Which set contains non equivalent members

See solutions Exercise 7 above illustrates a general fact about subsets: Every set is a subset of itself. Notation: Examples: The last example illustrates this basic fact: No set is a proper subset of itself. Are we supposed to list Unless we establish some boundaries on the scope of this exercise, we cannot finish it. To establish a frame of reference for a set problem, we can define a universal set U for the problem: For any set problem or discussion, a universal set U is a "larger" set that contains all of the elements that may be of interest in the discussion; in particular, the universal set at least contains all of the elements of all of the sets in the discussion.

So researchers need a way to decide between them. Although there are many specific null hypothesis testing techniques, they are all based on the same general logic. The steps are as follows:. Following this logic, we can begin to understand why Mehl and his colleagues concluded that there is no difference in talkativeness between women and men in the population. Therefore, they retained the null hypothesis—concluding that there is no evidence of a sex difference in the population.

We can also see why Kanner and his colleagues concluded that there is a correlation between hassles and symptoms in the population. Therefore, they rejected the null hypothesis in favor of the alternative hypothesis—concluding that there is a positive correlation between these variables in the population.

A crucial step in null hypothesis testing is finding the probability of the sample result or a more extreme result if the null hypothesis were true Lakens, A low p value means that the sample or more extreme result would be unlikely if the null hypothesis were true and leads to the rejection of the null hypothesis.

A p value that is not low means that the sample or more extreme result would be likely if the null hypothesis were true and leads to the retention of the null hypothesis. But how low must the p value criterion be before the sample result is considered unlikely enough to reject the null hypothesis?

When this happens, the result is said to be statistically significant. This does not necessarily mean that the researcher accepts the null hypothesis as true—only that there is not currently enough evidence to reject it. The p value is one of the most misunderstood quantities in psychological research Cohen, [4]. Even professional researchers misinterpret it, and it is not unusual for such misinterpretations to appear in statistics textbooks!

The most common misinterpretation is that the p value is the probability that the null hypothesis is true—that the sample result occurred by chance. For example, a misguided researcher might say that because the p value is. But this is incorrect. The p value is really the probability of a result at least as extreme as the sample result if the null hypothesis were true. So a p value of. You can avoid this misunderstanding by remembering that the p value is not the probability that any particular hypothesis is true or false.

Instead, it is the probability of obtaining the sample result if the null hypothesis were true. Specifically, the stronger the sample relationship and the larger the sample, the less likely the result would be if the null hypothesis were true.

That is, the lower the p value. This should make sense. If there were really no sex difference in the population, then a result this strong based on such a large sample should seem highly unlikely. If there were no sex difference in the population, then a relationship this weak based on such a small sample should seem likely.

And this is precisely why the null hypothesis would be rejected in the first example and retained in the second. Of course, sometimes the result can be weak and the sample large, or the result can be strong and the sample small. In these cases, the two considerations trade off against each other so that a weak result can be statistically significant if the sample is large enough and a strong relationship can be statistically significant even if the sample is small.

Table The columns of the table represent the three levels of relationship strength: weak, medium, and strong. The rows represent four sample sizes that can be considered small, medium, large, and extra large in the context of psychological research.

Thus each cell in the table represents a combination of relationship strength and sample size. If it contains the word No , then it would not be statistically significant for either. There is one cell where the decision for d and r would be different and another where it might be different depending on some additional considerations, which are discussed in Section If you keep this lesson in mind, you will often know whether a result is statistically significant based on the descriptive statistics alone.

It is extremely useful to be able to develop this kind of intuitive judgment. One reason is that it allows you to develop expectations about how your formal null hypothesis tests are going to come out, which in turn allows you to detect problems in your analyses.

For example, if your sample relationship is strong and your sample is medium, then you would expect to reject the null hypothesis. If for some reason your formal null hypothesis test indicates otherwise, then you need to double-check your computations and interpretations. A second reason is that the ability to make this kind of intuitive judgment is an indication that you understand the basic logic of this approach in addition to being able to do the computations.

A statistically significant result is not necessarily a strong one. Even a very weak result can be statistically significant if it is based on a large enough sample. The differences between women and men in mathematical problem solving and leadership ability are statistically significant.

But the word significant can cause people to interpret these differences as strong and important—perhaps even important enough to influence the college courses they take or even who they vote for.

This is why it is important to distinguish between the statistical significance of a result and the practical significance of that result. Practical significance refers to the importance or usefulness of the result in some real-world context. Many sex differences are statistically significant—and may even be interesting for purely scientific reasons—but they are not practically significant.

Yet this effect still might not be strong enough to justify the time, effort, and other costs of putting it into practice—especially if easier and cheaper treatments that work almost as well already exist.

Although statistically significant, this result would be said to lack practical or clinical significance. The term observational research is used to refer to several different types of non-experimental studies in which behavior is systematically observed and recorded.

The goal of observational research is to describe a variable or set of variables. More generally, the goal is to obtain a snapshot of specific characteristics of an individual, group, or setting. As described previously, observational research is non-experimental because nothing is manipulated or controlled, and as such we cannot arrive at causal conclusions using this approach. The data that are collected in observational research studies are often qualitative in nature but they may also be quantitative or both mixed-methods.

There are several different types of observational methods that will be described below. Thus naturalistic observation is a type of field research as opposed to a type of laboratory research. Jane Goodall's famous research on chimpanzees is a classic example of naturalistic observation.

Goodall spent three decades observing chimpanzees in their natural environment in East Africa. However, naturalistic observation could more simply involve observing shoppers in a grocery store, children on a school playground, or psychiatric inpatients in their wards. Researchers engaged in naturalistic observation usually make their observations as unobtrusively as possible so that participants are not aware that they are being studied.

Such an approach is called disguised naturalistic observation. Ethically, this method is considered to be acceptable if the participants remain anonymous and the behavior occurs in a public setting where people would not normally have an expectation of privacy. Grocery shoppers putting items into their shopping carts, for example, are engaged in public behavior that is easily observable by store employees and other shoppers.

For this reason, most researchers would consider it ethically acceptable to observe them for a study. In cases where it is not ethical or practical to conduct disguised naturalistic observation, researchers can conduct undisguised naturalistic observation where the participants are made aware of the researcher presence and monitoring of their behavior. However, one concern with undisguised naturalistic observation is reactivity.

In the case of undisguised naturalistic observation, the concern with reactivity is that when people know they are being observed and studied, they may act differently than they normally would. This type of reactivity is known as the Hawthorne effect. For instance, you may act much differently in a bar if you know that someone is observing you and recording your behaviors and this would invalidate the study.

So disguised observation is less reactive and therefore can have higher validity because people are not aware that their behaviors are being observed and recorded. In other words, over time people habituate to being observed. Think about reality shows like Big Brother or Survivor where people are constantly being observed and recorded. While they may be on their best behavior at first, in a fairly short amount of time they are flirting, having sex, wearing next to nothing, screaming at each other, and occasionally behaving in ways that are embarrassing.

Another approach to data collection in observational research is participant observation. In participant observation , researchers become active participants in the group or situation they are studying. As with naturalistic observation, the data that are collected can include interviews usually unstructured , notes based on their observations and interactions, documents, photographs, and other artifacts. The only difference between naturalistic observation and participant observation is that researchers engaged in participant observation become active members of the group or situations they are studying.

The basic rationale for participant observation is that there may be important information that is only accessible to, or can be interpreted only by, someone who is an active participant in the group or situation.

Like naturalistic observation, participant observation can be either disguised or undisguised. In disguised participant observation , the researchers pretend to be members of the social group they are observing and conceal their true identity as researchers.

In a famous example of disguised participant observation, Leon Festinger and his colleagues infiltrated a doomsday cult known as the Seekers, whose members believed that the apocalypse would occur on December 21, Interested in studying how members of the group would cope psychologically when the prophecy inevitably failed, they carefully recorded the events and reactions of the cult members in the days before and after the supposed end of the world.

Unsurprisingly, the cult members did not give up their belief but instead convinced themselves that it was their faith and efforts that saved the world from destruction. In contrast with undisguised participant observation , the researchers become a part of the group they are studying and they disclose their true identity as researchers to the group under investigation. Once again there are important ethical issues to consider with disguised participant observation.

First no informed consent can be obtained and second deception is being used. The researcher is deceiving the participants by intentionally withholding information about their motivations for being a part of the social group they are studying.

But sometimes disguised participation is the only way to access a protective group like a cult. Further, disguised participant observation is less prone to reactivity than undisguised participant observation. The staff and other patients were unaware of their true identities as researchers.

Another example of participant observation comes from a study by sociologist Amy Wilkins on a university-based religious organization that emphasized how happy its members were Wilkins, [8]. One of the primary benefits of participant observation is that the researchers are in a much better position to understand the viewpoint and experiences of the people they are studying when they are a part of the social group.

The primary limitation with this approach is that the mere presence of the observer could affect the behavior of the people being observed. Similarly, if the researcher acts as a participant observer there can be concerns with biases resulting from developing relationships with the participants.

Concretely, the researcher may become less objective resulting in more experimenter bias. Another observational method is structured observation. Here the investigator makes careful observations of one or more specific behaviors in a particular setting that is more structured than the settings used in naturalistic or participant observation. Often the setting in which the observations are made is not the natural setting. Instead, the researcher may observe people in the laboratory environment.

Alternatively, the researcher may observe people in a natural setting like a classroom setting that they have structured some way, for instance by introducing some specific task participants are to engage in or by introducing a specific social situation or manipulation.

Structured observation is very similar to naturalistic observation and participant observation in that in all three cases researchers are observing naturally occurring behavior; however, the emphasis in structured observation is on gathering quantitative rather than qualitative data.

Researchers using this approach are interested in a limited set of behaviors. This allows them to quantify the behaviors they are observing. In other words, structured observation is less global than naturalistic or participant observation because the researcher engaged in structured observations is interested in a small number of specific behaviors.

Therefore, rather than recording everything that happens, the researcher only focuses on very specific behaviors of interest. One of their measures involved observing pedestrians in a large city to see how long it took them to walk 60 feet. They found that people in some countries walked reliably faster than people in other countries. For example, people in Canada and Sweden covered 60 feet in just under 13 seconds on average, while people in Brazil and Romania took close to 17 seconds.

Levine and Norenzayan described their sampling process as follows:. Measurements were taken during main business hours on clear summer days. All locations were flat, unobstructed, had broad sidewalks, and were sufficiently uncrowded to allow pedestrians to move at potentially maximum speeds. To control for the effects of socializing, only pedestrians walking alone were used. Children, individuals with obvious physical handicaps, and window-shoppers were not timed.

Thirty-five men and 35 women were timed in most cities. Precise specification of the sampling process in this way makes data collection manageable for the observers, and it also provides some control over important extraneous variables. They simply measured out a foot distance along a city sidewalk and then used a stopwatch to time participants as they walked over that distance. The observers committed this list to memory and then practiced by coding the reactions of bowlers who had been videotaped.

During the actual study, the observers spoke into an audio recorder, describing the reactions they observed. Among the most interesting results of this study was that bowlers rarely smiled while they still faced the pins.

They were much more likely to smile after they turned toward their companions, suggesting that smiling is not purely an expression of happiness but also a form of social communication. In yet another example this one in a laboratory environment , Dov Cohen and his colleagues had observers rate the emotional reactions of participants who had just been deliberately bumped and insulted by a confederate after they dropped off a completed questionnaire at the end of a hallway.

The confederate was posing as someone who worked in the same building and who was frustrated by having to close a file drawer twice in order to permit the participants to walk past them first to drop off the questionnaire at the end of the hallway and once again on their way back to the room where they believed the study they signed up for was taking place.

The two observers were positioned at different ends of the hallway so that they could read the participants' body language and hear anything they might say. When the observations require a judgment on the part of the observers—as in the studies by Kraut and Johnston and Cohen and his colleagues—a process referred to as codingno post is typically required.

Coding generally requires clearly defining a set of target behaviors. The observers then categorize participants individually in terms of which behavior they have engaged in and the number of times they engaged in each behavior. The observers might even record the duration of each behavior. The target behaviors must be defined in such a way that guides different observers to code them in the same way. This difficulty with coding illustrates the issue of interrater reliability, as mentioned in Chapter 4.

Researchers are expected to demonstrate the interrater reliability of their coding procedure by having multiple raters code the same behaviors independently and then showing that the different observers are in close agreement.

One of the primary benefits of structured observation is that it is far more efficient than naturalistic and participant observation. Since the researchers are focused on specific behaviors this reduces time and expense. Also, often times the environment is structured to encourage the behaviors of interest which again means that researchers do not have to invest as much time in waiting for the behaviors of interest to naturally occur.

Finally, researchers using this approach can clearly exert greater control over the environment. However, when researchers exert more control over the environment it may make the environment less natural which decreases external validity.

It is less clear for instance whether structured observations made in a laboratory environment will generalize to a real world environment. Furthermore, since researchers engaged in structured observation are often not disguised there may be more concerns with reactivity. A case study is an in-depth examination of an individual.

Sometimes case studies are also completed on social units e. Most commonly in psychology, however, case studies provide a detailed description and analysis of an individual. Often the individual has a rare or unusual condition or disorder or has damage to a specific region of the brain. Like many observational research methods, case studies tend to be more qualitative in nature.

Case study methods involve an in-depth, and often a longitudinal examination of an individual. Depending on the focus of the case study, individuals may or may not be observed in their natural setting. Also, the bulk of the case study report will focus on in-depth descriptions of the person rather than on statistical analyses. With that said some quantitative data may also be included in the write-up of a case study.

For instance, an individual's depression score may be compared to normative scores or their score before and after treatment may be compared. When participants are not randomly assigned to conditions, however, the resulting groups are likely to be dissimilar in some ways. For this reason, researchers consider them to be nonequivalent. A nonequivalent groups design , then, is a between-subjects design in which participants have not been randomly assigned to conditions.

There are several types of nonequivalent groups designs we will consider. The first nonequivalent groups design we will consider is the posttest only nonequivalent groups design.

In this design, participants in one group are exposed to a treatment, a nonequivalent group is not exposed to the treatment, and then the two groups are compared. Imagine, for example, a researcher who wants to evaluate a new method of teaching fractions to third graders.

One way would be to conduct a study with a treatment group consisting of one class of third-grade students and a control group consisting of another class of third-grade students. This design would be a nonequivalent groups design because the students are not randomly assigned to classes by the researcher, which means there could be important differences between them.

For example, the parents of higher achieving or more motivated students might have been more likely to request that their children be assigned to Ms. Of course, researchers using a posttest only nonequivalent groups design can take steps to ensure that their groups are as similar as possible. In the present example, the researcher could try to select two classes at the same school, where the students in the two classes have similar scores on a standardized math test and the teachers are the same sex, are close in age, and have similar teaching styles.

Taking such steps would increase the internal validity of the study because it would eliminate some of the most important confounding variables. But without true random assignment of the students to conditions, there remains the possibility of other important confounding variables that the researcher was not able to control.

Another way to improve upon the posttest only nonequivalent groups design is to add a pretest. In the pretest-posttest nonequivalent groups design t here is a treatment group that is given a pretest, receives a treatment, and then is given a posttest. But at the same time there is a nonequivalent control group that is given a pretest, does not receive the treatment, and then is given a posttest. The question, then, is not simply whether participants who receive the treatment improve, but whether they improve more than participants who do not receive the treatment.

Imagine, for example, that students in one school are given a pretest on their attitudes toward drugs, then are exposed to an anti-drug program, and finally, are given a posttest. Students in a similar school are given the pretest, not exposed to an anti-drug program, and finally, are given a posttest. Again, if students in the treatment condition become more negative toward drugs, this change in attitude could be an effect of the treatment, but it could also be a matter of history or maturation.

If it really is an effect of the treatment, then students in the treatment condition should become more negative than students in the control condition. But if it is a matter of history e. This type of design does not completely eliminate the possibility of confounding variables, however. Something could occur at one of the schools but not the other e. The changes in scores from pretest to posttest would then be evaluated and compared across conditions to determine whether one group demonstrated a bigger improvement in knowledge of fractions than another.

Once again, differential history also represents a potential threat to internal validity. If asbestos is found in one of the schools causing it to be shut down for a month then this interruption in teaching could produce a difference across groups on posttest scores. If participants in this kind of design are randomly assigned to conditions, it becomes a true between-groups experiment rather than a quasi-experiment.