From 491a90406aee585f59bab0bcaaf5f71c0fe3ca67 Mon Sep 17 00:00:00 2001 From: Andrew Lorimer Date: Thu, 5 Sep 2019 12:59:13 +1000 Subject: [PATCH] [spec] p-values table and one/two-tail tests --- spec/statistics.pdf | Bin 212441 -> 247971 bytes spec/statistics.tex | 63 +++++++++++++++++++++++++++++++++++++++++--- 2 files changed, 59 insertions(+), 4 deletions(-) diff --git a/spec/statistics.pdf b/spec/statistics.pdf index 9a63860bef32597f9e026bdcc95c2915da78773d..6e08307749499c6965118c5929c022903d8b5efe 100644 GIT binary patch delta 148836 zcmaHyQ*bW8(yn7CE4FRhwwXl@#HxACL4^0 zR89f*(<6&lx<9UGl(}e?`(a^@M2F8`x%-kG-t6fZ8jqRJ_M}Aswj_bSvKoyH^GC~$ zY`r^aZ^0mov8_Wc?o*GeerMbvZ5h#AwWiW zQRMt^ciJY%R|DN{J=liUD+Ls>55TF@zqDw z%d!a@L*zv>v#Y}ZY>ycV4yIiDQUkrq2l<@nFqTW}uJd&$5^ zsI>a91}Qjega&m-rE^+{t9JjIudPQc$E*dJU;Kc}tERbO*WmPwl!WEmR*aYe z#K0NSE?U<6tJkUMLj+saQyzWd0;uE!2&c$Of37yvH%CB79F-FP3V4BMJk&jTCi0U> zpYZ`+bO)XWs7S>uKRnH!QHP?o0|WkrGN?_+t-)pRxg}7;K;Qc~(YwReL3w3%P(TL| 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z*9Nld(u~+YL>e3rwf|d-h%H5Z_$Lv-z^)5JuWaFILdeL#4nwa_sLe>oOvw0OQpwKF z`M)F!Ap@Zj485eSsoj4O>;H^I2(@{HnAll`L>buFIapW)S;d3}S%sKHS%jIHg*e5S z8QDbm2>;(p5dJ$z#>Cdl*_@DxzeMtd{HXDcBxV;la5e2 zm$M>8o-t6kg9+L87X&8>3rfXcyhZYXVvI{{lIi}ee;HddJ+9zxFh{i@r!_sU`G-tc zO#8Qw#<K98ZTrv_&fADM5cm@@4D?7f*yjGElo2 zj1bj39bT@eK~{ww*oCfgPxsJs}={{U#ic}4&L diff --git a/spec/statistics.tex b/spec/statistics.tex index da1ec9b..d7e7723 100644 --- a/spec/statistics.tex +++ b/spec/statistics.tex @@ -256,13 +256,26 @@ \subsection*{\(p\)-value} - Probability of observing a value of the sample statistic as significant as the one observed, assuming null hypothesis is true. - % table of p-values for strength of evidence + \begin{align*} + p &= \Pr(\overline{X} \lessgtr \mu(H_1)) \\ + &= 2 \cdot \Pr(\overline{X} <> \mu(H_1) | \mu = 8) + \end{align*} - \subsection*{Distribution of sample mean} + Probability of observing a value of the sample statistic as significant as the one observed, assuming null hypothesis is true. - If \(X \sim \operatorname{N}(\mu, \sigma)\), then distribution of sample mean \(\overline{X}\) is also normal with \(\overline{X} \sim \operatorname{N}(\mu, \frac{\sigma}{\sqrt{n}}\). + \vspace{0.5em} + \begin{tabularx}{23em}{|l|X|} + \hline + \rowcolor{cas} + \(\boldsymbol{p}\) & \textbf{Conclusion} \\ + \hline + \(> 0.05\) & insufficient evidence against \(H_0\) \\ + \(< 0.05\) (5\%) & good evidence against \(H_0\) \\ + \(< 0.01\) (1\%) & strong evidence against \(H_0\) \\ + \(< 0.001\) (0.1\%) & very strong evidence against \(H_0\) \\ + \hline + \end{tabularx} \subsection*{Statistical significance} @@ -287,4 +300,46 @@ \end{description} \end{cas} + \subsection*{One-tail and two-tail tests} + + \subsubsection*{One tail} + + \begin{itemize} + \item \(\mu\) has changed in one direction + \item State ``\(H_1: \mu \lessgtr \) known population mean'' + \end{itemize} + + \subsubsection*{Two tail} + + \begin{itemize} + \item Direction of \(\Delta \mu\) is ambiguous + \item State ``\(H_1: \mu \ne\) known population mean'' + \end{itemize} + + For two tail tests: + \begin{align*} + p\text{-value} &= \Pr(|\overline{X} - \mu| \ge |\overline{x}_0 - \mu|) \\ + &= \left( |Z| \ge \left|\dfrac{\overline{x}_0 - \mu}{\sigma \div \sqrt{n}} \right| \right) + \end{align*} + + \subsection*{Modulus notation for two tail} + + \(\Pr(|\overline{X} - \mu| \ge a) \implies\) ``the probability that the distance between \(\overline{\mu}\) and \(\mu\) is \(\ge a\)'' + + \subsection*{Inverse normal} + + \begin{cas} + \verb;invNormCdf("L", ;\(\alpha\)\verb;, ;\(\dfrac{\sigma}{n^\alpha}\)\verb;, ;\(\mu\)\verb;); + \end{cas} + + \subsection*{Errors} + + \begin{description}[labelwidth=2.5cm, labelindent=0.5cm] + \item [Type I error] \(H_0\) is rejected when it is \textbf{true} + \item [Type II error] \(H_0\) is \textbf{not} rejected when it is \textbf{false} + \end{description} + +% \subsection*{Using c.i. to find \(p\)} +% need more here + \end{document} -- 2.47.1