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©2011 Brooks/Cole, Cengage Learning Elementary Statistics: Looking at the Big Picture 1 Lecture 5: Chapter 4, Section 1 Single Variables (Focus on Categorical Variables) Displays and Summaries Data Production Issues Looking Ahead to Inference Details about Displays and Summaries
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©2011 Brooks/Cole, Cengage Learning Elementary Statistics: Looking at the Big Picture L5.2 Looking Back: Review 4 Stages of Statistics Data Production (discussed in Lectures 1-4) Displaying and Summarizing Single variables: 1 categorical, 1 quantitative Relationships between 2 variables Probability Statistical Inference
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©2011 Brooks/Cole, Cengage Learning Elementary Statistics: Looking at the Big Picture L5.3 First Process of Statistics SAMPLE POPULATION SAMPLE 1. Data Production: Take sample data from the population, with sampling and study designs that avoid bias.
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©2011 Brooks/Cole, Cengage Learning Elementary Statistics: Looking at the Big Picture L5.4 Second Process of Statistics POPULATION SAMPLE 2. Displaying and Summarizing: Use appropriate displays and summaries of the sample data, according to variable types and roles. 1. Data Production: Take sample data from the population, with sampling and study designs that avoid bias. CQ CQCQ CCCCQQQQ
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©2011 Brooks/Cole, Cengage Learning Elementary Statistics: Looking at the Big Picture L5.5 Third Process of Statistics: POPULATION SAMPLE PROBABILITY 3. Probability: Assume we know what’s true for the population; how should random samples behave? 2. Displaying and Summarizing: Use appropriate displays and summaries of the sample data, according to variable types and roles. 1. Data Production: Take sample data from the population, with sampling and study designs that avoid bias. C CQ QQQQ
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©2011 Brooks/Cole, Cengage Learning Elementary Statistics: Looking at the Big Picture L5.6 Fourth Process of Statistics 4. Statistical Inference: Assume we only know what’s true about sampled values of a single variable or relationship; what can we infer about the larger population? POPULATION SAMPLE PROBABILITY INFERENCE 3. Probability: Assume we know what’s true for the population; how should random samples behave? 2. Displaying and Summarizing: Use appropriate displays and summaries of the sample data, according to variable types and roles. 1. Data Production: Take sample data from the population, with sampling and study designs that avoid bias. CQ CQCQCCCCQQQQ
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©2011 Brooks/Cole, Cengage Learning Elementary Statistics: Looking at the Big Picture L5.7 Focus on Displaying and Summarizing
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©2011 Brooks/Cole, Cengage Learning Elementary Statistics: Looking at the Big Picture L5.8 Handling Single Categorical Variables Display: Pie chart Bar graph Summary: Count Percent Proportion
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©2011 Brooks/Cole, Cengage Learning Elementary Statistics: Looking at the Big Picture L5.9 Definitions and Notation Statistic: number summarizing sample Parameter: number summarizing population : sample proportion (a statistic) [“p-hat”] p: population proportion (a parameter)
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©2011 Brooks/Cole, Cengage Learning Elementary Statistics: Looking at the Big Picture L5.10 Example: Issues to Consider Background: 246 of 446 students at a certain university had eaten breakfast on survey day. Questions: Are intro stat students representative of all students at that university? Would they respond without bias? Responses: Yes, if years and class times are representative. Yes (not sensitive). Looking Back: These are data production issues. Practice: 4.4e p.79
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©2011 Brooks/Cole, Cengage Learning Elementary Statistics: Looking at the Big Picture L5.11 Example: More Issues to Consider Background: 246 of 446 students at a certain university had eaten breakfast on survey day. Questions: How do we display and summarize the info? Can we conclude that a majority of all students at that university eat breakfast? Responses: Display: pie chart Summary: 246/446= 55% or 0.55 ate breakfast. Can’t yet say if majority eat breakfast overall. Looking Ahead: This would be statistical inference.
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©2011 Brooks/Cole, Cengage Learning Elementary Statistics: Looking at the Big Picture L5.12 Example: Statistics vs. Parameters Background: 246 of 446 students at a certain university had eaten breakfast on survey day. Questions: Is 246/446=0.55 a statistic or a parameter? How do we denote it? Is the proportion of all students eating breakfast a statistic or a parameter? How do we denote it? Responses: 246/446=0.55 is a statistic denoted. Proportion of all students eating breakfast is a parameter denoted p. Practice: 4.3c p.79
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©2011 Brooks/Cole, Cengage Learning Elementary Statistics: Looking at the Big Picture L5.13 Example: Summary Issues Background: Location (state) for all 1,696 TV series in 2004 with known settings: 601 in California (601/1696=0.35) 412 in New York (412/1696=0.24) 683 in other states (683/1696=0.40) Questions: 0.35+0.24+0.40=0.99 mistake? Why is it not appropriate to use this info to draw conclusions about a larger population in 2004? Responses: No mistake; roundoff error There is no larger population (data is for all series in 2004).
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©2011 Brooks/Cole, Cengage Learning Elementary Statistics: Looking at the Big Picture L5.14 Definitions Mode: most common value Majority: more common of two possible values (same as mode) Minority: less common of two possible values
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©2011 Brooks/Cole, Cengage Learning Elementary Statistics: Looking at the Big Picture L5.15 Two or More Possible Values Looking Ahead: In Probability and Inference, most categorical variables discussed have just two possibilities. Still, we often summarize and display categorical data with more than two possibilities.
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©2011 Brooks/Cole, Cengage Learning Elementary Statistics: Looking at the Big Picture L5.16 Example: Proportions in Three Categories Background: Student wondered if she should resist changing answers in multiple choice tests. “Ask Marilyn” replied: 50% of changes go from wrong to right 25% of changes go from right to wrong 25% of changes go from wrong to wrong Question: How to display information? Response: Use a pie chart: Instructor may survey students: To which category do your changes typically belong ?
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©2011 Brooks/Cole, Cengage Learning Elementary Statistics: Looking at the Big Picture L5.17 Definition Bar graph: shows counts, percents, or proportions in various categories (marked on horizontal axis) with bars of corresponding heights
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©2011 Brooks/Cole, Cengage Learning Elementary Statistics: Looking at the Big Picture L5.18 Example: Bar Graph Background: Instructor can survey students to find proportion in each year (1 st, 2 nd, 3 rd, 4 th, Other). Questions: How can we display the information? What should we look for in the display? Responses: Construct a bar graph: years 1, 2, 3, 4, Other on horizontal axis counts or proportions in each year graphed vertically Look for mode (tallest bar) to tell what year is most common; compare heights of all 5 bars. Practice: 4.6a p.80
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©2011 Brooks/Cole, Cengage Learning Elementary Statistics: Looking at the Big Picture L5.19 Example: Bar Graph Background: Instructor can survey students to find proportion in each year (1 st, 2 nd, 3 rd, 4 th, Other). Questions: How can we display the information? What should we look for in the display? Responses: For the particular sample whose data accompany the book, mode is 2 nd year, next is 3 rd, then 1 st and 4 th, then Other. Practice: 4.6a p.80
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©2011 Brooks/Cole, Cengage Learning Elementary Statistics: Looking at the Big Picture L5.20 Overlapping Categories If more than two categorical variables are considered at once, we must note the possibility that categories overlap. Looking Ahead: In Probability, we will need to distinguish between situations where categories do and do not overlap.
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©2011 Brooks/Cole, Cengage Learning Elementary Statistics: Looking at the Big Picture L5.21 Example: Overlapping Categories Background: Report by ResumeDoctor.com on over 160,000 resumes: 13% said applicant had “communication skills” 7% said applicant was a “team player” Question: Can we conclude that 20% claimed communication skills or team player? Response: No: adding 13%+7% counts those in both categories twice.
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©2011 Brooks/Cole, Cengage Learning Elementary Statistics: Looking at the Big Picture L5.22 Processing Raw Categorical Data Small categorical data sets are easily handled without software.
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©2011 Brooks/Cole, Cengage Learning Elementary Statistics: Looking at the Big Picture L5.23 Example: Proportion from Raw Data Background: Harvard study claimed 44% of college students are binge drinkers. Agree on survey design and have students self-report: on one occasion in past month, alcoholic drinks more than 5 (males) or 4 (females)? Or use these data: Question: Are data consistent with claim of 44%? Response: % “yes”=28/66=42%, very close to 44% Practice: 4.7a p.80
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©2011 Brooks/Cole, Cengage Learning Elementary Statistics: Looking at the Big Picture L5.24 Example: Proportion from Raw Data Background: Harvard study claimed 44% of college students are binge drinkers. Agree on survey design and have students self-report: on one occasion in past month, alcoholic drinks more than 5 (males) or 4 (females)? Or use these data: Looking Ahead: How different would the sample percentage have to be, to convince you that your sample is significantly different from Harvard’s? This is an inference question. Question: Are data consistent with claim of 44%? Response: % “yes”=28/66=42%, very close to 44% Practice: 4.7a p.80
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©2011 Brooks/Cole, Cengage Learning Elementary Statistics: Looking at the Big Picture L5.25 Lecture Summary (Categorical Variables) Display: pie chart, bar graph Summarize: count, percent, proportion Sampling: data unbiased (representative)? Design: produced unbiased summary of data? Inference: will we ultimately draw conclusion about population based on sample? Mode, Majority: most common values Larger samples: provide more info Other issues: Two or more possibilities? Categories overlap? How to handle raw data?
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