\setcounter{ExampleCounter}{1}
\paragraph{Point estimate:} \text{}
\vspace{0.6in}
\paragraph{Confidence interval:} \text{}
\vspace{0.9in}
In other words, instead of saying
\begin{center}
``I think the average width of our iPhone cases is 67 mm''
\end{center}
you could say
\begin{center}
``I am 95\% confident that the average width of our iPhone\\
cases is between 66.8 and 67.2 mm.''
\end{center}
Notice how the second statement is much more precise (if less natural, perhaps). Also,
\begin{itemize}
\item \text{}
\vspace{1in}
\item \text{}
\vspace{0.75in}
\item \text{}
\vspace{1.75in}
\end{itemize}
\subsection{Constructing a Confidence Interval}
\paragraph{Assumption:} We know the population standard deviation $\sigma$. Also,
\vfill
\paragraph{Recall:} Central Limit Theorem
\vfill
\begin{example}{Battery Life}
A battery manufacturer claims that the lifetime of a certain type of battery has a population mean of $\mu=40$ hours and a standard deviation of $\sigma=5$ hours. Let $\overline{x}$ represent the mean lifetime of the batteries in a simple random sample of size 100.
\begin{enumerate}[(a)]
\item If the claim is true, what is $P(\overline{x} \leq 39.8)$?
\begin{center}
\texttt{normalcdf(-1000000,39.8,40,0.5)} $= 0.3446$
\end{center}
\item Based on the answer to part (a), if the claim is true, is a sample mean lifetime of 39.8 hours unusually short?
\begin{center}
Not really.
\end{center}
\item If the sample mean lifetime of the 100 batteries were 39.8 hours, would you find the manufacturer's claim to be plausible?
\begin{center}
Yeah, I think so.
\end{center}
\item If the claim is true, what is $P(\overline{x} \leq 38.5)$?
\begin{center}
\texttt{normalcdf(-1000000,38.5,40,0.5)} $= 0.0013$
\end{center}
\item Based on the answer to part (d), if the claim is true, is a sample mean lifetime of 38.5 hours unusually short?
\begin{center}
Yes.
\end{center}
\item If the sample mean lifetime of the 100 batteries were 38.5 hours, would you find the manufacturer's claim to be plausible?
\begin{center}
No.
\end{center}
\end{enumerate}
\end{example}
\pagebreak
Finding a confidence interval essentially means finding all the values for the population mean that would not make our sample mean \textbf{unusual} (where here ``unusual'' depends on our confidence level).
\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=5cm, width=12cm,
xtick={-2,0,1,2},
ytick=\empty,
xticklabels={$z=-2$,$\mu$,{\large $\overline{x}$},$z=2$},
enlargelimits=false, clip=false, %axis on top,
%grid = major
]
\addplot [fill=cyan!40, draw=none, domain=-2:2] {gauss(0,1)} \closedcycle;
%\addplot [fill=cyan!40, draw=none, domain=2:3] {gauss(0,1)} \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)};
\draw [yshift=-0.75cm, latex-latex](axis cs:-2,0) -- node [fill=white] {95\%} (axis cs:2,0);
\end{axis}
\end{tikzpicture}
\end{center}
For instance, in the sampling distribution above, we have a good idea of how likely it is that the sample mean will fall into a given range. Based on the Empirical Rule, we know that there is a 68\% chance that the sample mean will be within one standard deviation of the population mean, a 95\% chance that it will be within two standard deviations of the population mean, and a 99.7\% chance that it will be within three standard deviations. For any other probabilities, we can consult the z table or our calculators.\\
Okay, let's try an example.
\begin{example}{Confidence Interval}
If you get a sample mean of 23, and you know that the sampling distribution has standard deviation \[\dfrac{\sigma}{\sqrt{n}} = 1.5,\] find the 95\% confidence interval for the population mean $\mu$.\\
The population mean is unknown, but we know that whatever it is, our sample mean is 95\% likely to be within two standard deviations of it (two standard deviations equals 3 in this case). The sample mean could be 3 lower or 3 higher, so our confidence interval goes from $23-3$ to $23+3$:
\[23 \pm 3 = (20,26).\]
Either notation is acceptable for a confidence interval. Note that the point estimate is the sample mean, 23, and the margin of error is the standard deviation ($\sigma/\sqrt{n}$) times the number of standard deviations that correspond to a 95\% confidence level.
\end{example}
Finding a confidence interval, then, consists of three pieces:
\begin{enumerate}
\item \text{}
\vspace{0.5in}
\item \text{}
\vspace{0.5in}
\item \text{}
\vspace{0.75in}
\end{enumerate}
Then the confidence interval is
\vspace{0.5in}
\subsection{Finding $z_{\alpha/2}$}
Okay, with a 95\% confidence interval, the z value was pretty easy, because we know that 95\% of the data is within two standard deviations based on the Empirical Rule. But what if we wanted a 90\% confidence interval or a 99\% confidence interval? The Empirical Rule has nothing to say about those, so we need to use the z table or our calculator.
\begin{example}{Finding z}
How many standard deviations do you need to go out to cover 90\% of the data?\\
\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=5cm, width=12cm,
xtick={-2,0,2},
ytick=\empty,
xticklabels={$z=?$,$\mu$,$z=?$},
enlargelimits=false, clip=false, %axis on top,
%grid = major
]
\addplot [fill=cyan!40, draw=none, domain=-3:-2] {gauss(0,1)} \closedcycle;
\addplot [fill=cyan!40, draw=none, domain=2:3] {gauss(0,1)} \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)};
\draw [yshift=-0.75cm, latex-latex](axis cs:-2,0) -- node [fill=white] {90\%} (axis cs:2,0);
\end{axis}
\end{tikzpicture}
\end{center}
\vspace*{2in}
\end{example}
\begin{example}{Finding z}
Find $z_{\alpha/2}$ for a 98\% confidence interval.\\
\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=5cm, width=12cm,
xtick={-2,0,2},
ytick=\empty,
xticklabels={$z=?$,$\mu$,$z=?$},
enlargelimits=false, clip=false, %axis on top,
%grid = major
]
\addplot [fill=cyan!40, draw=none, domain=-3:-2] {gauss(0,1)} \closedcycle;
\addplot [fill=cyan!40, draw=none, domain=2:3] {gauss(0,1)} \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)};
\draw [yshift=-0.75cm, latex-latex](axis cs:-2,0) -- node [fill=white] {98\%} (axis cs:2,0);
\end{axis}
\end{tikzpicture}
\end{center}
\vspace{2.5in}
\end{example}
\vfill
\pagebreak
\subsection{Full Examples}
\begin{example}{Cereal Box Weight}
A machine that fills cereal boxes is supposed to put 20 ounces of cereal in each box. A simple random sample of 6 boxes is found to contain a sample mean of 20.25 ounces of cereal. It is known from past experience that fill weights are normally distributed with a standard deviation of 0.2 ounces. Construct a 92\% confidence interval for the mean fill weight.\\
\vspace{3in}
\end{example}
\begin{example}{SAT Scores}
A college admissions officer takes a simple random sample of 100 entering freshmen and computes their mean mathematics SAT score to be 458. Assume the population standard deviation is $\sigma=116$. Construct a 99\% confidence interval for the population mean score.\\
\vspace{2.5in}
\end{example}
\begin{example}{Baby Weight}
According to the National Health Statistics Reports, a sample of 360 one-year-old baby boys in the US had a mean weight of 25.5 pounds. Assume the population standard deviation is $\sigma=5.3$ pounds. Construct a 94\% confidence interval.\\
\vspace{4in}
\end{example}
\vfill
\pagebreak
\subsection{Changing the Confidence Level}
What does changing the confidence level do?
\begin{example}{Component Lifetimes}
In a simple random sample of 100 electronic components produced by a certain method, the mean lifetime was 125 hours. Assume that component lifetimes are normally distributed with population standard deviation $\sigma=20$ hours. Construct 90\%, 95\%, and 98\% confidence intervals.\\
\vspace{5in}
\end{example}
\paragraph{Note:} As the confidence level increases, the confidence intervals get wider (still centered at the same place).
\vfill
\pagebreak
\subsection{Calculating the Sample Size}
Suppose we want a given margin of error: what sample size do we need in order to make that happen?
\paragraph{Note:} A larger sample size leads to a smaller margin of error.
\begin{example}{College Student Age}
The population standard deviation for the age of students at a college is 15 years. If we want to be 95\% confident that the sample mean age is within two years of the true population mean age of these students, how many randomly selected students must be surveyed?\\
\vspace{5in}
\end{example}
\vfill
\pagebreak
\subsection{Using Your Calculator}
There is also a built-in function in your calculator that can find confidence intervals for problems like this one.
\begin{enumerate}
\item Press \includegraphics[height=0.3in]{CalcStatButton} and scroll over to the \texttt{TESTS} menu.
\begin{center}
\includegraphics[width=3in]{CalcStatTests}
\end{center}
\item Select \texttt{7:ZInterval} and you'll have two options: using \texttt{Data} or \texttt{Stats}.
\begin{center}
\begin{tabular}{c c}
\includegraphics[width=2in]{CalcZIntervalData}
&
\includegraphics[width=2in]{CalcZIntervalStats}
\end{tabular}
\end{center}
\item In either case, enter the population standard deviation as $\sigma$ and the confidence level (as a decimal).
\end{enumerate}
\vfill
\pagebreak
\begin{example}{Blackberry Prices}
A random sample of 11 BlackBerry Bold 9000 smartphones being sold over the Internet in 2010 had the following prices, in dollars:
\begin{center}
\begin{tabular}{c c c c c c}
230 & 484 & 379 & 300 & 239 & 350\\
300 & 395 & 230 & 410 & 460
\end{tabular}
\end{center}
Assume the population standard deviation is $\sigma=71$. Calculate a 95\% confidence interval for the population mean price.\\
After entering the data, press \includegraphics[height=0.3in]{CalcStatButton}, scroll over to \texttt{TESTS} menu, and select \texttt{7:ZInterval}. Enter $\sigma=71$, leave \texttt{C-Level} as 0.95, and press \texttt{Calculate}. You'll see the following:
\begin{center}
\includegraphics[width=3in]{CalcZIntervalEx}
\end{center}
The confidence interval, then, is \[(301.41, 385.32).\]
\end{example}
\vfill
\pagebreak