S.3 Hypothesis Testing

In reviewing hypothesis tests, we start first with the general idea. Then, we keep returning to the basic procedures of hypothesis testing, each time adding a little more detail.

The general idea of hypothesis testing involves:

  1. Making an initial assumption.
  2. Collecting evidence (data).
  3. Based on the available evidence (data), deciding whether to reject or not reject the initial assumption.

Every hypothesis test — regardless of the population parameter involved — requires the above three steps.

Example S.3.1

Is Normal Body Temperature Really 98.6 Degrees F? Section

Consider the population of many, many adults. A researcher hypothesized that the average adult body temperature is lower than the often-advertised 98.6 degrees F. That is, the researcher wants an answer to the question: "Is the average adult body temperature 98.6 degrees? Or is it lower?" To answer his research question, the researcher starts by assuming that the average adult body temperature was 98.6 degrees F.

Then, the researcher went out and tried to find evidence that refutes his initial assumption. In doing so, he selects a random sample of 130 adults. The average body temperature of the 130 sampled adults is 98.25 degrees.

Then, the researcher uses the data he collected to make a decision about his initial assumption. It is either likely or unlikely that the researcher would collect the evidence he did given his initial assumption that the average adult body temperature is 98.6 degrees:

  • If it is likely, then the researcher does not reject his initial assumption that the average adult body temperature is 98.6 degrees. There is not enough evidence to do otherwise.
  • If it is unlikely, then:
    • either the researcher's initial assumption is correct and he experienced a very unusual event;
    • or the researcher's initial assumption is incorrect.

In statistics, we generally don't make claims that require us to believe that a very unusual event happened. That is, in the practice of statistics, if the evidence (data) we collected is unlikely in light of the initial assumption, then we reject our initial assumption.

Example S.3.2

Criminal Trial Analogy Section

One place where you can consistently see the general idea of hypothesis testing in action is in criminal trials held in the United States. Our criminal justice system assumes "the defendant is innocent until proven guilty." That is, our initial assumption is that the defendant is innocent.

In the practice of statistics, we make our initial assumption when we state our two competing hypotheses -- the null hypothesis (H0) and the alternative hypothesis (HA). Here, our hypotheses are:

  • H0: Defendant is not guilty (innocent)
  • HA: Defendant is guilty

In statistics, we always assume the null hypothesis is true. That is, the null hypothesis is always our initial assumption.

The prosecution team then collects evidence — such as finger prints, blood spots, hair samples, carpet fibers, shoe prints, ransom notes, and handwriting samples — with the hopes of finding "sufficient evidence" to make the assumption of innocence refutable.

In statistics, the data are the evidence.

The jury then makes a decision based on the available evidence:

  • If the jury finds sufficient evidence — beyond a reasonable doubt — to make the assumption of innocence refutable, the jury rejects the null hypothesis and deems the defendant guilty. We behave as if the defendant is guilty.
  • If there is insufficient evidence, then the jury does not reject the null hypothesis. We behave as if the defendant is innocent.

In statistics, we always make one of two decisions. We either "reject the null hypothesis" or we "fail to reject the null hypothesis."

Errors in Hypothesis Testing Section

Did you notice the use of the phrase "behave as if" in the previous discussion? We "behave as if" the defendant is guilty; we do not "prove" that the defendant is guilty. And, we "behave as if" the defendant is innocent; we do not "prove" that the defendant is innocent.

This is a very important distinction! We make our decision based on evidence not on 100% guaranteed proof. Again:

  • If we reject the null hypothesis, we do not prove that the alternative hypothesis is true.
  • If we do not reject the null hypothesis, we do not prove that the null hypothesis is true.

We merely state that there is enough evidence to behave one way or the other. This is always true in statistics! Because of this, whatever the decision, there is always a chance that we made an error.

Let's review the two types of errors that can be made in criminal trials:

Table S.3.1
    Truth  
    Not Guilty Guilty
Jury Decision Not Guilty OK ERROR
  Guilty ERROR OK

Table S.3.2 shows how this corresponds to the two types of errors in hypothesis testing.

Table S.3.2
    Truth  
    Null Hypothesis Alternative Hypothesis
Decision Do not Reject Null OK Type II Error
  Reject Null Type I Error OK

 

Note that, in statistics, we call the two types of errors by two different  names -- one is called a "Type I error," and the other is called  a "Type II error." Here are the formal definitions of the two types of errors:

Type I Error
The null hypothesis is rejected when it is true.
Type II Error
The null hypothesis is not rejected when it is false.

There is always a chance of making one of these errors. But, a good scientific study will minimize the chance of doing so!

Making the Decision Section

Recall that it is either likely or unlikely that we would observe the evidence we did given our initial assumption. If it is likely, we do not reject the null hypothesis. If it is unlikely, then we reject the null hypothesis in favor of the alternative hypothesis. Effectively, then, making the decision reduces to determining "likely" or "unlikely."

In statistics, there are two ways to determine whether the evidence is likely or unlikely given the initial assumption:

  1. We could take the "critical value approach" (favored in many of the older textbooks).
  2. Or, we could take the "P-value approach" (what is used most often in research, journal articles, and statistical software).

In the next two sections, we review the procedures behind each of these two approaches. To make our review concrete, let's imagine that μ is the average grade point average of all American students who major in mathematics. We first review the critical value approach for conducting each of the following three hypothesis tests about the population mean $\mu$:

Type
Null
Alternative
Right-tailed
H0 : μ = 3
HA : μ > 3
Left-tailed
H0 : μ = 3
HA : μ < 3
Two-tailed
H0 : μ = 3
HA : μ ≠ 3

In Practice

  • We would want to conduct the first hypothesis test if we were interested in concluding that the average grade point average of the group is more than 3.
  • We would want to conduct the second hypothesis test if we were interested in concluding that the average grade point average of the group is less than 3.
  • And, we would want to conduct the third hypothesis test if we were only interested in concluding that the average grade point average of the group differs from 3 (without caring whether it is more or less than 3).

Upon completing the review of the critical value approach, we review the P-value approach for conducting each of the above three hypothesis tests about the population mean \(\mu\). The procedures that we review here for both approaches easily extend to hypothesis tests about any other population parameter.