Goal: make decisions about the value of a parameter.

We have confidence intervals, but we might also want to ask questions like

• Do cans of soda actually contain 12 oz?
• Is Medicine A better than Medicine B?

A hypothesis is a statement that something is true.

A hypothesis test involves two (competing) hypotheses:

1. The null hypothesis, denoted \(H_0\), is the hypothesis to be tested. This is the “default” assumption.
2. The alternative hypothesis, denoted \(H_A\) is the alternative to the null.

A hypothesis test helps us decide whether the null hypothesis should be rejected in favor of the alternative.

Cans of soda are labeled with “12 FL OZ”. Is this accurate?

The default, or uninteresting, assumption is that cans of soda contain 12 oz.

• \(H_0\): the mean volume of soda in a can is 12 oz.
• \(H_A\): the mean volume of soda in a can is NOT 12 oz.

We can write these hypotheses in words or in statistical notation.

The null specifies a single value of \(\mu\)

• \(H_0\): \(\mu=\mu_0\)

We call \(\mu_0\) the null value. When we run a hypothesis test, \(\mu_0\) will be replaced by some number.

The alternative specifies a range of possible values for \(\mu\):

• \(H_A\): \(\mu\ne\mu_0\). “The true mean is different from the null value.”

• \(H_0\): the mean volume of soda in a can is 12 oz.
  • The null value is 12. In statistical notation, \(H_0: \mu = 12\).
• \(H_A\): the mean volume of soda in a can is NOT 12 oz.
  • In statistical notation, \(H_A: \mu \ne 12\).

• Take a random sample from the population.
• If the data area consistent with the null hypothesis, do not reject the null hypothesis.
• If the data are inconsistent with the null hypothesis and supportive of the alternative hypothesis, reject the null in favor of the alternative.

In the US court system, jurors are told to assume the defendant is “innocent until proven guilty”.

Innocence is the default assumption, so

• \(H_0\): the defendant is innocent.
• \(H_A\): the defendant is guilty.

• It is not the jury’s job to decide if the defendant is innocent!
  • That should be their default assumption.
• They are only there to decide if the defendant is guilty or if there is not enough evidence to override that default assumption.

The burden of proof lies on the alternative hypothesis.

Notice the careful language in the logic of hypothesis testing: we either reject, or fail to reject, the null hypothesis.

We never “accept” a null hypothesis.

• A Type I Error is rejecting the null when it is true. (Null is true, but we conclude null is false.)
• A Type II Error is not rejecting the null when it is false. (Null is false, but we do not conclude it is false.)

In our jury trial,

• \(H_0\): the defendant is innocent.
• \(H_A\): the defendant is guilty.

A Type I error is concluding guilt when the defendant is innocent.

A Type II error is failing to convict when the person is guilty.

\(P(\)Type I Error\()=\alpha\), the significance level.

• This is the same \(\alpha\) we saw in confidence intervals!

\(P(\)Type II Error\()=\beta\).

• This is something we don’t have time to cover in detail.

We would like both \(\alpha\) and \(\beta\) to be small but,

• If we decrease \(\alpha\), then \(\beta\) will increase.
• If we increase \(\alpha\), then \(\beta\) will decrease.

In practice, we set \(\alpha\) (as we did in confidence intervals).

We can improve \(\beta\) by increasing sample size.

Consider two possible criminal charges:

1. Defendant is accused of stealing a loaf of bread. If found guilty, they may face some jail time and will have a criminal record.
2. Defendant is accused of murder. If found guilty, they will have a felony and may spend decades in prison.

Since these are moral questions, I will let you consider the consequences of each type of error.

However, keep in mind that we do make scientific decisions that have lasting impacts on people’s lives.

• If the null hypothesis is rejected, we say the result is statistically significant. We can interpret this result with:
  • At the \(\alpha\) level of significance, the data provide sufficient evidence to support the alternative hypothesis.
• If the null hypothesis is not rejected, we say the result is not statistically significant. We can interpret this result with:
  • At the \(\alpha\) level of significance, the data do not provide sufficient evidence to support the alternative hypothesis.

When we write these types of conclusions, we will write them in the context of the problem.