This page covers content from Sections 9.1 and 9.2

9.1 Null and Alternative Hypotheses

9.2 Outcomes and the Type I and Type II Errors

Introduction

In a hypothesis test, sample data is used to arrive at a decision about some type of claim that has been made. If certain conditions about the sample are satisfied, then the claim can be evaluated for a population.

Notation for Hypothesis Testing

We will always have two hypotheses to work with. One is called the null hypothesis, typically denoted by H₀. The null hypothesis will use some form of equality (i.e., =, ≤ or ≥)

The other hypothesis is the alternative hypothesis, typically denoted with H_a or H₁, using less than, greater than, or not equals symbols (i.e., ≠, >, or <)

What You Will Do

Make sure you read through all of the information below and watch the videos that are embedded within this page. The information written here is meant to help introduce concepts, and the videos are meant to help show examples and reinforce ideas about these concepts, so both are necessary to help you learn!

What is a hypothesis test?

Let's say you are reading a newspaper and see the claim, "Only 30% of registered voters in San Bernardino County voted in the last election." That is, the newspaper is claiming that LaTeX: p=0.30 for the population proportion of all San Bernardino registered voters voted in the last election

Let's say you and I believe there was a lot more than 30% who voted... We may want to investigate if that is truly accurate or not by collecting some samples and seeing what percent of those samples of registered voters actually voted.

In this case, we are essentially challenging the claim that "30% of registered voters in San Bernardino County voted in the last election."

So, our null hypothesis, H₀, will be that the population proportion of all San Bernardino registered voters voted in the last election is equal to 0.30.

That is, LaTeX: H_0:p=0.30

Then, our alternative hypothesis will come from what we believe may actually be the case or what we are wanting to test. In this scenario, we thought that percentage was actually much higher than 30%, so we will translate that into our hypothesis that LaTeX: p>0.30 .

That is, LaTeX: H_a:p>0.30 .

So, for our hypothesis test in this problem, we would have:

LaTeX: H_0:p=0.30

LaTeX: H_a:p>0.30

Our hypotheses will always be contradictory. That is, there is no situation in which both the null and the alternative hypotheses can be true at the same time. Once we collect data to investigate trends in samples, we can determine if we want to reject the null hypothesis or fail to reject the null hypothesis.

To clarify, we never actually "accept" anything! We just say we do or do not have sufficient evidence to reject or fail to reject the null hypothesis.

Another example of setting up hypotheses based on a claim:

We want to test if community college students take less than three years to graduate from college, on average.

Notice this is not a proportion anymore... we are wanting to investigate the average amount of time it takes for students to graduate from college. So, our hypotheses will involve $LaTeX: \mu$ since we are estimating the population mean time it takes students to graduate.

The null and alternative hypotheses here would be:

$LaTeX: H_0:\mu\ge3$

$LaTeX: H_a:\mu<3$

Remember, our alternative hypothesis will never involve a form of equality, so our alternative in this case is the same as what we want to test, which uses the "<" sign. It will be directly contradictory to the null hypothesis, so that is why the null hypothesis above uses the "≥" sign.

In summary:

We will always have two hypotheses to work with. One is called the null hypothesis, typically denoted by H₀. The null hypothesis will use some form of equality (i.e., =, ≤ or ≥)

The other hypothesis is the alternative hypothesis, typically denoted with H_a or H₁, using less than, greater than, or not equals symbols (i.e., ≠, >, or <). It will never involve an "equals" or a form of equality.

From our textbook:

H₀: The null hypothesis: This can often be considered the status quo and as a result if you cannot accept the null it requires some action.

H_a: The alternative hypothesis: It is a claim about the population that is contradictory to H₀ and what we conclude when we reject H₀. This is usually what the researcher is trying to prove.

Since the null and alternative hypotheses are contradictory, you must examine evidence to decide if you have enough evidence to reject the null hypothesis or not. The evidence is in the form of sample data.

After you have determined which hypothesis the sample supports, you make a decision. There are two options for a decision. They are "reject H₀" if the sample information favors the alternative hypothesis or "do not reject H₀" or "decline to reject H₀" if the sample information is insufficient to reject the null hypothesis.

Mathematical Symbols Used in H₀ and H_a:

H₀	H_a
equal (=)	not equal (≠) or greater than (>) or less than (<)
greater than or equal to (≥)	less than (<)
less than or equal to (≤)	more than (>)

Now You!

Watch this Khan Academy video to see more examples of the null and alternative hypotheses:

Outcomes and Types of Errors

Simplified, the types of errors we face here are if we end up being "wrong" about our outcome of a hypothesis test. For example, if we end up rejecting a null hypothesis when in fact, it was true... what does that mean? That would indicate a Type 1 Error.

From our textbook:

When you perform a hypothesis test, there are four possible outcomes depending on the actual truth (or falseness) of the null hypothesis H₀ and the decision to reject the null hypothesis or not. The outcomes are summarized in the following table:

ACTION	H₀ IS ACTUALLY	H₀ IS ACTUALLY
	True	False
Do not reject H₀	Correct Outcome	Type II error
Reject H₀	Type I Error	Correct Outcome

The four possible outcomes in the table are:

The decision is not to reject H₀ when H₀ is true (correct decision).
The decision is to reject H₀ when H₀ is true (incorrect decision known as aType I error).
The decision is not to reject H₀ when, in fact, H₀ is false (incorrect decision known as a Type II error).
The decision is to reject H₀ when H₀ is false (correct decision whose probability is called the Power of the Test).

Examples (from our text):

Example 1:

Suppose the null hypothesis, H₀, is: Navah's rock climbing equipment is safe.

Type I error: Navah thinks that her rock climbing equipment may not be safe when, in fact, it really is safe. Type II error: Navah thinks that her rock climbing equipment may be safe when, in fact, it is not safe.

α = probability that Navah thinks her rock climbing equipment may not be safe when, in fact, it really is safe. β = probability that Navah thinks her rock climbing equipment may be safe when, in fact, it is not safe.

Notice that, in this case, the error with the greater consequence is the Type II error. (If Navah thinks her rock climbing equipment is safe, she will go ahead and use it.)

Example 2:

A certain experimental drug claims a cure rate of at least 75% for males with prostate cancer. Which type error is the more serious?

Type I: A cancer patient believes the cure rate for the drug is less than 75% when it actually is at least 75%.

Type II: A cancer patient believes the experimental drug has at least a 75% cure rate when it has a cure rate that is less than 75%.

In this scenario, the Type II error contains the more severe consequence. If a patient believes the drug works at least 75% of the time, this most likely will influence the patient’s (and doctor’s) choice about whether to use the drug as a treatment option.

Now you!

Watch this Khan Academy video introducing Type 1 and Type 2 Errors (we will introduce the idea of a p-value and significance level soon, so it is okay to not follow that part!):

Watch this Khan Academy video going through several examples of Type 1 and Type 2 Errors:

Power of the Test

Each of the errors occurs with a particular probability. The Greek letters α and β represent the probabilities.

α = probability of a Type I error = P(Type I error) = probability of rejecting the null hypothesis when the null hypothesis is true.

β = probability of a Type II error = P(Type II error) = probability of not rejecting the null hypothesis when the null hypothesis is false.

α and β should be as small as possible because they are probabilities of errors. They are rarely zero.

The Power of the Test is 1 – β. Ideally, we want a high power that is as close to one as possible. Increasing the sample size can increase the Power of the Test.

Now you!

Watch this video from Khan Academy walking through examples of the Power of the Test:

Make Connections in our Textbook

Read through Sections 9.1 and 9.2 and the examples provided to help your understanding!

Read about the Null and Alternative Hypotheses in our text

Read about Outcomes and Type 1 and Type 2 Errors in our text

What's Next

You will complete a Progress Check for the material so far and a couple of CYUs.