1 

     2 ============

     3  Statistics

     4 ============

     5 .. contents::

     6    :local:

     7 

     8 .. role:: def

     9    :class: def

    10 

    11 Markov inequality

    12 =================

    13 

    14 :def:`Markov inequality`: :math:`P(X ≥ a) ≤ E[P]/a` for all :math:`a > 0`.

    15 

    16 Chebyshev inequality

    17 ====================

    18 

    19 :def:`Chebyshev inequality`: if :math:`X` is a random variable with mean

    20 :math:`μ` and variance :math:`σ²` then

    21 

    22 .. math::

    23 

    24   P(|X-μ| ≥ c) ≤ σ²/c²

    25 

    26 for all :math:`c > 0`.

    27 

    28 Central limit theorem

    29 =====================

    30 

    31 :def:`Central limit theorem`: let :math:`X_1, ..., X_n, ...` be a sequence of

    32 independent identically distributed random variables with common mean :math:`μ`

    33 and variance :math:`σ²` and let:

    34 

    35 .. math::

    36 

    37   Z_n = ((∑_{1≤i≤n} X_i) - n·μ) / (σ·sqrt(n))

    38 

    39 Then CDF of :math:`Z_n` converge to standard normal CDF:

    40 

    41 .. math::

    42 

    43   Φ(z) = 1/(2·π)·∫_{(-∞;z]} exp(-x²/2) 𝑑x

    44 

    45   lim_{n → ∞} P(Z_n ≤ z) = Φ(z)

    46 

    47 Null hypothesis

    48 ===============

    49 

    50 :def:`Null hypothesis` a statement that the phenomenon being studied produces no

    51 effect or makes no difference, assumption that effect actually due to chance.

    52 

    53 p-value

    54 =======

    55 

    56 :def:`p-value` is the probability of the apparent effect under the null

    57 hypothesis.

    58 

    59   https://en.wikipedia.org/wiki/P-value

    60     Wikipedia page

    61 

    62 Significance level

    63 ==================

    64 

    65 If the p-value is less than or equal to the chosen :def:`significance level`

    66 (:math:`α`), the test suggests that the observed data are inconsistent with the

    67 null hypothesis, so the null hypothesis should be rejected.

    68 

    69 Hypothesis testing

    70 ==================

    71 

    72 :def:`Hypothesis testing` is process of interpretation of statistical

    73 significance of given null hypothesis based on observed p-value from sample with

    74 choosen significance level.

    75 

    76 After finishing hypothesis testing we should reject null hypothesis or fail to

    77 reject due to lack of enough evidence or ...

    78 

    79 Hypothesis testing only take into account:

    80 

    81 * that effect might be due to chance; that is, the difference might appear in a

    82   random sample, but not in the general population

    83 

    84 But it doesn't cover cases:

    85 

    86 * The effect might be real; that is, a similar difference would be seen in the

    87   general population.

    88 * The apparent effect might be due to a biased sampling process, so it would not

    89   appear in the general population.

    90 * The apparent effect might be due to measurement errors.

    91 

    92 Asymptotic approximation

    93 ========================

    94 

    95 CLT say that sample mean distribution is approximated by normal distribution.

    96 

    97 With fair enough number of samples approximation is quite good.

    98 

    99 So during hypothesis testing usually researcher makes assumption that is is safe

   100 to replace unknown distribution of means for independent and identicaly

   101 distributed individual samples with approximation.

   102 

   103 For really small number of samples Student distribution is used instead of

   104 normal distribution. But again it means that researcher made assumption and you

   105 may not agree with it, so it is your right to reject any subsequent decision

   106 based on "wrong" assumption.

   107 

   108 Type I error

   109 ============

   110 

   111 :def:`Type I error` is the incorrect rejection of a true null hypothesis (a

   112 *false positive*).

   113 

   114 Type I error rate is at most :math:`α` (significant level).

   115 

   116 The p-value of a test is the maximum false positive risk you would take by

   117 rejecting the null hypothesis.

   118 

   119 Type II error

   120 =============

   121 

   122 :def:`Type II error` is failing to reject a false null hypothesis (a *false

   123 negative*).

   124 

   125 Probability of type II error usually called :math:`β`.

   126 

   127 Power

   128 =====

   129 

   130 :def:`Power` is a probability to reject null hypothesis when it's false. So

   131 power probability is :math:`1-β`.

   132 

   133 Confidence interval

   134 ===================

   135 

   136 :def:`Confidence interval` 

   137 

   138   https://en.wikipedia.org/wiki/Confidence_interval

   139     Wikipedia page

   140 

   141 

   142 Question

   143 ========

   144 

   145 What to do with null hypothesis in classical inference?

   146 

   147 I successfully shirked stat classes 10 years ago (last night reading help me

   148 actually to pass exam) and now when I take several Coursera stat classes I have

   149 difficulties with understanding **null hypothesis**. Somehow with unclear

   150 intuition I passed quizzes but want to understand subject.

   151 

   152 Suppose we have population and sample some data from population. Reasonable

   153 question: is some property of sample make evidence to be true on population?

   154 

   155 Statistic is a real number that can be derived from population or sample.

   156 Classical example is a mean value.

   157 

   158 We ask is it statistically significant that statistic of population is near to

   159 statistic of sample.

   160