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Solved: Prove or disprove each of these statements about

Discrete Mathematics and Its Applications | 7th Edition | ISBN: 9780073383095 | Authors: Kenneth Rosen ISBN: 9780073383095 37

Solution for problem 74E Chapter 2.3

Discrete Mathematics and Its Applications | 7th Edition

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Discrete Mathematics and Its Applications | 7th Edition | ISBN: 9780073383095 | Authors: Kenneth Rosen

Discrete Mathematics and Its Applications | 7th Edition

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Problem 74E

Prove or disprove each of these statements about the floor and ceiling functions.

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Tuesday, April 19, 2016 Week 4 Lecture 7 - April 19, 2016 - Reminder: Midterm next Tuesday • Group office hours: Thursday, April 21 from 1:30-2:30 (SSMS 4105) • TA Q&A session (SSMS 1009) on Monday, April 25, TBA • Review guide and practice questions on GauchoSpace (and seating chart for midterm) - Measurement: Operationalizing Variables (both IVs and DVs) - Recall from last time… • Using Questionnaire Items as Measures - Common for IVs and DVs in surveys - Common for DVs in experiments (IV is a manipulation into groups) • Types of Questionnaire Items - Open-ended • Respondents give their own answers to Qs - Closed-ended • Respondents select from list of choices - Choices must be mutually exclusive! (Can’t choose more than one) - Choices must be exhaustive! (There is an option for any attitude/opinion a person may have) • Some closed-ended formats: - Likert-Type items: • Respondents indicate their agreement with a particular statement - Example: “I feel uncomfortable when people are arguing.” Strongly disagree = 1, Strongly agree = 5 - Other response options also possible (oppose/favor; not at all/very much; almost never/almost always) 1 Tuesday, April 19, 2016 - Semantic Differential • Respondents make ratings between two opposite (bipolar) adjectives - Example: This candidate seems: honest 7:6:5:4:3:2:1 dishonest - Composite Measures • Use multiple items combined to measure one variable (i.e., create an index/scale) - Example: This candidate seems: honest 7:6:5:4:3:2:1 dishonest, trustworthy 7:6:5:4:3:2:1 untrustworthy, unbiased 7:6:5:4:3:2:1 biased - Add together (or average) scores on all three items into an overall credibility score - How Good Is Your Measurement Reliability and Validity • - Textbook note: SKIP pages 188-193 (save for final) • Reliability of Measurement - Does your measure (of the variable) have consistency • Assessing Reliability - For measures using questionnaire items: • Inter-item reliability (consistency between items) - Look at internal consistency of similar items in a scale/index • Want a high “item-total” score - Example: Candidate credibility - honest 7:6:5:4:3:2:1 dishonest - trustworthy 7:6:5:4:3:2:1 untrustworthy - unbiased 7:6:5:4:3:2:1 biased - funny 7:6:5:4:3:2:1 not funny 2 Tuesday, April 19, 2016 - What’s wrong with the “funny” item Doesn’t go with the rest; will lower the reliability of the scale. If you remove the funny item, then all the items are indicators of the SAME basic quality of the candidate (credibility/trustworthiness) and will probably have a high internal consistency (high item-total score)…so, a good “Uni- dimentional” variable/concept - What if credibility involves more than just trustworthiness (like expertise) • Would need “Multi-dimensional” scale: - honest 7:6:5:4:3:2:1 dishonest - trustworthy 7:6:5:4:3:2:1 untrustworthy - unbiased 7:6:5:4:3:2:1 biased - knowledgeable 7:6:5:4:3:2:1 not knowledgeable - experienced 7:6:5:4:3:2:1 inexperienced - competent 7:6:5:4:3:2:1 incompetent • Multidimensional “credibility”: 2 dimensions - trust + honest + unbiased = “trustworthiness” dimension - knowledge + experience + competence = “expertise” dimension - —> Evaluate reliability separately for each sub-scale - For measures using coders (e.g., behavioral observations): Inter-coder reliability • - Compare multiple coders • Intra-coder reliability - Compare multiple observations of the same coder • Validity of Measurement - Does your measure really capture the concept you intend to be measuring • Assessing Validity 3 Tuesday, April 19, 2016 - Subjective types of validity: • Face Validity - The measure looks/sounds good “on the face of it” • Content Validity - The measure captures the full range of meanings/dimensions of the concept • Criterion-related/predictive validity: - The measure is shown to predict scores on an appropriate future measure • Ex: SAT scores (your “potential” to achieve) —> college GPA (your achievement) • Construct validity: - The measure is shown to be related to measures of other concepts that should be related (and not to ones that shouldn’t) • Ex: aggression scale hostility scale Lecture 8 - April 21, 2016 - Sampling • How to select participants for a study - Or other units • Sample-subset of target population - Voters, FB users, married couples, juries, football fans, etc. - Or TV shows, magazine ads, blog posts, etc. • Sampling units - Individual persons - Groups - Social artifacts (ads, TV scenes, tweets, etc.) - Representative Sampling • *NOTE: NEED TO KNOW REPRESENTATIVE AND NON-REPRESENTATIVE 4 Tuesday, April 19, 2016 • Probability Sampling • Sample should be mini-versos of target population - Equal sample proportions to target populations (Democrats, Republicans, men, women, etc.) - The larger the sample, the more representative it is normally - Allows you to generalize for the larger population - The key is random selection: • Everyone in population has to have an equal chance of being in the sample • How representative is it - You won’t always be dead on with your findings - There is always sampling error • Sample data will be slightly different from population because of chance alone • Random error/chance error/sample error • Statistically called margin of error - National poll ~1,000 people —> +/- 3% • Larger sample size = smaller margin of error - After about 1,000 people, your margin of error plateaus and doesn’t change much - Representative Sampling Techniques • Simple random sampling - select elements randomly from population - Assign each element a number and randomly select a number • Listed populations - random # from table • Phones - random digit dialing • Systematic random sampling - form a list of population and take random starting point, pick an interval, select every “nth” person off that list, cycle through the entire list - Similar results to simple random sampling 5 Tuesday, April 19, 2016 - Close to right proportions, but some sampling error • Stratified Sampling - for getting population proportions even more accurate - Divide population into subsets (strata) of particular variables • Usually for demographic variables (sex, race, political party) • Anything where you think groups will have different opinions/views on something - Randomly select from each strata to get right proportions of population • Need prior knowledge of population proportions - Increases representativeness because it reduces sampling error • But very time consuming Multistage Cluster Sampling - useful for populations not listed as individuals • - First randomly sample groups (clusters) then randomly sample elements within each cluster - Reduces costs - Sampling error at each stage and margin of error goes up - All Representative Sampling: • Will always have sampling error, but can generalize findings to larger target population (assuming the sampling is done properly) • What to avoid: - Systematic error (sampling bias) • Systematically over/under representing certain segments of a population - Caused by: • Wrong sampling face (where you are finding people) • Very low response rate • Importer weighting (wrong proportions in stratified sample) - Non-Representative Sampling: *READ ABOUT THESE IN BOOK • Cannot generalize results 6 Tuesday, April 19, 2016 • Typical of experiments and qualitative research Section - April 21, 2016 - Sampling • Subset of the population • Representative sampling - Random selection, sampling error - Simple Random, Systematic, Stratified, multistage closet • Sampling bias • Non-representative sampling - Convenience, purposive, volunteer, quota, network/snowball • Convenience sample: choosing people close to you/easily available to sample from, nearby • Purposive sample: purpose exists behind why you choose a certain sample, usually related to a specific reason as to why you are looking at the purposive sample in a certain way (people in a relationship, homosexual people, etc.) • Volunteer sample: sample members volunteer to take part in the study, people choose to participate for some reason (an incentive/reward is sometimes the reason) - *NOTE: ALL studies are voluntary, but not all studies use volunteer samples • Quota sample: selecting individuals to match a certain demographic, similar to a stratified sample but not the same - Taking a certain number of people who match a certain criteria needed for the study, finding a certain number of people to match the demographic RANDOMLY (asking a random person who is African American to participate just because they are African American) • Network/snowball sample: taking a certain participant and asking if they know anyone else who would like to/be able to participate in the study, asking if the participant knows anyone else who is in the same situation/criteria needed for the study 7 Tuesday, April 19, 2016 - Upsides of Non-Representative sampling: • Easier, cheaper • Not necessarily “bad”, just different - Remember, it IS NOT generalizable - Review of homework exercise: Hypothesizing Relationships Between Variables • Causality (cause and effect) MUST be found by an experiment, not a survey/ observation - Midterm Review - Brief Overview of Topics • Ways of Knowing (Epistemology) - “Everyday” ways of knowing Tradition/Tenacity • • Authority • Experience • Intuition/Logic - Problem overall • Overgeneralization - Science! • Characteristics of science (public, peer-review, etc.) • Goals of scientific research - Description, explanation, prediction • Research Process - The Wheel of Science Going from theory to observation is deduction, while going from observation to • theory is induction - Quantitative vs. Qualitative research, Basic vs. Applied - Theory in research 8 Tuesday, April 19, 2016 • Concepts and relationships - Relationships between variables • Associational vs. Causal • Survey vs. Experiment • Variables - Independent vs Dependent Variables Conceptual and operational definitions • - Measuring variables • Types of measures, levels of measurement - Questionnaire items • Open vs. closed ended items • Composite measures • Reliability and Validity • Sampling - Subset of the population - Representative sampling • Random selection, sampling error • Simple Random, Systematic, Stratified, multistage cluster - Sampling bias - Non-Representative sampling • Convenience, purposive, volunteer, quota, network/snowball 9

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Chapter 2.3, Problem 74E is Solved
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Textbook: Discrete Mathematics and Its Applications
Edition: 7
Author: Kenneth Rosen
ISBN: 9780073383095

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