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Statistics
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STAB22H3
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Michael Krashinsky
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Lecture 20

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Statistics

STAB22H3

Michael Krashinsky

Fall

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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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