\setcounter{ExampleCounter}{1}
What we did in the last section is rarely, if ever, done.
\paragraph{Note:} \text{}
\vspace{0.5in}
\begin{itemize}
\item \text{}
\vspace{0.75in}
\item \text{}
\vspace{0.5in}
\end{itemize}
Instead of a CI that looks like
\vspace{0.5in}
we'll have one know that looks like
\vspace{0.5in}
Notice:
\begin{itemize}
\item \text{}
\vspace{0.75in}
\item \text{}
\vspace{0.5in}
\end{itemize}
\subsection{The $t$ Distribution}
The $t$ distribution is necessary when $\sigma$ is unknown and the sample size is small, but nowadays, it's used pretty much all the time (since $\sigma$ is always unknown).
When the sample size is large, the $t$ distribution is nearly indistinguishable from the normal distribution.
\begin{center}
\begin{tikzpicture}
\begin{axis}[
no markers, domain=-4:4, samples=100,
axis lines*=none, xlabel=$x$,
hide y axis,
every axis y label/.style={at=(current axis.above origin),anchor=south},
every axis x label/.style={at=(current axis.right of origin),anchor=west},
height=4cm, width=12cm,
xtick={0}, ytick=\empty,
xticklabels={0},
enlargelimits=false, clip=false, %axis on top,
%grid = major
]
%\addplot [fill=cyan!40, draw=none, domain=-4:1] {gauss(0,1.3)} \closedcycle;
%\addplot [fill=yellow!20, draw=none, domain=-2:-1] {gauss(0,1)} \closedcycle;
%\addplot [fill=yellow!20, draw=none, domain=1:2] {gauss(0,1)} \closedcycle;
%\addplot [fill=green!20, draw=none, domain=-3:-2] {gauss(0,1)} \closedcycle;
%\addplot [fill=green!20, draw=none, domain=2:3] {gauss(0,1)} \closedcycle;
\addplot [very thick,cyan!40!black] {gauss(0,1.3)};
%\draw [yshift=2.5cm, latex-latex](axis cs:-1,0) -- node [fill=white] {68\%} (axis cs:1,0);
\end{axis}
\end{tikzpicture}
\end{center}
The $t$ distribution:
\begin{itemize}
\item \text{}
\vspace{0.75in}
\item \text{}
\vspace{0.5in}
\item \text{}
\vspace{0.5in}
\item \text{}
\vspace{0.5in}
\item \text{}
\vspace{0.5in}
\end{itemize}
To find $t_{\alpha/2}$, we can use either a table or a calculator (TI-84+ and up). The table looks like this (the full table is in the appendix):
\begin{center}
\includegraphics[width=0.8\textwidth]{tDist}
\end{center}
Be careful: different books record the $t$ table in different ways; all the results will be equal, but you may need to read the table differently.
\begin{example}{Finding t}
Find $t_{\alpha/2}$ to construct a 90\% confidence interval based on a sample of 7 items.\\
\vspace{3in}
\end{example}
\vfill
\begin{example}{Finding t with a Calculator}
Find the same $t$ value using a calculator.\\
The TI-84+ and later models have a \texttt{invT} function located directly beneath \texttt{invNorm}.
Press \includegraphics[height=0.3in]{Calc2ndVars} to access the \texttt{DISTR} menu, then select \text{4: invT}. This function requires two inputs: \texttt{area} and \texttt{df}. Enter 0.95 and 6, respectively:
\begin{center}
\includegraphics[width=3in]{CalcinvT}
\end{center}
The answer is the same.
\end{example}
\pagebreak
\subsection{Confidence Intervals with $t$}
Now that we can find $t_{\alpha/2}$, we can find $t$ confidence intervals:
\vspace{0.5in}
All we have to do is
\begin{enumerate}
\item \text{}
\vspace{0.5in}
\item \text{}
\vspace{0.5in}
\item \text{}
\vspace{0.5in}
\item \text{}
\vspace{0.5in}
\end{enumerate}
\begin{example}{Potato Chip Bags}
A potato chip company wants to evaluate the accuracy of its potato chip bag-filling machine. Bags are labeled as containing 8 ounces of potato chips. A simple random sample of 12 bags had mean weight 8.12 ounces with a sample standard deviation of 0.1 ounce. Construct a 99\% confidence interval for the population mean weight of bags of potato chips.\\
\vspace{3in}
\end{example}
\vfill
\pagebreak
\subsection{Using Your Calculator}
The process for finding a $t$ interval is identical to that for finding a $z$ interval, except that you need to select \texttt{8: TInterval} in the \texttt{STAT} \texttt{TESTS} menu.
\begin{example}{Movie Lengths}
A random sample of 45 Hollywood movies made since the year 2000 had a mean length of 111.7 minutes, with a standard deviation of 13.8 minutes. Construct a 92\% confidence interval for the population mean.\\
On your calculator, press \texttt{STAT} and scroll over to \texttt{TESTS}. Scroll down or press the 8 key to select \texttt{8: TInterval}.
\begin{center}
\includegraphics[width=3in]{CalcTIntervalEx}
\end{center}
Enter the given information and click \texttt{Calculate}.
\begin{center}
\includegraphics[width=3in]{CalcTIntervalEx2}
\end{center}
Therefore, the 92\% confidence interval is
\[(108.01, 115.39).\]
\end{example}
\vfill
\pagebreak
\begin{example}{Cereal Box Weights}
Boxes of cereal are labeled as containing 14 ounces. Following are the weights, in ounces, of a sample of 12 boxes. It is reasonable to assume that the population is approximately normal.
\begin{center}
\begin{tabular}{c c c c c c}
14.02 & 13.97 & 14.11 & 14.12 & 14.10 & 14.02\\
14.15 & 13.97 & 14.05 & 14.04 & 14.11 & 14.12
\end{tabular}
\end{center}
Construct a 95\% confidence interval. Based on this confidence interval, are the boxes labeled correctly?\\
\vspace{2.5in}
\end{example}
\begin{example}{Online Course Satisfaction}
A sample of 263 students who were taking online courses were asked to describe their overall impression of online learning on a scale of 1--7, with 7 representing the most favorable impression. The average score was 5.53, and the standard deviation was 0.92. Construct a 99\% confidence interval for the population mean score.\\
\vspace{2.5in}
\end{example}
\vfill
\pagebreak