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: Every hypothesis test — regardless of the population parameter involved — requires the above three steps. Is Normal Body Temperature Really 98.6 Degrees F? SectionConsider 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:
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.3.2Criminal Trial Analogy SectionOne 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:
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:
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 SectionDid 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:
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
Table S.3.2 shows how this corresponds to the two types of errors in hypothesis testing. Table S.3.2
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 SectionRecall 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:
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 $\mu$:
In Practice
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.
Which of the approaches has focused on the way people learn from watching role models?Social learning theory is the philosophy that people can learn from each other through observation, imitation and modeling. The concept was theorized by psychologist Albert Bandura and combined ideas behind behaviorist and cognitive learning approaches.
Is the Big Five personality test projective?Examples of objective personality tests include the Big Five Inventory and the MMPI. Projective personality tests present test-takers with ambiguous stimuli under the assumption that they will project unconscious personality traits onto these ambiguous stimuli.
When personality researchers use the term individual differences they are referring to ______?When personality researchers use the term individual differences they are referring to: the consistent behavior patterns individuals display across situations. Although Sally is well behaved and polite at home, she can be really mean to the other children at school.
What significance level would indicate that a researcher is 5 percent sure that they would get the result they found if the null hypothesis were true?A p-value less than 0.05 (typically ≤ 0.05) is statistically significant. It indicates strong evidence against the null hypothesis, as there is less than a 5% probability the null is correct (and the results are random).
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