# 8.2 Null and Alternative Hypotheses

## LEARNING OBJECTIVES

• Describe hypothesis testing in general and in practice.

A hypothesis test begins by considering two hypotheses.  They are called the null hypothesis and the alternative hypothesis.  These hypotheses contain opposing viewpoints and only one of these hypotheses is true.  The hypothesis test determines which hypothesis is most likely true.

• The null hypothesis is denoted $H_0$.  It is a statement about the population that either is believed to be true or is used to put forth an argument unless it can be shown to be incorrect beyond a reasonable doubt.
• The null hypothesis is a claim that a population parameter equals some value.  For example, $H_0: \mu=5$.
• The alternative hypothesis is denoted $H_a$.  It is a claim about the population that is contradictory to the null hypothesis and is what we conclude is true when we reject $H_0$.
• The alternative hypothesis is a claim that a population parameter is greater than, less than, or not equal to some value.  For example, $H_a: \mu>5$, $H_a: \mu<5$, or $H_a: \mu \neq 5$.  The form of the alternative hypothesis depends on the wording of the hypothesis test.
• An alternative notation for $H_a$ is $H_1$.

Because the null and alternative hypotheses are contradictory, we must examine evidence to decide if we have enough evidence to reject the null hypothesis or not reject the null hypothesis.  The evidence is in the form of sample data.  After we have determined which hypothesis the sample data supports, we make a decision.  There are two options for a decision. They are “reject $H_0$” if the sample information favors the alternative hypothesis or “do not reject $H_0$” if the sample information is insufficient to reject the null hypothesis.

Watch this video: Simple hypothesis testing | Probability and Statistics | Khan Academy by Khan Academy [6:24]

## EXAMPLE

A candidate in a local election claims that 30% of registered voters voted in a recent election.  Information provided by the returning office suggests that the percentage is higher than the 30% claimed.

Solution:

The parameter under study is the proportion of registered voters, so we use $p$ in the statements of the hypotheses.  The hypotheses are

$\begin{eqnarray*} \\ H_0: & & p=30\% \\ \\ H_a: & & p \gt 30\% \\ \\ \end{eqnarray*}$

### NOTES

1. The null hypothesis $H_0$ is the claim that the proportion of registered voters that voted equals 30%.
2. The alternative hypothesis $H_a$ is the claim that the proportion of registered voters that voted is greater than (i.e. higher) than 30%.

## TRY IT

A medical researcher believes that a new medicine reduces cholesterol by 25%.  A medical trial suggests that the percent reduction is different than claimed.  State the null and alternative hypotheses.

Click to see Solution

$\begin{eqnarray*} H_0: & & p=25\% \\ \\ H_a: & & p \neq 25\% \end{eqnarray*}$

## EXAMPLE

We want to test whether the mean GPA of students in American colleges is different from 2.0 (out of 4.0).  State the null and alternative hypotheses.

Solution:

$\begin{eqnarray*} H_0: & & \mu=2 \mbox{ points} \\ \\ H_a: & & \mu \neq 2 \mbox{ points} \end{eqnarray*}$

## EXAMPLE

We want to test whether or not the mean height of eighth graders is 66 inches.  State the null and alternative hypotheses.

Solution:

$\begin{eqnarray*} H_0: & & \mu=66 \mbox{ inches} \\ \\ H_a: & & \mu \neq 66 \mbox{ inches} \end{eqnarray*}$

## EXAMPLE

We want to test if college students take less than five years to graduate from college, on the average.  The null and alternative hypotheses are:

Solution:

$\begin{eqnarray*} H_0: & & \mu=5 \mbox{ years} \\ \\ H_a: & & \mu \lt 5 \mbox{ years} \end{eqnarray*}$

## TRY IT

We want to test if it takes fewer than 45 minutes to teach a lesson plan.  State the null and alternative hypotheses.

Click to see Solution

$\begin{eqnarray*} H_0: & & \mu=45 \mbox{ minutes} \\ \\ H_a: & & \mu \lt 45 \mbox{ minutes} \end{eqnarray*}$

## EXAMPLE

In an issue of 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 is more than 6.6%.  State the null and alternative hypotheses.

Solution:

$\begin{eqnarray*} H_0: & & p=6.6\% \\ \\ H_a: & & p \gt 6.6\% \end{eqnarray*}$

## TRY IT

On a state driver’s test, about 40% pass the test on the first try.  We want to test if more than 40% pass on the first try.   State the null and alternative hypotheses.

Click to see Solution

$\begin{eqnarray*} H_0: & & p=40\% \\ \\ H_a: & & p \gt 40\% \end{eqnarray*}$

## Concept Review

In a hypothesis test, sample data is evaluated in order to arrive at a decision about some type of claim.  If certain conditions about the sample are satisfied, then the claim can be evaluated for a population.  In a hypothesis test, we evaluate the null hypothesis, typically denoted with $H_0$. The null hypothesis is not rejected unless the hypothesis test shows otherwise.  The null hypothesis always contain an equal sign ($=$).  Always write the alternative hypothesis, typically denoted with $H_a$ or $H_1$, using less than, greater than, or not equals symbols ($\lt$, $\gt$, $\neq$).  If we reject the null hypothesis, then we can assume there is enough evidence to support the alternative hypothesis.  But we can never state that a claim is proven true or false.  All we can conclude from the hypothesis test is which of the hypothesis is most likely true.  Because the underlying facts about hypothesis testing is based on probability laws, we can talk only in terms of non-absolute certainties.