\setcounter{ExampleCounter}{1}
Remember this example from the section on confidence intervals?
\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.\\
Confidence interval:
\[(20.12,20.38)\]
\end{example}
At the end of the problem, we have a confidence interval, but we also have a conclusion about the claim that was made: we can conclude that the average weight of the boxes is \emph{more than} 20 ounces, since the entire interval is above 20. If we'd gotten something like
\[(19.88,20.56)\] we would not have been able to conclude that the average weight is more than 20 ounces or less than 20 ounces.
\paragraph{Note:} we also wouldn't say that the average weight \emph{equals} 20 ounces; we just haven't found evidence that it is greater or smaller than 20 ounces.\\
If we did a hypothesis test with this example, we'd find (with 92\% confidence) the same conclusion. Again, a hypothesis test is a different way to go about it, but the hypothesis test and the confidence interval will draw the same conclusions.
\vfill
\pagebreak
\subsection{Kinds of Hypothesis Tests}
There are many hypothesis tests that can be done, but we'll stick to ones that are similar to what we did with confidence intervals:
\begin{itemize}
\item Test a claim about what the population mean is, given the population standard deviation.
\item Test a claim about what the population mean is, without knowing the population standard deviation.
\item Test a claim about what the population proportion is.
\item Test a claim about the difference between the means of two populations.
\item Test a claim about the difference between the proportions in two populations.
\end{itemize}
In general, a hypothesis test tests a claim about a population parameter based on a sample.
\subsection{Hypotheses}
At the heart of a hypothesis test are the two contradictory hypotheses.
\begin{itemize}
\item \textbf{Null hypothesis:} $H_0$ is the null hypothesis. This is what we assume unless we can prove otherwise.
\item \textbf{Alternate hypothesis:} $H_a$ or $H_1$ is the alternate hypothesis. This is usually what we're trying to prove; if we reject $H_0$, we conclude that $H_1$ is true.
\end{itemize}
\vfill
\pagebreak
\paragraph{Note: Terminology} At the end of a hypothesis test, we'll either say
\begin{center}
``Reject $H_0$''
\end{center}
or
\begin{center}
``Fail to reject $H_0$.''
\end{center}
We'll never say ``Accept $H_0$.'' (look back at the cereal weights example)
\paragraph{Options:} Compare the parameter (mean or proportion) to some value
\begin{center}
\begin{tabular}{r l | r l}
& \textbf{$H_0$} & & \textbf{$H_1$}\\
\hline
& & & \\
$=$ & Equal & $\neq$ & Not equal (greater than or less than)\\
$\geq$ & Greater than or equal to & $<$ & Less than\\
$\leq$ & Less than or equal to & $>$ & Greater than
\end{tabular}
\end{center}
\paragraph{Note:} the equals sign is always on the null hypothesis. Some people always use $=$ as the null hypothesis in every case.
\begin{example}{Restaurant Bills}
Last year, the mean amount spent by customers at a restaurant was \$35. The restaurant owner believes that the mean may be higher this year.
\begin{align*}
H_0:\ &\mu \leq 35\\
H_1:\ &\mu > 35
\end{align*}
\end{example}
\vfill
\pagebreak
\begin{example}{Newborn Weight}
In a recent year, the mean weight of newborn boys in a certain country was 6.6 pounds. A doctor wants to know whether the mean weight of newborn girls differs from this.
\begin{align*}
H_0:\ &\mu = 6.6\\
H_1:\ &\mu \neq 6.6
\end{align*}
\end{example}
\vfill
\begin{example}{Gas Mileage}
A certain model of car can be ordered with either a large or small engine. The mean number of miles per gallon for cars with a small engine is 25.5. An automotive engineer thinks that the mean for cars with the larger engine will be less than this.
\begin{align*}
H_0:\ &\mu \geq 25.5\\
H_1:\ &\mu < 25.5
\end{align*}
\end{example}
\vfill
\begin{example}{Registered Voters}
A pollster thinks that less than 30\% of registered voters in the county voted.
\begin{align*}
H_0:\ &p \geq 0.3\\
H_1:\ &p < 0.3
\end{align*}
\end{example}
\pagebreak
\begin{example}{Mean GPA}
We want to test whether the mean GPA of American college students differs from 2.0.
\begin{align*}
H_0:\ &\mu = 2.0\\
H_1:\ &\mu \neq 2.0
\end{align*}
\end{example}
\vfill
\begin{example}{Placement Tests}
In an issue of \emph{U.S. News and World Report}, an article on school standards stated that about half of all students in France, Germany, and Israel take advanced placement exams and a third pass. The same article stated that 6.6\% of U.S. students take advanced placement exams and 4.4\% pass. Test if the percentage of U.S. students who take advanced placement exams differs from 6.6\%.
\begin{align*}
H_0:\ &p = 0.066\\
H_1:\ &p \neq 0.066
\end{align*}
\end{example}
\vfill
\begin{example}{Driver's Test}
On a state driver's test, about 40\% pass on the first try. We want to test if more than 40\% pass on the first try in a different state.
\begin{align*}
H_0:\ &p \leq 0.4\\
H_1:\ &p > 0.4
\end{align*}
\end{example}
\pagebreak