Tutor profile: Nikki T.
Translate the following statement into Predicate Logic: “No manager who earns more than some staff member is eligible to apply for any of the job openings.” Explain each step in translating from English to Predicate Logic as it applies to the statement above.
Translation: (∀x)[(Mx & (∃y)(Sy & Exy)) -> (∀z)(Jz -> ~Lxz)] Step 1: Identify the statement as an A, E, I, or O statement (A: All Ss are Ps, E: No Ss are Ps, I: Some Ss are Ps, O: Some Ss are not Ps). This is an E-statement. Step 2: Set up the corresponding structure. Since we have an E-statement, the structure is '(∀x)(_____-> ~_____)' Step 3: Fill in the blanks with English phrases. In this case, we get the following (∀x)[(x is a manager who earns more than some staff member) -> (x is not eligible to apply for any of the job openings.)] Step 4: Translate what you wrote in the blanks into Predicate Logic. (∀x)[(Mx and x earns more than some staff member) -> (x is not eligible to apply for any of the job openings.)] (∀x)[(Mx & (∃y)(y is the staff member and x earns more than y)) -> (x is not eligible to apply for any of the job openings.)] (∀x)[(Mx & (∃y)(Sy & Exy)) -> (x is not eligible to apply for any of the job openings.)] (∀x)[(Mx & (∃y)(Sy & Exy)) -> (∀z)(z is a job opening -> x is not eligible for z)] (∀x)[(Mx & (∃y)(Sy & Exy)) -> (∀z)(Jz -> ~Lxz)]
In 1959, Festinger & Carlsmith conducted a study on Cognitive Dissonance: Subject A and Subject B are waiting to be called in by the researchers to participate in a study. Subject A is called first. Subject A is asked to do a silly, and very uninteresting task for a duration of one hour. Once the task is completed, Subject A is asked to return to the waiting room and tell Subject B that she will really enjoy the task. Subject A is paid either $1 or $20 to tell this lie (this was 1959, so $20.00 was a lot of money!). The experiment was repeated with many subjects, splitting "Subject A's" into two groups, those who were paid $1 to lie, and those who were paid $20 to lie. Later on, the "Subject A's" were asked how much they actually enjoyed the task. The results showed that subjects in the $1 condition most often answered by saying they really did enjoy the task. Subjects in the $20 condition most often answered by saying they really did not enjoy the task at all. Explain the results of this study.
The results of the study can be explained by Insufficient Justification. Insufficient Justification is a phenomenon that occurs when individuals are experiencing Cognitive Dissonance. Cognitive Dissonance is a highly aversive mental state that occurs when two of our cognitions are in conflict with one another. For example, when one's behavior does not align with their beliefs, this is likely to result in Cognitive Dissonance. There are two ways to reduce this dissonance. First, we can attempt to justify our behavior. If we can't find a justification, then we change our attitude/cognition/belief. Insufficient Justification occurs when we are unable to justify our behavior with some external cause. In this study, subjects in the $20 condition were able to justify lying because they would receive a large sum of money for doing so. Subjects in the $1 condition could not use an external justification to resolve their cognitive dissonance, as $1 is not enough for most people to justify lying and is, therefore, an insufficient external justification. So, participants in the $1 condition had to instead justify this behavior internally by changing their attitude/cognition. By believing that they really did enjoy the task, subjects in the $1 condition were able to reduce their dissonance internally.
Assume that we have data on a population of individuals. We know that, in this population, individuals are exposed to 7,000 advertisements per day on average. The population standard deviation of advertisement exposure is 1,500. Exposure to advertisements per day is normally distributed. There are two researchers who conduct studies where they sample individuals from this population and record how many advertisements they are exposed to each day. Researcher A samples 40 individuals for their study. Researcher B samples 80 individuals for their study. I. Is it more likely that the sample average will be 6500 ads per day or fewer for Researcher A's study, or for Researcher B's study? II. Which study (Researcher A's study or Researcher B's study) will have a lower standard error if we assume that they are sampling from the same population (a population with the same population mean and standard deviation)?
I. Researcher A (N = 40) The standard error for a sampling distribution (not a sample distribution!) of a smaller sample is more spread out. So, more of the distribution is likely to be less than 6500 if the sample is smaller (N=40 vs N=80). II. Researcher B (N = 80) The formula for standard error is influenced by sample size N (the standard deviation divided by the square root of N). As N increases, the standard error decreases. 80 (Researcher B's N) is greater than 40 (Researcher A's N), so Researcher B will have the smaller standard error.
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