\documentclass[methods-collated.tex]{subfiles}
+
\begin{document}
- \section{Statistics}
- \subsection*{Probability}
+\section{Statistics}
+
+\subsection*{Probability}
+
+\begin{align*}
+ \Pr(A \cup B) &= \Pr(A) + \Pr(B) - \Pr(A \cap B) \\
+ \Pr(A \cap B) &= \Pr(A|B) \times \Pr(B) \\
+ \Pr(A|B) &= \frac{\Pr(A \cap B)}{\Pr(B)} \\
+ \Pr(A) &= \Pr(A|B) \cdot \Pr(B) + \Pr(A|B^{\prime}) \cdot \Pr(B^{\prime})
+\end{align*}
+
+Mutually exclusive: \(\Pr(A \cap B) = 0\) \\
+
+Independent events:
+\begin{flalign*}
+ \quad \Pr(A \cap B) &= \Pr(A) \times \Pr(B)& \\
+ \Pr(A|B) &= \Pr(A) \\
+ \Pr(B|A) &= \Pr(B)
+\end{flalign*}
+
+\subsection*{Combinatorics}
+
+\begin{align*}
+ \text{Arrangements} && {n \choose k} & = \frac{n!}{(n-k)} \\
+ \text{Combinations} && {n \choose k} & = \frac{n!}{k!(n-k)!}
+\end{align*}
+
+Note \({n \choose k} = {n \choose k-1}\)
+
+\begin{cas}
+ Keyboard \(\rightarrow\) Advance \(\rightarrow\) \keystroke{nCr}/\keystroke{nPr} \\
+ \-\hspace{1em} \texttt{nCr(n, r)} or \texttt{nPr(n, r)}
+\end{cas}
+
+\subsection*{Distributions}
+
+\begin{tikzpicture}
+ \begin{axis}[axis lines=left,
+ ticks=none,
+ xmin=0,
+ ymax=0.5,
+ enlargelimits=upper,
+ ylabel={\(\Pr(X=x)\)},
+ xlabel={\(x\)},
+ every axis x label/.style={at={(current axis.right of origin)},anchor=north west},
+ every axis y label/.style={at={(axis description cs:-0.02,0.5)}, anchor=south west, rotate=90},
+ ]
+ \fill[pattern=north east lines, pattern color=orange] (0,0) -- plot[domain=0:1.68, samples=50] function {abs(x)*exp(-x)} -- (1.68,0) -- cycle;
+ \fill[pattern=north west lines, pattern color=red] (1.68,0) -- plot[domain=1.68:5, samples=50] function {abs(x)*exp(-x)} -- (5,0) -- cycle;
+ \draw[dashed, blue, very thick] (axis cs:1.68,0) -- (axis cs:1.68,0.31) node [above, anchor=south west, black] {Median};
+ \draw[dashed, blue, very thick] (axis cs:2,0) -- (axis cs:2,0.27) node [above, anchor=west, black] {Mean};
+ \draw[dashed, blue, very thick] (axis cs:1,0) -- (axis cs:1,0.365) node [above, black] {Mode};
+ \node at (1,0.18) {\textbf{50\%}};
+ \node at (3.1,0.08) {\textbf{50\%}};
+ \addplot[thick, black, no markers, samples=200, domain=0:5] {abs(x)*exp(-x)};
+ \end{axis}
+\end{tikzpicture}
+
+\subsubsection*{Mean \(\mu\)}
+
+\begin{align*}
+ E(X) &= \frac{\Sigma \left[ x \cdot f(x) \right]}{\Sigma f} \tag{\(f =\) absolute frequency} \\
+ &= \sum_{i=1}^n \left[ x_i \cdot \Pr(X=x_i) \right] \tag{discrete}\\
+ &= \int_\textbf{X} (x \cdot f(x)) \> dx
+\end{align*}
+
+\subsubsection*{Mode}
+
+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}
+
+Value separating lower and upper half of distribution area
+
+\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 \(\sigma^2\)}
+
+\begin{align*}
+ \operatorname{Var}(x) &= \sum_{i=1}^n p_i (x_i-\mu)^2 \\
+ &= \sum (x-\mu)^2 \times \Pr(X=x) \\
+ &= \sum x^2 \times p(x) - \mu^2 \\
+ &= \operatorname{E}(X^2) - [\operatorname{E}(X)]^2 \\
+ &= E\left[(X-\mu)^2\right]
+\end{align*}
+
+\subsubsection*{Standard deviation \(\sigma\)}
+
+\begin{align*}
+ \sigma &= \operatorname{sd}(X) \\
+ &= \sqrt{\operatorname{Var}(X)}
+\end{align*}
+
+\subsection*{Binomial distributions}
+
+Conditions for a \textit{binomial distribution}:
+\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
+\end{enumerate}
+
+
+\subsubsection*{Properties of \(X \sim \operatorname{Bi}(n,p)\)}
+
+\begin{align*}
+ \mu(X) &= np \\
+ \operatorname{Var}(X) &= np(1-p) \\
+ \sigma(X) &= \sqrt{np(1-p)} \\
+ \Pr(X=x) &= {n \choose x} \cdot p^x \cdot (1-p)^{n-x}
+\end{align*}
+
+\begin{cas}
+ Interactive \(\rightarrow\) Distribution \(\rightarrow\) \verb;binomialPdf;
+ \begin{description}[nosep, style=multiline, labelindent=0.5cm, leftmargin=3cm, font=\normalfont]
+ \item [x:] no. of successes
+ \item [numtrial:] no. of trials
+ \item [pos:] probability of success
+ \end{description}
+\end{cas}
- \begin{align*}
- \Pr(A \cup B) &= \Pr(A) + \Pr(B) - \Pr(A \cap B) \\
- \Pr(A \cap B) &= \Pr(A|B) \times \Pr(B) \\
- \Pr(A|B) &= \frac{\Pr(A \cap B)}{\Pr(B)} \\
- \Pr(A) &= \Pr(A|B) \cdot \Pr(B) + \Pr(A|B^{\prime}) \cdot \Pr(B^{\prime})
- \end{align*}
+\subsection*{Continuous random variables}
- Mutually exclusive \(\implies \Pr(A \cup B) = 0\) \\
+A continuous random variable \(X\) has a pdf \(f\) such that:
- Independent events:
- \begin{flalign*}
- \quad \Pr(A \cap B) &= \Pr(A) \times \Pr(B)& \\
- \Pr(A|B) &= \Pr(A) \\
- \Pr(B|A) &= \Pr(B)
- \end{flalign*}
+\begin{enumerate}
+ \item \(f(x) \ge 0 \forall x \)
+ \item \(\int^\infty_{-\infty} f(x) \> dx = 1\)
+\end{enumerate}
- \subsection*{Combinatorics}
+\begin{align*}
+ E(X) &= \int_\textbf{X} (x \cdot f(x)) \> dx \\
+ \operatorname{Var}(X) &= E\left[(X-\mu)^2\right]
+\end{align*}
- \begin{itemize}
- \item Arrangements \({n \choose k} = \frac{n!}{(n-k)}\)
- \item \colorbox{important}{Combinations} \({n \choose k} = \frac{n!}{k!(n-k)!}\)
- \item Note \({n \choose k} = {n \choose k-1}\)
- \end{itemize}
-
- \subsection*{Distributions}
+\[ \Pr(X \le c) = \int^c_{-\infty} f(x) \> dx \]
- \subsubsection*{Mean \(\mu\)}
+\subsection*{Two random variables \(X, Y\)}
- \textbf{Mean} \(\mu\) or \textbf{expected value} \(E(X)\)
+If \(X\) and \(Y\) are independent:
+\begin{align*}
+ \operatorname{E}(aX+bY) & = a\operatorname{E}(X)+b\operatorname{E}(Y) \\
+ \operatorname{Var}(aX \pm bY \pm c) &= a^2 \operatorname{Var}(X) + b^2 \operatorname{Var}(Y)
+\end{align*}
- \begin{align*}
- E(X) &= \frac{\Sigma \left[ x \cdot f(x) \right]}{\Sigma f} \tag{\(f =\) absolute frequency} \\
- &= \sum_{i=1}^n \left[ x_i \cdot \Pr(X=x_i) \right] \tag{discrete}\\
- &= \int_\textbf{X} (x \cdot f(x)) \> dx
- \end{align*}
+\subsection*{Linear functions \(X \rightarrow aX+b\)}
- \subsubsection*{Mode}
+\begin{align*}
+ \Pr(Y \le y) &= \Pr(aX+b \le y) \\
+ &= \Pr\left(X \le \dfrac{y-b}{a}\right) \\
+ &= \int^{\frac{y-b}{a}}_{-\infty} f(x) \> dx
+\end{align*}
- Most popular value (has highest probability of all \(X\) values). Multiple modes can exist if \(>1 \> X\) value have equal-highest probability. Number must exist in distribution.
+\begin{align*}
+ \textbf{Mean:} && \operatorname{E}(aX+b) & = a\operatorname{E}(X)+b \\
+ \textbf{Variance:} && \operatorname{Var}(aX+b) &= a^2 \operatorname{Var}(X) \\
+\end{align*}
- \subsubsection*{Median}
+\subsection*{Expectation theorems}
- 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.
+For some non-linear function \(g\), the expected value \(E(g(X))\) is not equal to \(g(E(X))\).
- \[ m = X \> \text{such that} \> \int_{-\infty}^{m} f(x) dx = 0.5 \]
+\begin{align*}
+ E(X^2) &= \operatorname{Var}(X) - \left[E(X)\right]^2 \\
+ E(X^n) &= \Sigma x^n \cdot p(x) \tag{non-linear} \\
+ &\ne [E(X)]^n \\
+ E(aX \pm b) &= aE(X) \pm b \tag{linear} \\
+ E(b) &= b \tag{\(\forall b \in \mathbb{R}\)}\\
+ E(X+Y) &= E(X) + E(Y) \tag{two variables}
+\end{align*}
- \subsubsection*{Variance \(\sigma^2\)}
+\begin{figure*}[hb]
+ \centering
+ \include{../spec/normal-dist-graph}
+\end{figure*}
- \begin{align*}
- \operatorname{Var}(x) &= \sum_{i=1}^n p_i (x_i-\mu)^2 \\
- &= \sum (x-\mu)^2 \times \Pr(X=x) \\
- &= \sum x^2 \times p(x) - \mu^2 \\
- &= \operatorname{E}(X^2) - [\operatorname{E}(X)]^2
- &= E\left[(X-\mu)^2\right]
- \end{align*}
+\subsection*{Sample mean}
- \subsubsection*{Standard deviation \(\sigma\)}
+Approximation of the \textbf{population mean} determined experimentally.
- \begin{align*}
- \sigma &= \operatorname{sd}(X) \\
- &= \sqrt{\operatorname{Var}(X)}
- \end{align*}
+\[ \overline{x} = \dfrac{\Sigma x}{n} \]
- \subsection*{Binomial distributions}
+where
+\begin{description}[nosep, labelindent=0.5cm]
+ \item \(n\) is the size of the sample (number of sample points)
+ \item \(x\) is the value of a sample point
+\end{description}
- Conditions for a \textit{binomial distribution}:
- \begin{enumerate}
- \item Two possible outcomes: \textbf{success} or \textbf{failure}
- \item \(\Pr(\text{success})\) is constant across trials (also denoted \(p\))
- \item Finite number \(n\) of independent trials
+\begin{cas}
+ \begin{enumerate}[leftmargin=3mm]
+ \item Spreadsheet
+ \item In cell A1:\\ \path{mean(randNorm(sd, mean, sample size))}
+ \item Edit \(\rightarrow\) Fill \(\rightarrow\) Fill Range
+ \item Input range as A1:An where \(n\) is the number of samples
+ \item Graph \(\rightarrow\) Histogram
\end{enumerate}
+\end{cas}
+\subsubsection*{Sample size of \(n\)}
- \subsubsection*{Properties of \(X \sim \operatorname{Bi}(n,p)\)}
-
- \begin{align*}
- \mu(X) &= np \\
- \operatorname{Var}(X) &= np(1-p) \\
- \sigma(X) &= \sqrt{np(1-p)} \\
- \Pr(X=x) &= {n \choose x} \cdot p^x \cdot (1-p)^{n-x}
- \end{align*}
+\[ \overline{X} = \sum_{i=1}^n \frac{x_i}{n} = \dfrac{\sum x}{n} \]
- \begin{cas}
- Interactive \(\rightarrow\) Distribution \(\rightarrow\) \verb;binomialPdf; then input
- \begin{description}[nosep, style=multiline, labelindent=0.5cm, leftmargin=3cm, font=\normalfont]
- \item [x:] no. of successes
- \item [numtrial:] no. of trials
- \item [pos:] probability of success
- \end{description}
- \end{cas}
+Sample mean is distributed with mean \(\mu\) and sd \(\frac{\sigma}{\sqrt{n}}\) (approaches these values for increasing sample size \(n\)).
- \subsection*{Continuous random variables}
+For a new distribution with mean of \(n\) trials, \(\operatorname{E}(X^\prime) = \operatorname{E}(X), \quad \operatorname{sd}(X^\prime) = \dfrac{\operatorname{sd}(X)}{\sqrt{n}}\)
- A continuous random variable \(X\) has a pdf \(f\) such that:
+\begin{cas}
- \begin{enumerate}
- \item \(f(x) \ge 0 \forall x \)
- \item \(\int^\infty_{-\infty} f(x) \> dx = 1\)
- \end{enumerate}
+ \begin{itemize}
+ \item Spreadsheet \(\rightarrow\) Catalog \(\rightarrow\) \verb;randNorm(sd, mean, n); where \verb;n; is the number of samples. Show histogram with Histogram key in top left
+ \item To calculate parameters of a dataset: Calc \(\rightarrow\) One-variable
+ \end{itemize}
- \begin{align*}
- E(X) &= \int_\textbf{X} (x \cdot f(x)) \> dx \\
- \operatorname{Var}(X) &= E\left[(X-\mu)^2\right]
- \end{align*}
+\end{cas}
- \[ \Pr(X \le c) = \int^c_{-\infty} f(x) \> dx \]
+\subsection*{Population sampling}
+\subsubsection*{Population proportion}
- \subsection*{Two random variables \(X, Y\)}
+\[ p = \dfrac{n \text{ with attribute in population}}{\text{population size}} \]
- If \(X\) and \(Y\) are independent:
- \begin{align*}
- \operatorname{E}(aX+bY) & = a\operatorname{E}(X)+b\operatorname{E}(Y) \\
- \operatorname{Var}(aX \pm bY \pm c) &= a^2 \operatorname{Var}(X) + b^2 \operatorname{Var}(Y)
- \end{align*}
+Constant for a given population.
- \subsection*{Linear functions \(X \rightarrow aX+b\)}
+\subsection*{Sample proportion}
- \begin{align*}
- \Pr(Y \le y) &= \Pr(aX+b \le y) \\
- &= \Pr\left(X \le \dfrac{y-b}{a}\right) \\
- &= \int^{\frac{y-b}{a}}_{-\infty} f(x) \> dx
- \end{align*}
+\[ \hat{p} = \dfrac{n \text{ with attribute in sample}}{\text{sample size}} \]
- \begin{align*}
- \textbf{Mean:} && \operatorname{E}(aX+b) & = a\operatorname{E}(X)+b \\
- \textbf{Variance:} && \operatorname{Var}(aX+b) &= a^2 \operatorname{Var}(X) \\
- \end{align*}
+Varies with each sample.
- \subsection*{Expectation theorems}
+\subsection*{Normal distributions}
- For some non-linear function \(g\), the expected value \(E(g(X))\) is not equal to \(g(E(X))\).
- \begin{align*}
- E(X^2) &= \operatorname{Var}(X) - \left[E(X)\right]^2 \\
- E(X^n) &= \Sigma x^n \cdot p(x) \tag{non-linear} \\
- &\ne [E(X)]^n \\
- E(aX \pm b) &= aE(X) \pm b \tag{linear} \\
- E(b) &= b \tag{\(\forall b \in \mathbb{R}\)}\\
- E(X+Y) &= E(X) + E(Y) \tag{two variables}
- \end{align*}
+\[ Z = \frac{X - \mu}{\sigma} \]
- \subsection*{Sample mean}
+Normal distributions must have area (total prob.) of 1 \(\implies \int^\infty_{-\infty} f(x) \> dx = 1\) \\
+\(\text{mean} = \text{mode} = \text{median}\)
- Approximation of the \textbf{population mean} determined experimentally.
+\begin{warning}
+ Always express \(z\) as +ve. Express confidence \textit{interval} as ordered pair.
+\end{warning}
- \[ \overline{x} = \dfrac{\Sigma x}{n} \]
+\subsection*{Confidence intervals}
- where
- \begin{description}[nosep, labelindent=0.5cm]
- \item \(n\) is the size of the sample (number of sample points)
- \item \(x\) is the value of a sample point
- \end{description}
+\begin{itemize}
+ \item \textbf{Point estimate:} single-valued estimate of the population mean from the value of the sample mean \(\overline{x}\)
+ \item \textbf{Interval estimate:} confidence interval for population mean \(\mu\)
+ \item \(C\)\% confidence interval \(\implies\) \(C\)\% of samples will contain population mean \(\mu\)
+\end{itemize}
- \begin{cas}
- \begin{enumerate}[leftmargin=3mm]
- \item Spreadsheet
- \item In cell A1:\\ \path{mean(randNorm(sd, mean, sample size))}
- \item Edit \(\rightarrow\) Fill \(\rightarrow\) Fill Range
- \item Input range as A1:An where \(n\) is the number of samples
- \item Graph \(\rightarrow\) Histogram
- \end{enumerate}
- \end{cas}
-
- \subsubsection*{Sample size of \(n\)}
-
- \[ \overline{X} = \sum_{i=1}^n \frac{x_i}{n} = \dfrac{\sum x}{n} \]
-
- Sample mean is distributed with mean \(\mu\) and sd \(\frac{\sigma}{\sqrt{n}}\) (approaches these values for increasing sample size \(n\)).
-
- For a new distribution with mean of \(n\) trials, \(\operatorname{E}(X^\prime) = \operatorname{E}(X), \quad \operatorname{sd}(X^\prime) = \dfrac{\operatorname{sd}(X)}{\sqrt{n}}\)
-
- \begin{cas}
-
- \begin{itemize}
- \item Spreadsheet \(\rightarrow\) Catalog \(\rightarrow\) \verb;randNorm(sd, mean, n); where \verb;n; is the number of samples. Show histogram with Histogram key in top left
- \item To calculate parameters of a dataset: Calc \(\rightarrow\) One-variable
- \end{itemize}
-
- \end{cas}
-
- \subsection*{Normal distributions}
-
-
- \[ Z = \frac{X - \mu}{\sigma} \]
-
- Normal distributions must have area (total prob.) of 1 \(\implies \int^\infty_{-\infty} f(x) \> dx = 1\) \\
- \(\text{mean} = \text{mode} = \text{median}\)
-
- \begin{warning}
- Always express \(z\) as +ve. Express confidence \textit{interval} as ordered pair.
- \end{warning}
-
- \pgfmathdeclarefunction{gauss}{2}{%
- \pgfmathparse{1/(#2*sqrt(2*pi))*exp(-((x-#1)^2)/(2*#2^2))}%
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- \begin{figure*}[hb]
- \centering
- \begin{tikzpicture}
- \begin{axis}[every axis plot post/.style={
- mark=none,domain=-3:3,samples=50,smooth},
- axis x line=bottom,
- axis y line=left,
- enlargelimits=upper,
- x=\textwidth/10,
- ytick={0.55},
- yticklabels={\(\frac{1}{\sigma \sqrt{2\pi}}\)},
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- x tick label style = {font=\footnotesize},
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- \addplot {gauss(0,0.75)};
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- hide y axis,
- x tick label style = {font=\footnotesize},
- xlabel={\(Z\)},
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- \addplot {gauss(0,0.75)};
- \end{axis}
- \end{tikzpicture}
- \end{figure*}
-
- \subsection*{Confidence intervals}
+\begin{cas}
+ Menu \(\rightarrow\) Stats \(\rightarrow\) Calc \(\rightarrow\) Interval \\
+ Set \textit{Type = One-Sample Z Int} \\ \-\hspace{1em} and select \textit{Variable}
+\end{cas}
- \begin{itemize}
- \item \textbf{Point estimate:} single-valued estimate of the population mean from the value of the sample mean \(\overline{x}\)
- \item \textbf{Interval estimate:} confidence interval for population mean \(\mu\)
- \item \(C\)\% confidence interval \(\implies\) \(C\)\% of samples will contain population mean \(\mu\)
- \end{itemize}
+\subsubsection*{95\% confidence interval}
- \subsubsection*{95\% confidence interval}
+For 95\% c.i. of population mean \(\mu\):
- For 95\% c.i. of population mean \(\mu\):
+\[ x \in \left(\overline{x} \pm 1.96 \dfrac{\sigma}{\sqrt{n}} \right)\]
- \[ x \in \left(\overline{x} \pm 1.96 \dfrac{\sigma}{\sqrt{n}} \right)\]
+where:
+\begin{description}[nosep, labelindent=0.5cm]
+ \item \(\overline{x}\) is the sample mean
+ \item \(\sigma\) is the population sd
+ \item \(n\) is the sample size from which \(\overline{x}\) was calculated
+\end{description}
- where:
- \begin{description}[nosep, labelindent=0.5cm]
- \item \(\overline{x}\) is the sample mean
- \item \(\sigma\) is the population sd
- \item \(n\) is the sample size from which \(\overline{x}\) was calculated
- \end{description}
+\subsubsection*{Confidence interval of \(p\) from \(\hat{p}\)}
- \begin{cas}
- Menu \(\rightarrow\) Stats \(\rightarrow\) Calc \(\rightarrow\) Interval \\
- Set \textit{Type = One-Sample Z Int} \\ \-\hspace{1em} and select \textit{Variable}
- \end{cas}
+\[ x \in \left( \hat{p} \pm Z \sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}} \right) \]
- \subsection*{Margin of error}
+\subsection*{Margin of error}
- For 95\% confidence interval of \(\mu\):
- \begin{align*}
- M &= 1.96 \times \dfrac{\sigma}{\sqrt{n}} \\
- &= \dfrac{1}{2} \times \text{width of c.i.} \\
- \implies n &= \left( \dfrac{1.96 \sigma}{M} \right)^2
- \end{align*}
+For 95\% confidence interval of \(\mu\):
+\begin{align*}
+ M &= 1.96 \times \dfrac{\sigma}{\sqrt{n}} \\
+ &= \dfrac{1}{2} \times \text{width of c.i.} \\
+ \implies n &= \left( \dfrac{1.96 \sigma}{M} \right)^2
+\end{align*}
- Always round \(n\) up to a whole number of samples.
+Always round \(n\) up to a whole number of samples.
- \subsection*{General case}
+\subsection*{General case}
- For \(C\)\% c.i. of population mean \(\mu\):
+For \(C\)\% c.i. of population mean \(\mu\):
- \[ x \in \left( \overline{x} \pm k \dfrac{\sigma}{\sqrt{n}} \right) \]
- \hfill where \(k\) is such that \(\Pr(-k < Z < k) = \frac{C}{100}\)
+\[ x \in \left( \overline{x} \pm k \dfrac{\sigma}{\sqrt{n}} \right) \]
+\hfill where \(k\) is such that \(\Pr(-k < Z < k) = \frac{C}{100}\)
- \begin{cas}
- Menu \(\rightarrow\) Stats \(\rightarrow\) Calc \(\rightarrow\) Interval \\
- Set \textit{Type = One-\colorbox{important}{Prop} Z Int} \\
- Input x \(= \hat{p} * n\)
- \end{cas}
+\begin{cas}
+ Menu \(\rightarrow\) Stats \(\rightarrow\) Calc \(\rightarrow\) Interval \\
+ Set \textit{Type = One-\colorbox{important}{Prop} Z Int} \\
+ Input x \(= \hat{p} * n\)
+\end{cas}
- \subsection*{Confidence interval for multiple trials}
+\subsection*{Confidence interval for multiple trials}
- For a set of \(n\) confidence intervals (samples), there is \(0.95^n\) chance that all \(n\) intervals contain the population mean \(\mu\).
+For a set of \(n\) confidence intervals (samples), there is \(0.95^n\) chance that all \(n\) intervals contain the population mean \(\mu\).
- \end{document}
+\end{document}