Suppose you wanted to show that a coin was fair, what would be involved with setting up the experiment(s) to validate or deny the claim?

### The Null Hypothesis

One of the main uses for statistics is the testing of hypothesis. Commonly, a statement is made, such as “I think that the probability of a student in my class preferring red candy is 70%,” and experiments are conducted to determine the validity of the claim. However, to strengthen the results, specific steps should be followed as the experiment is set up, carried out, and reviewed or reported. The first step is the establishment of the **null hypothesis** notated as \begin{align*}H_0\end{align*} and **alternative hypothesis**, notated as \begin{align*}H_1\end{align*}.

- The
**null hypothesis**, \begin{align*}H_0\end{align*}, is the opposite of what you are hoping to claim. In the case of the claim above, the null hypothesis would be “The probability of a student in class preferring red candy is less than 70%” The null hypothesis may also be referred to as the “no-change” value, as it is the default conclusion. - The
**alternative****hypothesis**, \begin{align*}H_1\end{align*}, is the clear and concise statement of the initial claim. In the case of the candy above, the hypothesis could be: “An average of 70% or greater of the students in school preferred candy to green candy,” or simply \begin{align*}H_1 : p \ge 70\%\end{align*}, where \begin{align*}p\end{align*} stands for “population percent”. - It is extremely important that the alternative hypothesis and null hypothesis be mutually exclusive, meaning that if one is true, the other must be false.

In this lesson, you will practice identifying the null hypothesis for a number of claims.

**Assuming Null Hypotheses **

The claim is made that a certain medication relives headaches for more than 75% of patients who take it. What null hypothesis could be assumed during the investigation of this claim?

The null hypothesis for the claim “More than 75% of patients who take medication Z experience headache relief,” would be the mutually exclusive statement “75% or fewer of the patients who take medication Z experience headache relief”. This could be written as \begin{align*}H_0:p \le 75\%\end{align*}.

**Creating Alternative and Null Hypotheses **

A researcher wants to demonstrate that an average of more than 8 out of 10 dogs prefer the taste of a new dog food. What alternative hypothesis and null hypothesis might he work with? Use notation.

The researcher’s hypothesis could be stated as: “On average, greater than 80% of dogs prefer the taste of dog food X.” In notation that would be:

\begin{align*}H_1:p>80\%\end{align*}

The null hypothesis would be: “On average, 80% or fewer of dogs prefer the taste of dog food X.” In notation: \begin{align*}H_0:p \le 80\%\end{align*}.

#### Finding an Alternative Hypothesis That Correlates to a Null Hypothesis

What alternative hypothesis would correlate to the null hypothesis \begin{align*}H_0:p<60\%\end{align*}?

If the null hypothesis is \begin{align*}H_0: \mu < 60\%\end{align*}, then the alternative hypothesis is \begin{align*}H_1:p \ge 60\%\end{align*}.

#### Earlier Problem Revisited

*Suppose you wanted to show that a coin was fair, what would be the first steps involved with setting up the experiment(s) to validate or deny the claim?*

The first step would be to identify the hypothesis and null hypothesis:

\begin{align*}H_1:p &= 50\%\\ H_0:p & \neq 50\%\end{align*}

### Examples

#### Example 1

What is the null hypothesis to the claim that more people like cereal X than cereal Y?

The null hypothesis is the opposite of what you are hoping to claim, so \begin{align*}H_0\end{align*}: More people like cereal Y than cereal X.

#### Example 2

What alternative hypothesis would correlate to the null hypothesis \begin{align*}H_0:0 \le \sigma \le 5\end{align*}?

\begin{align*}H_1:0 \ge \sigma\end{align*} or \begin{align*}\sigma \ge 5\end{align*}

#### Example 3

What alternative hypothesis and null hypothesis could be stated to define the claim that more people own blue cars than red cars? Use notation.

\begin{align*}H_0:R \ge B\end{align*} and \begin{align*}H_1:R < B\end{align*}

### Review

For questions 1-8, state the null hypothesis for the given alternative hypothesis.

1. The number of sales of mp3 players would not go down if the price were raised by $10.

2. The average dog owner owns 2 or more cats.

3. The average cat owner does not own a dog.

4. The average subcompact car gets more than 30 mpg.

5. The average computer gamer owns 4 or more games.

6. The average temperature in Northern CO during the month of March is less than 70 degrees.

7. At least 28% of astronomers can name more than 50 stars.

8. Less than 35% of astronomers can name at least 75 stars.

For questions 9-16, use notation to state the alternative hypothesis and null hypothesis.

9. Less than 1% of U.S. citizens participate in a militia.

10. Between 20% and 35% of students watch cartoons on Saturday morning.

11. More than 71% of high school students can name a favorite fantasy book.

12. Less than 1 in 5 U.S. college students ride a bike to school.

13. Less than 2% of U.S. adults have milked a cow.

14. Between 2% and 10% of Americans are vegetarians.

15. Less than 4% of high school students take Statistics.

16. More than 80% of students say they learn better through video than with a textbook.

### Review (Answers)

To view the Review answers, open this PDF file and look for section 10.1.