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zOON!Ls`(*Jy94Q7!JtQaP0_3=XyaiaY_mP-Zi3&XoYwTrl$3p;W@&rM-7LFDc}>e~ ze`!|SO3CaUEO|fW?a^M-GFx8kXjh{4+tXgpOMBGAt^J6-1L*z4utz-I+L+%#)p~hR zmo)xY8aL2h3EbVmjr3BOK4o)juJ2$?dwE5lw!0nI_71An&j9+Q*EG$#AuniyWe zPd(ju!#ly+9pa^a+}oqB)|4%lw=5RKJU2Ko{33jYpM+l{QP`<}8-JWt@8k#7T7IRi zdSxInF*OP=S0Gz4ATuB_T?#K!Z*O!UF)=bAFd$M2FG+4@Zy+%-GaxV^QVK6g zL?Bx{I5b2xFgQ0vIYvZAML0P%G(k5vK{hf$Mm0A!L^L=-J|H|eG(Q4HSt46xe~;9X8c66`_gvy79!Sj9StDtHq=I_fAS+~t_(>CKCIQkyT1guT zl6KNTLL^K&Nf+rRJ*1cPkqGH0Q8EB=9`pw>$Yn`u5K_))#UZants%(Ay;cGUziSNx zxq?;_IIO%F0j|q@(*F;F4vb|s{1 \> X\) value have equal-highest probability. Number must exist in distribution. +Value of \(X\) which has the highest probability + +\begin{itemize} \tightlist + \item Most popular value in discrete distributions + \item Must exist in distribution + \item Represented by local max in pdf + \item Multiple modes exist when \(>1 \> X\) value have equal-highest probability +\end{itemize} \subsubsection*{Median} -If \(m > 0.5\), then value of \(X\) that is reached is the median of \(X\). If \(m = 0.5 = 0.5\), then \(m\) is halfway between this value and the next. To find \(m\), add values of \(X\) from smallest to alrgest until the sum reaches 0.5. +Value separating lower and upper half of distribution area -\[ m = X \> \text{such that} \> \int_{-\infty}^{m} f(x) dx = 0.5 \] +\textbf{Continuous:} +\[ m = X \> \text{such that} \> \int_{-\infty}^{m} f(x) \> dx = 0.5 \] + +\textbf{Discrete:} (not in course) +\begin{itemize} \tightlist + \item Does not have to exist in distribution + \item Add values of \(X\) smallest to largest until sum is \(\ge 0.5\) + \item If \(X_1 < 0.5 < X_2\), then median is the average of \(X_1\) and \(X_2\) + \begin{itemize}\tightlist + \item If \(m > 0.5\), then value of \(X\) that is reached is the median of \(X\) + \end{itemize} +\end{itemize} \subsubsection*{Variance \(\sigma^2\)} @@ -72,7 +110,7 @@ If \(m > 0.5\), then value of \(X\) that is reached is the median of \(X\). If \ \subsection*{Binomial distributions} Conditions for a \textit{binomial distribution}: -\begin{enumerate} +\begin{enumerate} \tightlist \item Two possible outcomes: \textbf{success} or \textbf{failure} \item \(\Pr(\text{success})\) (=\(p\)) is constant across trials \item Finite number \(n\) of independent trials @@ -115,7 +153,17 @@ A continuous random variable \(X\) has a pdf \(f\) such that: \begin{cas} Define piecewise functions: \\ - Math3 \(\rightarrow\) + \-\hspace{1em}Math3 \(\rightarrow\) + \begin{tikzpicture}% + \draw rectangle (0.5,0.5); + \node at (0.08,0.25) {\(\{\)}; + \filldraw [black] (0.15, 0.4) rectangle(0.25, 0.3); + \draw (0.35, 0.4) rectangle(0.45, 0.3); + \node [font=\footnotesize] at (0.3,0.3) {\verb;,;}; + \draw (0.15, 0.2) rectangle(0.25, 0.1); + \node [font=\footnotesize] at (0.3,0.1) {\verb;,;}; + \draw (0.35, 0.2) rectangle(0.45, 0.1); + \end{tikzpicture} % TODO: finish this section \end{cas} @@ -197,6 +245,20 @@ For a new distribution with mean of \(n\) trials, \(\operatorname{E}(X^\prime) = \end{cas} +\subsection*{Population sampling} + +\subsubsection*{Population proportion} + +\[ p = \dfrac{n \text{ with attribute in population}}{\text{population size}} \] + +Constant for a given population. + +\subsection*{Sample proportion} + +\[ \hat{p} = \dfrac{n \text{ with attribute in sample}}{\text{sample size}} \] + +Varies with each sample. + \subsection*{Normal distributions} @@ -217,6 +279,11 @@ Normal distributions must have area (total prob.) of 1 \(\implies \int^\infty_{- \item \(C\)\% confidence interval \(\implies\) \(C\)\% of samples will contain population mean \(\mu\) \end{itemize} +\begin{cas} + Menu \(\rightarrow\) Stats \(\rightarrow\) Calc \(\rightarrow\) Interval \\ + Set \textit{Type = One-Sample Z Int} \\ \-\hspace{1em} and select \textit{Variable} +\end{cas} + \subsubsection*{95\% confidence interval} For 95\% c.i. of population mean \(\mu\): @@ -230,10 +297,9 @@ where: \item \(n\) is the sample size from which \(\overline{x}\) was calculated \end{description} -\begin{cas} - Menu \(\rightarrow\) Stats \(\rightarrow\) Calc \(\rightarrow\) Interval \\ - Set \textit{Type = One-Sample Z Int} \\ \-\hspace{1em} and select \textit{Variable} -\end{cas} +\subsubsection*{Confidence interval of \(p\) from \(\hat{p}\)} + +\[ x \in \left( \hat{p} \pm Z \sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}} \right) \] \subsection*{Margin of error} diff --git a/methods/statistics.pdf b/methods/statistics.pdf index 27e441e8af8d9423ddf41c60a083f33fe1be90cc..cf7a26c4f19e42829974531804206d555d8cf972 100644 GIT binary patch delta 78474 zcmY(pQ*fYN*LEA*HafO#chs?M+q`4j?670IW83Q3wvGS!cGY{Z=h2+=Y^_=~s;*UI zEgSx@4!*964u_qYgqg&_*cy(XAC5`R%-+J)l7xkYl{MXH7nB;9(vWpnmqh74*RXY1 z3_SM}f_pIRaNi+`RnbA3UlaWKBNwHvEG@+t`Xv6EKD}7l%1Tx;;c0LL$-<)h^=O}7 zBo9@@K~gGtuCH7KOJ@ipYlF1Bu7Ln|d^d^`wmq)q&XyyN=NQ!k8}&1-rkj3*WkE^# zjqCg6abn3eHMJ73MXGIH;^m!8W{Hh*C0ID=Tkq7tTK&N=+?w19U7F+Gauj1iiaDre zN4G;J`qS0-%isCw>+Iufp6|txB`kNQ?>f=N$J6!c%Jwe&o{Kj#=8H(lQ5eOJjkFFN 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