X ▯ N(▯;▯= n) Distribution of X depends on distribution of the X’s Xp▯ ▯ N(0;1) ▯= n X▯p S= n ▯ tn▯1 n large (let’s say bigger than 30) n large (let’s say bigger than 30) p approx p X ▯ N(▯;▯= n) X▯ ▯ N(▯;▯= n) ▯ ▯ approx Xp▯ ▯ N(0;1) Xp▯ ▯ N(0;1) ▯= n ▯= n Xp▯ Xp▯ approx S= n ▯ N(0;1) S= n ▯ N(0;1) 3 MooreIntro-3620056 ips_tables October 6, 2010 13:30 ▯ We have some (fewer than for the standard normal) probabilities for the t distribution with n ▯ 1 degrees of freedom tabulated for several n’s Tables T-11 Table entry for p and C is the critical value t with Probability p probability p lying to its right and probability C lying between ▯t and t . ▯ t* T A B L E D t distribution critical values Upper-tail probability p df .25 .20 .15 .10 .05 .025 .02 .01 .005 .0025 .001 .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 5.893 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 3.611 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
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