Order #365390 Topic: Null Hypothesis and Testing

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In Chapter 6, we saw how the inferential techniques of estimation can assist researchers when they use sample data to make educated guesses about the unknown values of population parameters. Now, we turn our attention to a second way in which researchers engage in inferential thinking. This procedure is called hypothesis testing.

Before we examine the half-dozen elements of hypothesis testing, let me reiterate something I said near the beginning of Chapter 5. In order for inferential statistics to begin, the researcher must first answer four preliminary questions: (1) What is/are the relevant population(s)? (2) How will a sample be extracted from the population(s) of interest? (3) What characteristic(s) of the sample people, animals, or objects will serve as the target of the measurement process? (4) What is the study’s statistical focus—or stated differently, how will the sample data be summarized so as to obtain a statistic that can be used to make an inferential statement concerning the unknown parameter? In this chapter, I assume that these four questions have been both raised and answered by the time the researcher starts to apply the hypothesis testing procedure. To help you understand the six-step version of hypothesis testing, I first simply list the various steps in their proper order (i.e., the order in which a researcher ought to do things when engaged in this form of statistical inference). After presenting an ordered list of the six steps, I then discuss the function and logic of each step. An Ordered List of the Six Steps Whenever researchers use the six-step version of the hypothesis testing procedure, they do the following: 1. State the null hypothesis. 2. State the alternative hypothesis.

3. Select a level of significance.

4. Collect and summarize the sample data.

5. Refer to a criterion for evaluating the sample evidence.

6. Make a decision to discard/retain the null hypothesis.

It should be noted that there is no version of hypothesis testing that involves fewer than six steps. Stated differently, it is outright impossible to eliminate any of these six ingredients and have enough left to test a statistical hypothesis.

A Detailed Look at Each of the Six Steps

As indicated previously, the list of steps just presented is arranged in an ordered fashion. In discussing these steps, however, we now look at these six component parts in a somewhat jumbled order: 1, 6, 2, 4, 5, and then 3. My motivation in doing this is not related to sadistic tendencies! Rather, I am convinced that the function and logic of these six steps can be understood far more readily if we purposely chart an unusual path through the hypothesis testing procedure. Please note, however, that the six steps are rearranged here for pedagogical reasons only. If I were asked to apply these six steps in an actual study, I would use the ordered list as my guide, not the sequence to which we now turn.

Step 1: The Null Hypothesis When engaged in hypothesis testing, a researcher begins by stating a null hypothesis. If there is just one population involved in the study, the null hypothesis is a pinpoint statement as to the unknown quantitative value of the parameter in the population of interest. To illustrate what this kind of null hypothesis might look like, suppose that (1) we conduct a study in which our population contains all full-time students enrolled in a particular university, (2) our variable of interest is intelligence, and (3) our statistical focus is the mean IQ score. Given this situation, we could set up a null hypothesis to say that 100. This statement deals with a population parameter, it is pinpoint in nature, and we made it. The symbol for null hypothesis is , and is usually followed by (1) a colon, (2) the parameter symbol that indicates the researcher’s statistical focus, (3) an equal sign, and (4) the pinpoint numerical value that the researcher has selected. Accordingly, we specify the null hypothesis for our imaginary study by stating H0 : 100. If our study’s statistical focus involves something other than the mean, we must change the parameter’s symbol to make consistent with the study’s focus. For example, if our imaginary study is concerned with the variance among students’ heights, the null hypothesis must contain the symbol rather than the symbol . Or, if we are concerned with the product–moment correlation between the students’ heights and weights, the symbol must appear in . r H0

With respect to the pinpoint numerical value that appears in the null hypothesis, researchers have the freedom to select any value that they wish to test. Thus, in our example dealing with the mean IQ of university students, the null hypothesis could be set up to say that , , , or any specific value of our choosing. Likewise, if our study focuses on the variance, we could set up , the null hypothesis, to say that 10 or that any other positive number of our choosing. And in a study having Pearson’s product–moment correlation coefficient as its statistical focus, the null hypothesis could be set up to say that or that or that or that any specific number between and

The only statistical restrictions on the numerical value that appears in are that it (1) must lie somewhere on the continuum of possible values that correspond to the parameter and (2) cannot be fixed at the upper or lower limit of that continuum, presuming that the parameter has a lowest or highest possible value. These restrictions rule out the following null hypotheses: because the variance has a lower limit of 0 whereas Pearson’s product–moment correlation coefficient has limits of 1.00.

Excerpts 7.1 and 7.2

show how researchers sometimes talk about their null hypotheses. In the first of these excerpts, the statistical focus of the null hypothesis is the mean, as is made clear by the inclusion of the symbol . As you can see, there are two s in this null hypothesis, because there are two populations involved in this study, boys and girls. The symbol of course, corresponds to the mean score m, m m H0 : s 2 = 0 H0 : r = -1.00 H0 : s 2 = -15 H0 : r = +1.30 H0 -1.00 +1.00. r = 0.00 r = -.50 r = +.92

on some variable of interest. As indicated in the excerpt, the researchers wanted to compare the two groups in terms of how much they valued reading.

Previously, I indicated that every null hypothesis must contain a pinpoint numerical value. From what is stated in Excerpt 7.1, it is clear that the pinpoint number in this excerpt’s is zero. This pinpoint number would still be in the null hypothesis (but slightly hidden from view) if the researchers had said . If two things are equal, there is no difference between them, and the notion of no difference is equivalent to saying that a zero difference exists.

Although researchers have the freedom to select any pinpoint number they wish for , a zero is often selected when the samples from two populations are being compared. When this is done, the null hypothesis becomes a statement that there is no difference between the populations. Because of the popularity of this kind of null hypothesis, people sometimes begin to think that a null hypothesis must be set up as a “no difference” statement. This is both unfortunate and wrong. When two populations are compared, the null hypothesis can be set up with any pinpoint value the researcher wishes to use. (For example, in comparing the mean height of men and women, we could set up a legitimate null hypothesis that stated inches.) When the hypothesis testing procedure is used with a single population, the notion of “no difference,” applied to parameters, simply does not make sense. How could there be a difference, zero or otherwise, when there is only one (or only one , or only one , etc.)? Excerpt 7.2 contains a null hypothesis that involves a correlation. The symbol in this represents the Pearson product–moment correlation in the study’s population. As you can see, is set equal to zero in this null hypothesis. Theoretically, the value of in could have been set equal to any pinpoint number, such as .20, .55, or any other value between 1.00 and 1.00. However, it is almost always the case that researchers set equal to 0.00 in when using the hypothesis testing procedure in a correlational study. In Excerpts 7.3 and 7.4, we see two additional null hypotheses. In the first of these excerpts, the null hypothesis stated that two percentages were equal. 1 Because of the wording in this excerpt, you might think that this stipulates that the percentage of intangible outputs from the 10 members of the T work group is identical to

EXCERPTS 7.3–7.4• Two Additional Null Hypotheses This research utilized a set of subjects, 19 in total (ten from T-work group and nine from the WT-work group). . . . H0 : P wto Pto , this means the percentage of “intangible” outputs by WT-work is equal to the percentage of “intangible” outputs by T-work. Source: Waters, N. M., & Beruvides, M. G. (2009). An empirical study analyzing traditional work schemes versus work teams. Engineering Management Journal, 21(4), 36–43.

EXCERPTS 7.3–7.4•(continued) More specifically, the article explores the perceived influence of adolescents on the purchase of various product groups across four family communication types, namely laissez-faire, protective, pluralistic, and consensual families. The following null hypothesis was formulated for the purposes of the study: Source: Tustin, D. (2009). Exploring the perceived influence of South African adolescents on product purchases by family communication type. Communicatio: South African Journal for Communication Theory & Research, 35(1), 165–183. H0 : m1 = m2 = m3 = m4 .

the percentage of such outputs from the 9 members of the WT work group. This is an incorrect conceptualization of this study’s , because the two percentages represented in the null hypothesis refer to the population of people in T work groups and the population of people in WT work groups. Without exception, null hypotheses always are focused on populations (and this is true for studies that focus on means, correlations, percentages, or anything else).

Excerpt 7.4 shows a null hypothesis that involves four population means. The s in this null hypothesis correspond to the means of the four populations represented by the different kinds of families involved in this study. In the stated , is connected to families with a laissez-faire style of communication, is connected to families having a protective style of communication, and so on. The data of the study came from the parents in each of the four samples who were asked to indicate how much influence their adolescent children had on their (the parents’) decisions to purchase various items (e.g., cell phones, fast food, clothing).

Before we leave our discussion of the null hypothesis, it should be noted that does not always represent the researcher’s personal belief, or hunch, as to the true state of affairs in the population(s) of interest. In fact, the vast majority of null hypotheses are set up by researchers in such a way as to disagree with what they actually believe to be the case. We return to this point later in the chapter; for now, however, I want to alert you to the fact that the associated with any given study probably is not an articulation of the researcher’s personal belief concerning the involved population(s).

Step 6: The Decision Regarding H0

At the end of the hypothesis testing procedure, the researcher does one of two things with One option is for the researcher to take the position that the null hypothesis is probably false. In this case, the researcher rejects The other option available to the researcher is to refrain from asserting that is probably false. In this case, a fail-to-reject decision is made. If, at the end of the hypothesis testing procedure, a conclusion is reached that H is probably false, the researcher communicates this decision by saying one of four things: (1) was rejected, (2) a statistically significant finding was obtained, (3) a reliable difference was observed, or (4) p is less than a small decimal value (e.g., p .05). In Excerpts 7.5 through 7.7, we see examples of how researchers sometimes communicate their decision to disbelieve H0 .

EXCERPTS 7.5–7.7• Rejecting the Null Hypothesis The authors were able to reject the null hypothesis that the program would have no effect on knowledge. Source: Rethlefsen, M. L., Piorun, M., & Prince, D. (2009). Teaching Web 2.0 technologies using Web 2.0 technologies. Journal of the Medical Library Association, 97(4), 253–259. ESPN Internet articles included a significantly higher proportion of descriptors about the positive skill level/accomplishments and family roles/personal relationships than CBS SportsLine articles. Source: Kian, E. T. M., Mondello, M., & Vincent, J. (2009). ESPN—The women’s sports network? A content analysis of Internet coverage of March Madness. Journal of Broadcasting & Electronic Media, 53(3), 477–495. Participants generated more original analogies of time following exposure to dual cultures or a fusion culture (vs. control) (t 2.08, p .05). Source: Leung, A. K., & Chiu, C. (2010). Multicultural experience, idea receptiveness, and creativity. Journal of Cross-Cultural Psychology, 41(5–6), 723–741.

Just as there are different ways for a researcher to tell us that is considered to be false, there are various mechanisms for expressing the other possible decision concerning the null hypothesis. Instead of saying that a fail-to-reject decision has been reached, the researcher may tell us (1) was tenable, (2) was accepted, (3) no reliable differences were observed, (4) no significant difference was found, (5) the result was not significant (often abbreviated as ns or NS), or (6) p is greater than a small decimal value (e.g., p .05). Excerpts 7.8 through 7.10 illustrate these different ways of communicating a fail-to-reject decision.

EXCERPTS 7.8–7.10• Failing to Reject the Null Hypothesis Hence, this null hypothesis was accepted. Source: Vinodh, S., Sundararaj, G., & Devadasan, S. R. (2010). Measuring organisational agility before and after implementation of TADS. International Journal of Advanced Manufacturing Technology, 47(5–8), 809–818.

EXCERPTS 7.8–7.10•(continued) The male participants were evenly split, with 51% choosing the true crime book and 49% choosing the war book, ns. Source: Vicary, A. M., & Fraley, R. C. (2010). Captured by true crime: Why Are women drawn to tales of rape, murder, and serial killers? Social Psychological and Personality Science, 1(1), 81–86. No significant age variance was found between Jewish and Muslim participants (t(215) 1.89, p .05). Source: Winstok, Z. (2010). The effect of social and situational factors on the intended response to aggression among adolescents. Journal of Social Psychology, 150(1), 57–76.

It is especially important to be able to decipher the language and notation used by researchers to indicate the decision made concerning , because most researchers neither articulate their null hypotheses nor clearly state that they used the hypothesis testing procedure. Often, the only way to tell that a researcher has used this kind of inferential technique is by noting what happened to the null hypothesis.

Step 2: The Alternative Hypothesis

Near the beginning of the hypothesis testing procedure, the researcher must state an alternative hypothesis. Referred to as (or as ), the alternative hypothesis takes the same form as the null hypothesis. For example, if the null hypothesis deals with the possible value of Pearson’s product–moment correlation in a single population (e.g., ), then the alternative hypothesis must also deal with the possible value of Pearson’s correlation in a single population. Or, if the null hypothesis deals with the difference between the means of two populations (perhaps indicating that ), then the alternative hypothesis must also say something about the difference between those populations’ means. In general, therefore, and are identical in that they must (1) deal with the same number of populations, (2) have the same statistical focus, and (3) involve the same variable(s).

The only difference between the null and alternative hypothesis is that the possible value of the population parameter included within always differs from what is specified in . If the null hypothesis is set up to say then the alternative hypothesis might be set up to say or, if a researcher speci- fies in Step 1 that we might find that the alternative hypothesis is set up to say .

Excerpt 7.11 contains an alternative hypothesis, labeled as well as the null hypothesis with which it was paired. Notice that both and deal with the same two populations and have the same statistical focus (a percentage). The null

EXCERPT 7.11• The Alternative Hypothesis The null and alternative hypotheses are as follows: and where p1 is the population proportion of administrators who select a certain outcome, and p2 is the population proportion of school social workers who also select that outcome (for example, school social work services lead to increased attendance). Source: Bye, L., Shepard, M., Patridge, J., & Alvarez, M. (2009). School social work outcomes: Perspectives of school social worker and school administrators. Children & Schools, 31(2), 97–108. H1 : p1 – p2 Z 0 H0 : p1 – p2 = 0

hypothesis states that the two populations—administrators and social workers—are identical in the proportion of the population choosing a particular outcome. The alternative hypothesis states the two populations are not identical.

As indicated in the previous section, the hypothesis testing procedure terminates (in Step 6) with a decision to either reject or fail to reject the null hypothesis. In the event that is rejected, represents the state of affairs that the researcher considers to be probable. In other words, and always represent two opposing statements as to the possible value of the parameter in the population(s) of interest. If, in Step 6, is rejected, then belief shifts from to Stated differently, if a reject decision is made at the end of the hypothesis testing procedure, the researcher will reject in favor of h0

Although researchers have flexibility in the way they set up alternative hypotheses, they normally will set up either in a directional fashion or in a nondirectional fashion. 2 To clarify the distinction between these options for the alternative hypothesis, let’s imagine that a researcher conducts a study to compare men and women in terms of intelligence. Further suppose that the statistical focus of this hypothetical study is on the mean, with the null hypothesis asserting that Now, if the alternative hypothesis is set up in a nondirectional fashion, the researcher simply states If, however, the alternative hypothesis is stated in a directional fashion, the researcher specifies a direction in This could be done by asserting or by asserting The directional/nondirectional nature of is highly important within the hypothesis testing procedure. The researcher must know whether was set up in Ha Ha Ha : m men 6 m women . Ha Ha : m men 7 m women . Ha : m men Z m women . H0 : m men = m women . Ha Ha a directional or nondirectional manner in order to decide whether to reject (or to fail to reject) the null hypothesis. No decision can be made about unless the directional/nondirectional character of is clarified. In most empirical studies, the alternative hypothesis is set up in a nondirectional fashion. Thus, if I were to guess what says in studies containing the null hypotheses presented as shown on the left, I would bet that the researchers had set up their alternative hypotheses as indicated on the right.

Possible H0 nondirectional Ha Researchers typically set up in a nondirectional fashion because they do not know whether the pinpoint number in is too large or too small. By specifying a nondirectional the researcher permits the data to point one way or the other in the event that is rejected. Hence, in our hypothetical study comparing men and women in terms of intelligence, a nondirectional alternative hypothesis allows us to argue that is probably higher than (in the event that we reject the because ); or, such an alternative hypothesis allows us to argue that is probably higher than (if we reject because ). Occasionally, a researcher believes so strongly (based on theoretical consideration or previous research) that the true state of affairs falls on one side of ’s pinpoint number that is set up in a directional fashion. So long as the researcher makes this decision prior to looking at the data, such a decision is fully legitimate. It is, however, totally inappropriate for the researcher to look at the data first and then subsequently decide to set up in a directional manner. Although a decision to reject or fail to reject could still be made after first examining the data and then articulating a directional , such a sequence of events would sabotage the fundamental logic and practice of hypothesis testing. Simply stated, decisions concerning how to state (and how to state ) must be made without peeking at any data. When the alternative hypothesis is set up in a nondirectional fashion, researchers sometimes use the phrase two-tailed test to describe their specific application of the hypothesis testing procedure. In contrast, directional lead to what researchers sometimes refer to as one-tailed tests. Inasmuch as researchers rarely specify the alternative hypothesis in their technical write-ups, the terms one-tailed and two-tailed help us to know exactly how was set up. For example, consider Excerpts 7.12 and 7.13. Here, we see how researchers sometimes use the term two-tailed or one-tailed to communicate their decisions to set up in a nondirectional or directional fashion. Ha Ha Ha s Ha H0 Ha H0 Ha Ha H0 m men m women H0 Mmen 7 Mwomen Mwomen 7 Mmen m women m men H0 H0 Ha , H0 Ha Ha H0 : m1 -m2 = 0 : m1 -m2 Z 0 Ha : s 2 H0 : s Z 4 2 = 4 Ha H0 : r = +.20 : r Z +.20 Ha H0 : m = 100 : m Z 100

EXCERPTS 7.12–7.13• Two-Tailed and One-Tailed Tests All tests of significance were two-tailed. Source: Miller, K. (2010). Using a computer-based risk assessment tool to identify risk for chemotherapy-induced febrile neutropenia. Clinical Journal of Oncology Nursing, 14(1), 87–91. To investigate what variables might be important predictors of company support for fathers taking leave, [Pearson] correlations were calculated. . . . One-tailed tests of significance were used. Source: Haas, L., & Hwang, P. C. (2010). Is fatherhood becoming more visible at work? Trends in corporate support for fathers taking parental leave in Sweden. Fathering: A Journal of Theory, Research, & Practice about Men as Fathers, 7(3), 303–321.

Step 4: Collection and Analysis of Sample Data So far, we have covered Steps 1, 2, and 6 of the hypothesis testing procedure. In the first two steps, the researcher states the null and alternative hypotheses. In Step 6, the researcher either (1) rejects in favor of or (2) fails to reject We now turn our attention to the principal “stepping stone” used to move from the beginning points of the hypothesis testing procedure to the final decision. Inasmuch as the hypothesis testing procedure is, by its very nature, an empirical strategy, it should come as no surprise that the researcher’s ultimate decision to reject or to retain is based on the collection and analysis of sample data.

No crystal ball is used, no Ouija board is relied on, and no eloquent argumentation is permitted. Once and are fixed, only scientific evidence is allowed to affect the disposition of The fundamental logic of the hypothesis testing procedure can now be laid bare because the connections between the data, and the final decision are as straightforward as what exists between the speed of a car, a traffic light at a busy intersection, and a lawful driver’s decision as the car approaches the intersection. Just as the driver’s decision to stop or to pass through the intersection is made after observing the color of the traffic light, the researcher’s decision to reject or to retain is made after observing the sample data. To carry this analogy one step further, the researcher looks at the data and asks, “Is the empirical evidence inconsistent with what one would expect if were true?” If the answer to this question is yes, then the researcher has a green light and rejects However, if the data turn out to be consistent with then the data set serves as a red light telling the researcher not to discard Because the logic of hypothesis testing is so important, let us briefly consider a hypothetical example. Suppose a valid intelligence test is given to a random sample of 100 males and a random sample of 100 females attending the same university. If the null hypothesis was first set up to say and if the data reveal that the two sample means (of IQ scores) differ by only two-tenths of a point, the sample data would be consistent with what we expect to happen when two samples are selected from populations having identical means.

Clearly, the notion of sampling error could fully explain why the two Ms might differ by two-tenths of an IQ point even if In this situation, there is no empirical justification for making the data-based claim that males at our hypothetical university have a different IQ, on average, than do their female classmates. Now, let’s consider what would happen if the difference between the two sample means turns out to be equal to 20 IQ points. If the empirical evidence turns out this way, we have a situation where the data are inconsistent with what one would expect if were true.

Although the concept of sampling error strongly suggests that neither sample mean will turn out exactly equal to its population parameter, the difference of 20 IQ points between and and are equal. With results such as this, the researcher would reject the arbitrarily selected null hypothesis. To drive home the point I am trying to make about the way the sample data influence the researcher’s decision concerning let’s shift our attention to a real study that had Pearson’s correlation as its statistical focus. In Excerpt 7.14, the hypothesis testing procedure was used to evaluate three bivariate correlations based on data that came from 90 men who had surgery after going to an infertility clinic. Each man was measured in terms of the number of left and right spermatic arteries as well the number of left and right lymphatic channels. Then, the left-right data were correlated for each of the two kinds of arteries and for the channels.

EXCERPT 7.14• Rejecting H0 When the Sample Data Are Inconsistent with H0 An analysis of the relationship between the right and left spermatic cord anatomy in the bilateral varicocelectomy cases ( ) revealed a significant correlation between the number of right and left internal spermatic arteries (r 0.42, P .05). However, we did not identify a significant correlation between the number of right and left external spermatic arteries ( ) or the number of right and left lymphatic channels ( ). Source: Libman, J. L., Segal, R., Baazeem, A., Boman, J., & Zini, A. (2010). Microanatomy of the left and right spermatic cords at subinguinal microsurgical varicocelectomy: Comparative study of primary and redo repairs. Urology, 75(6), 1324–1327.

In the study associated with Excerpt 7.14, the hypothesis testing procedure was used separately to evaluate each of the three sample rs. In each case, the null hypothesis stated The sample data, once analyzed, yielded correla H tions

of .42, .13, and .19. The first of these rs ended up being quite different from the null hypothesis number of 0.00. Statistically speaking, the r of .42 was so inconsistent with that sampling error alone was considered to be an inadequate explanation for why the observed correlation was so far away from the pinpoint number in the null hypothesis. Although we expect some discrepancy between 0.00 and the databased value of r even if were true, we do not expect this big difference. Accordingly, the null hypothesis concerning the internal spermatic arteries—that there was no relationship between the number of left and right arteries—was rejected, as indicated by the phrase significant correlation and the notation P .05. The second and third correlations in Excerpt 7.14 turned out to be much closer to the pinpoint number in The small differences between the null number and the rs of .13 and .19 could each be explained by sampling error. In other words, if the correlation in the population were truly equal to 0.00, it would not be surprising to have a sample r (with ) be anywhere between .20 and .20. Accordingly, the null hypotheses concerning external spermatic arteries and the lymphatic channels were not rejected, as indicated by the notation P .05. In Step 4 of the hypothesis testing procedure, the summary of the sample data always leads to a single numerical value. Being based on the data, this number is technically referred to as the calculated value (or the test statistic). Occasionally, the researcher’s task in obtaining the calculated value involves nothing more than computing a value that corresponds to the study’s statistical focus. This was the case in Excerpt 7.14, where the statistical focus was Pearson’s correlation coefficient and where the researcher needed to do nothing more than compute a value for r. In most applications of the hypothesis testing procedure, the sample data are summarized in such a way that the statistical focus becomes hidden from view. For example, consider Excerpts 7.15 and 7.16. In the first of these excerpts, the calculated

EXCERPTS 7.15–7.16• The Calculated Value [A] t-test found that overall satisfaction levels of male students ( ) were significantly higher than those of female students ( ), = 13.78, p 0.05 (two-tailed). Source: Kim, H., Lee, S., Goh, B., & Yuan, J. (2010). Assessing College Students’ Satisfaction with University Foodservice. Proceedings of the 15th Annual Graduate Student Research Conference in Hospitality and Tourism, Washington, DC, 34–46. There was no difference in girls’ (M 5 years, 8 months; SD 1.52) and boys’ (M 5 years, 10 months; SD 1.68) ages, F(1, 114) 0.25, p 05. Source: Tenenbaum, T. R., Hill, D. B., Joseph, N., & Roche, E. (2010). “It’s a boy because he’s painting a picture”: Age differences in children’s conventional and unconventional gender schemas. British Journal of Psychology, 101(1), 137–154. t(225)

value was labeled t and it turned out equal to 13.78. In Excerpt 7.16, the calculated value was F, and it was equal to 0.25. In each of these excerpts, the statistical focus was the mean. In each of these excerpts, two sample means were compared. In Excerpt 7.15, the mean of 4.14 was compared against the mean of 3.87. In Excerpt 7.16, the means were 5 years, 8 months and 5 years, 10 months. Within each of these studies, the researchers put their sample data into a formula that produced the calculated value. The important thing to notice in these excerpts is that in neither case does the calculated value equal the difference between the two means being compared. In Chapter 10, we consider t-tests and F-tests in more detail, so you should not worry now if you do not currently comprehend everything that is presented in these excerpts.

They are shown solely to illustrate the typical situation in which the statistical focus of a study is not reflected directly in the calculated value. Before computers were invented, researchers always had a single goal in mind when they turned to Step 4 of the hypothesis testing procedure: the computation of the data-based calculated value. Now that computers are widely available, researchers still are interested in the magnitude of the calculated value derived from the data analysis.

Contemporary researchers, however, are also interested in a second piece of information generated by the computer: the data-based p-value. Whenever researchers use a computer to perform the data analysis, they either (1) tell the computer what the null hypothesis is going to be or (2) accept the computer’s built-in default version of The researcher also specifies whether is directional or nondirectional in nature. Once the computer knows what the researcher’s and are, it can easily analyze the sample data and compute the probability of having a data set that deviates as much or more from as does the data set being analyzed. The computer informs the researcher as to this probability by means of a statement that takes the form , with the blank being filled by a single decimal value somewhere between 0 and 1. Excerpt 7.17 illustrates nicely how a p-value is like a calculated value in that either one can be used as a single-number summary of the sample data. As you can see, three Pearson correlation coefficients are in this excerpt. The researchers associated

EXCERPT 7.17• Using p as the Calculated Value Correlation analyses and inspection of scatterplots between the PA composite and speech production variables showed that there was no significant relationship between PA and distortions ( ), nor between PA and typical sound changes ( ). However, a significant relationship was found between PA and atypical sound changes ( ). Source: Preston, J., & Louise Edwards, M. (2010). Phonological awareness and types of sound errors in preschoolers with speech sound disorders. Journal of Speech, Language & Hearing Research, 53(1), 44–60. r = -.362,

with this passage used a p-value to determine how likely it would be, assuming the null hypothesis to be true, to end up with a sample correlation as large or larger than each of their computed rs. Each p functioned as a measure of how inconsistent the sample data were compared with what would be expected to happen if were true. Be sure to note in Excerpt 7.17 that there is an inverse relationship between the size of p and the degree to which the sample data deviate from the null hypothesis. The r that is furthest away from 0.00 (the pinpoint number in the null hypothesis) has the smallest p. In contrast, the smallest of the three rs has the largest p.

Step 5: The Criterion for Evaluating the Sample Evidence After

The researcher has summarized the study’s data, the next task involves asking the question, “Are the sample data inconsistent with what would likely occur if the null hypothesis were true?” If the answer to this question is “Yes,” then is rejected; however, a negative response to this query requires a fail-to-reject decision. Thus, as soon as the sample data can be tagged as consistent or inconsistent (with ), the decision in Step 6 is easily made. “But how,” you might ask, “does the researcher decide which of these labels should be attached to the sample data?” If the data from the sample(s) are in perfect agreement with the pinpoint numerical value specified in then it is obvious that the sample data are consistent with (This would be the case if the sample mean turned out equal to 100 when testing if the sample correlation coefficient turned out equal to 0.00 when testing etc.) Such a situation, however, is unlikely.

There is almost always a discrepancy between ’s parameter value and the corresponding sample statistic. In light of the fact that the sample statistic (produced by Step 4) is almost certain to be different from ’s pinpoint number (specified in Step 1), the concern over whether the sample data are inconsistent with actually boils down to the question, “Should the observed difference between the sample evidence and the null hypothesis be considered a big difference or a small difference?” If this difference (between the data and ) is judged to be large, then the sample data are looked on as being inconsistent with and, as a consequence, is rejected. If, however, this difference is judged to be small, the data and are looked on as consistent with each other and, therefore, is not rejected. To answer the question about the sample data’s being either consistent or inconsistent with what one would expect if were true, a researcher can use either of two simple procedures.

Note that both of these procedures involve comparing a single-number summary of the sample evidence against a criterion number. The single-number summary of the data can be either the calculated value or the p-value. Our job now is to consider what each of these data-based indices is compared against, and what kind of result allows researchers to consider their samples to represent a large or a small deviation from H0 . H0 H0 H0 H0 H0 H0 H0 H0 H0 H0 : r = 0.00, H0 : m = 100, H0 . H0 , H0

One available procedure for evaluating the sample data involves comparing the calculated value against something called the critical value. The critical value is nothing more than a number extracted from one of many statistical tables developed by mathematical statisticians. Applied researchers, of course, do not close their eyes and point to just any entry in a randomly selected table of critical values. Instead, they must learn which table of critical values is appropriate for their studies and also how to locate the single number within the table that constitutes the correct critical value. As a reader of research reports, you do not have to learn how to locate the proper table that contains the critical value for any given statistical test, nor do you have to locate, within the table, the single number that allows the sample data to be labeled as being consistent or inconsistent with The researcher does these things. Occasionally, the critical value is included in the research report, as exemplified in Excerpts 7.18 and 7.19.

The observed value of 63.22 is greater than the critical t value of 1.96, and this is significant at a 0.05 level; hence the rejection of the null hypothesis. Source: Oluwole, D. A. (2009). Spirituality, gender and age factors in cybergossip among Nigerian adolescents. CyberPsychology & Behavior, 12(3), 323–326. [T]he calculated chi square ( ) value of 44.35 was greater than the critical value of 9.49 at 0.05 level of significance with 4 degrees of freedom. This means that there is a significance relationship between the marital status of the nurses in Akwa-Ibom State and their being obese. Source: Ogunjimi, L. O., Maria M. Ikorok, M. M., & Yusuf, O. O. (2010). Prevalence of obesity among Nigeria nurses: The Akwa Ibom State experience. International NGO Journal, 5(2), 045–049.

Once the critical value is located, the researcher compares the data-based summary of the sample data against the scientific dividing line that has been extracted from a statistical table. The simple question being asked at this point is whether the calculated value is larger or smaller than the critical value. With most tests (such as t, F, chi-square, and tests of correlation coefficients), the researcher follows a decision rule that says to reject if the calculated value is at least as large as the critical value. With a few tests (such as U or W), the decision rule tells the researcher to reject if the calculated value is smaller than the critical value. You do not need to worry about which way the decision rule works for any given test, because this is the responsibility of the individual who performs the data analysis.

The only things you must know about the comparison of calculated and critical values are that (1) this comparison allows the researcher to decide easily whether to reject or fail to reject and (2) some tests use a decision rule that says to reject if the calculated value is larger than the critical value, whereas other tests involve a decision rule that says to reject if the calculated value is smaller than the critical value.

The researchers associated with Excerpts 7.18 and 7.19 helped the readers of their research reports by specifying not only the critical value but also the nature of the decision rule that was used when the calculated value was compared against the critical value. In most research reports, you see neither of these things; instead, you are given only the calculated value. (On rare occasions, you do not even see the calculated value.) As indicated previously, however, you should not be concerned about this, because it is the researcher’s responsibility to obtain the critical value and to know which way the decision rule operates. When reading most research reports, all you can do is trust that the researcher did these two things properly.

The second way a researcher can evaluate the sample evidence is to compare the data-based p-value against a preset point on the 0-to-1 scale on which the p must fall. This criterion is called the level of significance, and it functions much as does the critical value in the first procedure for evaluating sample evidence. Simply stated, the researcher compares his or her data-based p-value against the criterion point along the 0-to-1 continuum so as to decide whether the sample evidence ought to be considered consistent or inconsistent with The decision rule used in this second procedure is always the same: If the data-based p-value is equal to or smaller than the criterion, the sample is viewed as being inconsistent with however, if p is larger than the criterion, the data are looked on as being consistent with Excerpt 7.20 exemplifies the use of this second kind of criterion for evaluating sample evidence. Note that the data-based p-value of 0.006 was substantially smaller than the criterion number of 0.05. Accordingly, the null hypothesis (that the populations of men and women have equal levels of trust) was rejected. H0 . H0 ; H0 . H0 H0

Huck, Schuyler W.. Reading Statistics and Research (6th Edition) (Page 146). Pearson HE, Inc.. Kindle Edition.