Note hypotheses are always expressed in terms of population parameters
\end{warning}
- \subsection*{Null hypothesis \(H_0\)}
+ \subsection*{Null hypothesis \(\textbf{H}_0\)}
Sample drawn from population has same mean as control population, and any difference can be explained by sample variations.
- \subsection*{Alternative hypothesis \(H_1\)}
+ \subsection*{Alternative hypothesis \(\textbf{H}_1\)}
Amount of variation from control is significant, despite standard sample variations.
\subsection*{\(p\)-value}
+ Probability of observing a value of the sample statistic as significant as the one observed, assuming null hypothesis is true.
+ For one-tail tests:
\begin{align*}
- p &= \Pr(\overline{X} \lessgtr \mu(H_1)) \\
- &= 2 \cdot \Pr(\overline{X} <> \mu(H_1) | \mu = 8)
+ p\text{-value} &= \Pr\left( \> \overline{X} \lessgtr \mu(\textbf{H}_1) \> \given \> \mu = \mu(\textbf{H}_0)\> \right) \\
+ &= \Pr\left( Z \lessgtr \dfrac{\left( \mu(\textbf{H}_1) - \mu(\textbf{H}_0) \right) \cdot \sqrt{n} }{\operatorname{sd}(X)} \right) \\
+ &\text{then use \texttt{normCdf} with std. norm.}
\end{align*}
- Probability of observing a value of the sample statistic as significant as the one observed, assuming null hypothesis is true.
-
\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 \(H_0\) \\
+ \(> 0.05\) & insufficient evidence against \(\textbf{H}_0\) \\
+ \(< 0.05\) (5\%) & good evidence against \(\textbf{H}_0\) \\
+ \(< 0.01\) (1\%) & strong evidence against \(\textbf{H}_0\) \\
+ \(< 0.001\) (0.1\%) & very strong evidence against \(\textbf{H}_0\) \\
\hline
\end{tabularx}
- \subsection*{Statistical significance}
+ \subsection*{Significance level \(\alpha\)}
- Significance level is denoted by \(\alpha\).
+ The condition for rejecting the null hypothesis.
\-\hspace{1em} If \(p<\alpha\), null hypothesis is \textbf{rejected} \\
\-\hspace{1em} If \(p>\alpha\), null hypothesis is \textbf{accepted}
Menu \(\rightarrow\) Statistics \(\rightarrow\) Calc \(\rightarrow\) Test. \\
Select \textit{One-Sample Z-Test} and \textit{Variable}, then input:
\begin{description}[nosep, style=multiline, labelindent=0.5cm, leftmargin=2cm, font=\normalfont]
- \item[\(\mu\) cond:] same operator as \(H_1\)
+ \item[\(\mu\) cond:] same operator as \(\textbf{H}_1\)
\item[\(\mu_0\):] expected sample mean (null hypothesis)
\item[\(\sigma\):] standard deviation (null hypothesis)
\item[\(\overline{x}\):] sample mean
\end{cas}
\subsection*{One-tail and two-tail tests}
+
+ \[ p\text{-value (two-tail)} = 2 \times p\text{-value (one-tail)} \]
\subsubsection*{One tail}
\begin{itemize}
\item \(\mu\) has changed in one direction
- \item State ``\(H_1: \mu \lessgtr \) known population mean''
+ \item State ``\(\textbf{H}_1: \mu \lessgtr \) 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''
+ \item State ``\(\textbf{H}_1: \mu \ne\) 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)
+ &= \left( |Z| \ge \left|\dfrac{\overline{x}_0 - \mu}{\sigma \div \sqrt{n}} \right| \right) \\
\end{align*}
+ where
+ \begin{description}[nosep, labelindent=0.5cm]
+ \item [\(\mu\)] is the population mean under \(\textbf{H}_0\)
+ \item [\(\overline{x}_0\)] is the observed sample mean
+ \item [\(\sigma\)] is the population s.d.
+ \item [\(n\)] is the sample size
+ \end{description}
+
\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*{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}
+ \item [Type I error] \(\textbf{H}_0\) is rejected when it is \textbf{true}
+ \item [Type II error] \(\textbf{H}_0\) is \textbf{not} rejected when it is \textbf{false}
\end{description}
+ \begin{tabularx}{\columnwidth}{|X|l|l|}
+ \rowcolor{cas}\hline
+ \cellcolor{white}&\multicolumn{2}{c|}{\textbf{Actual result}} \\
+ \hline
+ \cellcolor{cas}\(\boldsymbol{z}\)\textbf{-test} & \cellcolor{light-gray}\(\textbf{H}_0\) true & \cellcolor{light-gray}\(\textbf{H}_0\) false \\
+ \hline
+ \cellcolor{light-gray}Reject \(\textbf{H}_0\) & Type I error & Correct \\
+ \hline
+ \cellcolor{light-gray}Do not reject \(\textbf{H}_0\) & Correct& Type II error \\
+ \hline
+ \end{tabularx}
+
% \subsection*{Using c.i. to find \(p\)}
% need more here