Inferential statistics

https://www.khanacademy.org/math/probability/statistics-inferential

Estimating the probability that the true population mean lies within a range around a sample mean.

Confidence Interval Example

Small sample size confidence intervals

Constructing small sample size confidence

Z-statistics vs. T-statistics

http://youtu.be/5ABpqVSx33I

T-Statistic Confidence Interval

http://youtu.be/hV4pdjHCKuA

Student's t-distribution

Probability density function[edit]

Student's t-distribution has the probability density function given by

$f(t) = \frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{t^2}{\nu} \right)^{-\frac{\nu+1}{2}},\!$

where $\nu$ is the number of degrees of freedom and $\Gamma$ is the gamma function.

CDF

$\begin{matrix} \frac{1}{2} + x \Gamma \left( \frac{\nu+1}{2} \right) \times\\[0.5em] \frac{\,_2F_1 \left ( \frac{1}{2},\frac{\nu+1}{2};\frac{3}{2}; -\frac{x^2}{\nu} \right)} {\sqrt{\pi\nu}\,\Gamma \left(\frac{\nu}{2}\right)} \end{matrix}$

One Sided 75% 80% 85% 90% 95% 97.5% 99% 99.5% 99.75% 99.9% 99.95%

Two Sided 50% 60% 70% 80% 90% 95% 98% 99% 99.5% 99.8% 99.9%

1 1.000 1.376 1.963 3.078 6.314 12.71 31.82 63.66 127.3 318.3 636.6

2 0.816 1.061 1.386 1.886 2.920 4.303 6.965 9.925 14.09 22.33 31.60

3 0.765 0.978 1.250 1.638 2.353 3.182 4.541 5.841 7.453 10.21 12.92

4 0.741 0.941 1.190 1.533 2.132 2.776 3.747 4.604 5.598 7.173 8.610

5 0.727 0.920 1.156 1.476 2.015 2.571 3.365 4.032 4.773 5.893 6.869

6 0.718 0.906 1.134 1.440 1.943 2.447 3.143 3.707 4.317 5.208 5.959

7 0.711 0.896 1.119 1.415 1.895 2.365 2.998 3.499 4.029 4.785 5.408

8 0.706 0.889 1.108 1.397 1.860 2.306 2.896 3.355 3.833 4.501 5.041

9 0.703 0.883 1.100 1.383 1.833 2.262 2.821 3.250 3.690 4.297 4.781

10 0.700 0.879 1.093 1.372 1.812 2.228 2.764 3.169 3.581 4.144 4.587

11 0.697 0.876 1.088 1.363 1.796 2.201 2.718 3.106 3.497 4.025 4.437

12 0.695 0.873 1.083 1.356 1.782 2.179 2.681 3.055 3.428 3.930 4.318

13 0.694 0.870 1.079 1.350 1.771 2.160 2.650 3.012 3.372 3.852 4.221

14 0.692 0.868 1.076 1.345 1.761 2.145 2.624 2.977 3.326 3.787 4.140

15 0.691 0.866 1.074 1.341 1.753 2.131 2.602 2.947 3.286 3.733 4.073

16 0.690 0.865 1.071 1.337 1.746 2.120 2.583 2.921 3.252 3.686 4.015

17 0.689 0.863 1.069 1.333 1.740 2.110 2.567 2.898 3.222 3.646 3.965

18 0.688 0.862 1.067 1.330 1.734 2.101 2.552 2.878 3.197 3.610 3.922

19 0.688 0.861 1.066 1.328 1.729 2.093 2.539 2.861 3.174 3.579 3.883

20 0.687 0.860 1.064 1.325 1.725 2.086 2.528 2.845 3.153 3.552 3.850

21 0.686 0.859 1.063 1.323 1.721 2.080 2.518 2.831 3.135 3.527 3.819

22 0.686 0.858 1.061 1.321 1.717 2.074 2.508 2.819 3.119 3.505 3.792

23 0.685 0.858 1.060 1.319 1.714 2.069 2.500 2.807 3.104 3.485 3.767

24 0.685 0.857 1.059 1.318 1.711 2.064 2.492 2.797 3.091 3.467 3.745

25 0.684 0.856 1.058 1.316 1.708 2.060 2.485 2.787 3.078 3.450 3.725

26 0.684 0.856 1.058 1.315 1.706 2.056 2.479 2.779 3.067 3.435 3.707

27 0.684 0.855 1.057 1.314 1.703 2.052 2.473 2.771 3.057 3.421 3.690

28 0.683 0.855 1.056 1.313 1.701 2.048 2.467 2.763 3.047 3.408 3.674

29 0.683 0.854 1.055 1.311 1.699 2.045 2.462 2.756 3.038 3.396 3.659

30 0.683 0.854 1.055 1.310 1.697 2.042 2.457 2.750 3.030 3.385 3.646

40 0.681 0.851 1.050 1.303 1.684 2.021 2.423 2.704 2.971 3.307 3.551

50 0.679 0.849 1.047 1.299 1.676 2.009 2.403 2.678 2.937 3.261 3.496

60 0.679 0.848 1.045 1.296 1.671 2.000 2.390 2.660 2.915 3.232 3.460

80 0.678 0.846 1.043 1.292 1.664 1.990 2.374 2.639 2.887 3.195 3.416

100 0.677 0.845 1.042 1.290 1.660 1.984 2.364 2.626 2.871 3.174 3.390

120 0.677 0.845 1.041 1.289 1.658 1.980 2.358 2.617 2.860 3.160 3.373

$\infty$ 0.674 0.842 1.036 1.282 1.645 1.960 2.326 2.576 2.807 3.090 3.291

One Sided	75%	80%	85%	90%	95%	97.5%	99%	99.5%	99.75%	99.9%	99.95%
Two Sided	50%	60%	70%	80%	90%	95%	98%	99%	99.5%	99.8%	99.9%
1	1.000	1.376	1.963	3.078	6.314	12.71	31.82	63.66	127.3	318.3	636.6
2	0.816	1.061	1.386	1.886	2.920	4.303	6.965	9.925	14.09	22.33	31.60
3	0.765	0.978	1.250	1.638	2.353	3.182	4.541	5.841	7.453	10.21	12.92
4	0.741	0.941	1.190	1.533	2.132	2.776	3.747	4.604	5.598	7.173	8.610
5	0.727	0.920	1.156	1.476	2.015	2.571	3.365	4.032	4.773	5.893	6.869
6	0.718	0.906	1.134	1.440	1.943	2.447	3.143	3.707	4.317	5.208	5.959
7	0.711	0.896	1.119	1.415	1.895	2.365	2.998	3.499	4.029	4.785	5.408
8	0.706	0.889	1.108	1.397	1.860	2.306	2.896	3.355	3.833	4.501	5.041
9	0.703	0.883	1.100	1.383	1.833	2.262	2.821	3.250	3.690	4.297	4.781
10	0.700	0.879	1.093	1.372	1.812	2.228	2.764	3.169	3.581	4.144	4.587
11	0.697	0.876	1.088	1.363	1.796	2.201	2.718	3.106	3.497	4.025	4.437
12	0.695	0.873	1.083	1.356	1.782	2.179	2.681	3.055	3.428	3.930	4.318
13	0.694	0.870	1.079	1.350	1.771	2.160	2.650	3.012	3.372	3.852	4.221
14	0.692	0.868	1.076	1.345	1.761	2.145	2.624	2.977	3.326	3.787	4.140
15	0.691	0.866	1.074	1.341	1.753	2.131	2.602	2.947	3.286	3.733	4.073
16	0.690	0.865	1.071	1.337	1.746	2.120	2.583	2.921	3.252	3.686	4.015
17	0.689	0.863	1.069	1.333	1.740	2.110	2.567	2.898	3.222	3.646	3.965
18	0.688	0.862	1.067	1.330	1.734	2.101	2.552	2.878	3.197	3.610	3.922
19	0.688	0.861	1.066	1.328	1.729	2.093	2.539	2.861	3.174	3.579	3.883
20	0.687	0.860	1.064	1.325	1.725	2.086	2.528	2.845	3.153	3.552	3.850
21	0.686	0.859	1.063	1.323	1.721	2.080	2.518	2.831	3.135	3.527	3.819
22	0.686	0.858	1.061	1.321	1.717	2.074	2.508	2.819	3.119	3.505	3.792
23	0.685	0.858	1.060	1.319	1.714	2.069	2.500	2.807	3.104	3.485	3.767
24	0.685	0.857	1.059	1.318	1.711	2.064	2.492	2.797	3.091	3.467	3.745
25	0.684	0.856	1.058	1.316	1.708	2.060	2.485	2.787	3.078	3.450	3.725
26	0.684	0.856	1.058	1.315	1.706	2.056	2.479	2.779	3.067	3.435	3.707
27	0.684	0.855	1.057	1.314	1.703	2.052	2.473	2.771	3.057	3.421	3.690
28	0.683	0.855	1.056	1.313	1.701	2.048	2.467	2.763	3.047	3.408	3.674
29	0.683	0.854	1.055	1.311	1.699	2.045	2.462	2.756	3.038	3.396	3.659
30	0.683	0.854	1.055	1.310	1.697	2.042	2.457	2.750	3.030	3.385	3.646
40	0.681	0.851	1.050	1.303	1.684	2.021	2.423	2.704	2.971	3.307	3.551
50	0.679	0.849	1.047	1.299	1.676	2.009	2.403	2.678	2.937	3.261	3.496
60	0.679	0.848	1.045	1.296	1.671	2.000	2.390	2.660	2.915	3.232	3.460
80	0.678	0.846	1.043	1.292	1.664	1.990	2.374	2.639	2.887	3.195	3.416
100	0.677	0.845	1.042	1.290	1.660	1.984	2.364	2.626	2.871	3.174	3.390
120	0.677	0.845	1.041	1.289	1.658	1.980	2.358	2.617	2.860	3.160	3.373
$\infty$	0.674	0.842	1.036	1.282	1.645	1.960	2.326	2.576	2.807	3.090	3.291

How the t-distribution arises

Sampling distribution

Let x₁, ..., x_n be the numbers observed in a sample

from a continuously distributed population with expected value μ.

The sample mean and sample variance are given by:

\begin{align} \bar{x} &= \frac{x_1+\cdots+x_n}{n} \\ s^2 &= \frac{1}{n-1}\sum_{i=1}^n (x_i - \bar{x})^2 \end{align}

The resulting t-value is

t = \frac{\bar{x} - \mu}{s/\sqrt{n}}.

Suppose X1, ..., Xn are independent realizations

of the normally-distributed, random variable X,

which has an expected value μ and variance σ2.

Let
$\overline{X}_n = \frac{1}{n}(X_1+\cdots+X_n)$

$S_n^{\;2} = \frac{1}{n-1}\sum_{i=1}^n\left(X_i-\overline{X}_n\right)^2$

==>

$Z = \left(\overline{X}_n-\mu\right)\frac{\sqrt{n}}{\sigma}$
is normally distributed with mean 0 and variance 1,

$V = (n-1)\frac{S_n^2}{\sigma^2}$
has a chi-squared distribution with v=n−1 degrees of freedom

$T \equiv \frac{Z}{\sqrt{V/v}} = \left(\overline{X}_n-\mu\right)\frac{\sqrt{n}}{S_n},$
has a Student's t-distribution with v=n−1 degrees of freedom

Parameters	ν = (n-1)> 0 degrees of freedom (real)

Mean	0 for ν > 1, otherwise undefined

Variance	$\textstyle\frac{\nu}{\nu-2}$ for ν > 2, ∞ for 1 < ν ≤ 2, otherwiseundefined

pdf	$\textstyle\frac{\Gamma \left(\frac{\nu+1}{2} \right)} {\sqrt{\nu\pi}\,\Gamma \left(\frac{\nu}{2} \right)} \left(1+\frac{x^2}{\nu} \right)^{-\frac{\nu+1}{2}}\!$

CDF	$\begin{matrix} \frac{1}{2} + x \Gamma \left( \frac{\nu+1}{2} \right) \times\\[0.5em] \frac{\,_2F_1 \left ( \frac{1}{2},\frac{\nu+1}{2};\frac{3}{2}; -\frac{x^2}{\nu} \right)} {\sqrt{\pi\nu}\,\Gamma \left(\frac{\nu}{2}\right)} \end{matrix}$ where ₂F₁ is the hypergeometric function

Density of the t-distribution (red) for 1, 5, and 30 degrees of freedom
compared to the standard normal distribution (blue).
Previous plots shown in green.

1df

Normal Distribution for Z value:

File:NormalDist1.96.png

T distribution for T value, with v= inf, v=10, v=1

blue: Normal, 95% ==> 1.96
red: T, v= 10, 95% ==> 2.28
green: T, v=1, 95% ==> 12.71

ryTeach

2014年5月23日星期五

Inferential statistics @ Khanacademy

Confidence interval 1

Estimating the probability that the true population mean lies within a range around a sample mean.

Confidence interval example

Confidence Interval Example

Small sample size confidence intervals

Constructing small sample size confidence

Z-statistics vs. T-statistics

T-Statistic Confidence Interval

Student's t-distribution

Probability density function[edit]

CDF

How the t-distribution arises

Sampling distribution

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張貼留言

2014年5月23日 星期五

Inferential statistics @ Khanacademy

Confidence interval 1

Estimating the probability that the true population mean lies within a range around a sample mean.

Confidence interval example

Confidence Interval Example

Small sample size confidence intervals

Constructing small sample size confidence

Z-statistics vs. T-statistics

T-Statistic Confidence Interval

Student's t-distribution

Probability density function[edit]

CDF

How the t-distribution arises

Sampling distribution

沒有留言:

張貼留言

2014年5月23日星期五