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Teaching Statistical Concepts with Activities, Data, and Technology Beth L. Chance and Allan J. Rossman Dept of Statistics, Cal Poly – San Luis Obispo

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Goals Acquaint you with recent recommendations and ideas for teaching introductory statistics Including some very “modern” approaches On top of some issues we consider essential Provide specific examples and activities that you might plug into your courses Point you toward online and print resources that might be helpful APSA Conference, Sept 20102

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Schedule Introductions Opening Activity Activity Sessions Data Collection Data Analysis > Randomness Statistical Inference Resources and Assessment Q&A, Wrap-Up APSA Conference, Sept 20103

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Requests Participate in activities 23 of them! We’ll skip/highlight some Play role of student Good student, not disruptive student! Feel free to interject comments, questions APSA Conference, Sept 20104

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GAISE Emphasize statistical literacy and develop statistical thinking Use real data Stress conceptual understanding rather than mere knowledge of procedures Foster active learning in the classroom Use technology for developing conceptual understanding and analyzing data Use assessments to improve and evaluate student learning APSA Conference, Sept 20105

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Opening Activity Naughty or nice? (Nature, 2007) Videos: Helper-Hinderer.html Helper-Hinderer.html Flip 16 coins, one for each infant, to decide which toy you want to play with (heads=helper) Coin Tossing Applet: APSA Conference, Sept 20106

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3S Strategy Statistic Simulate “Could have been” distribution of data for each repetition (under null model) “What if” distribution of statistics across repetitions (under null model) Strength of evidence Reject vs. plausible APSA Conference, Sept 20107

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Summary Use real data/scientific studies Emphasize the process of statistical investigation Stress conceptual understanding Idea of p-value on day 1/in one day! Foster active learning You are a dot on the board Use technology Could this have happened “by chance alone”? What if only 10 infants had picked the helper? APSA Conference, Sept 20108

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Data Collection Activities: Activity 2: Sampling Words Circle 10 representative words in the passage Record the number of letters in each word Calculate the mean number of letters in your sample Dotplot of results… APSA Conference, Sept 20109

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Sampling Words The population mean of all 268 words is letters How many sample means were too high? Why do you think so many sample means are too high? APSA Conference, Sept

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Sampling Words “Tactile” simulation Ask students to use computer or random number table to take simple random samples Determine the sample mean in each sample Compare the distributions APSA Conference, Sept

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Sampling Words Java applet Select “Sampling words” applet Select individual sample of 5 words Repeat Select 98 more samples of size 5 Explore the effect of sample size Explore the effect of population size APSA Conference, Sept

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Morals: Selecting a Sample Random Sampling eliminates human selection bias so the sample will be fair and unbiased/representative of the population. While increasing the sample size improves precision, this does not decrease bias. APSA Conference, Sept

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Activity 3: Night Lights and Near-Sightedness Quinn, Shin, Maguire, and Stone (1999) 479 children Did your child use a night light (or room light or neither) before age 2? Eyesight: Hyperopia (far-sighted), emmetropia (normal) or myopia (near- sighted)? APSA Conference, Sept

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Night Lights and Near-Sightedness DarknessNight lightRoom light Near- sighted Normal refraction Far-sighted APSA Conference, Sept

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Night Lights and Near-Sightedness APSA Conference, Sept

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Morals: Confounding Students can tell you that association is not the same as causation! Need practice clearly describing how confounding variable Is linked to both explanatory and response variables Provides an alternative explanation for observed association APSA Conference, Sept

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Activity 4: Have a Nice Trip Can instruction in a recovery strategy improve an older person’s ability to recover from a loss of balance?loss of balance 12 subjects have agreed to participate in the study Assign 6 people to use the lowering strategy and 6 people to use the elevating strategyelevating What does “random assignment” gain you? APSA Conference, Sept

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Have a Nice Trip Randomizing subjects applet How do the two groups compare? APSA Conference, Sept

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Morals Goal of random assignment is to be willing to consider the treatment groups equivalent prior to the imposition of the treatment(s). This allows us to eliminate all potential confounding variables as a plausible explanation for any significant differences in the response variable after the treatments are imposed. APSA Conference, Sept

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Activity 5: Cursive Writing Does using cursive writing cause students to score better on the SAT essay? APSA Conference, Sept

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Morals: Scope of Conclusions Allocation of units to groups By random assignmentNo random assignment Selection of units Random sampling A random sample is selected from one population; units are then randomly assigned to different treatment groups Random samples are selected from existing distinct populations Inferences to populations can be drawn Not random sampling A groups of study units is found; units are then randomly assigned to treatment groups Collections of available units from distinct groups are examined Cause and effect conclusions can be drawn The Statistical Sleuth, Ramsey and Schafer APSA Conference, Sept

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Activity 6: Memorizing Letters You will be asked to memorize as many letters as you can in 20 seconds, in order, from a sequence of 30 letters Variables? Type of study? Comparison? Random assignment? Blindness? Random sampling? More to come … APSA Conference, Sept

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Morals: Data Collection Quick, simple experimental data collection Highlighting critical aspects of effective study design Can return to the data several times in the course APSA Conference, Sept

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Data Analysis Activities Activity 7: Matching Variables to Graphs Which dotplot belongs to which variable? Justify your answer APSA Conference, Sept

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Morals: Graph-sense Learn to justify opinions Consistency, completeness Appreciate variability Be able to find and explain patterns in the data APSA Conference, Sept

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Activity 8: Rower Weights 2008 Men’s Olympic Rowing Team APSA Conference, Sept

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Rower Weights APSA Conference, Sept MeanMedian Full Data Set Without Coxswain Without Coxswain or lightweight rowers With heaviest at With heaviest at Resistance....

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Morals: Rower Weights Think about the context “Data are numbers with a context”-Moore Know what your numerical summary is measuring Investigate causes for unusual observations Anticipate shape APSA Conference, Sept

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Activity 9: Cancer Pamphlets Researchers in Philadelphia investigated whether pamphlets containing information for cancer patients are written at a level that the cancer patients can comprehend APSA Conference, Sept

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Cancer Pamphlets APSA Conference, Sept

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Morals: Importance of Graphs Look at the data Think about the question Numerical summaries don’t tell the whole story “median isn’t the message” - Gould APSA Conference, Sept

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Activity 10: Draft Lottery Draft numbers (1-366) were assigned to birthdates in the 1970 draft lottery Find your draft number Any 225s? APSA Conference, Sept

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Draft Lottery APSA Conference, Sept

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monthmedian January February March April May June month median July August September October November December Draft Lottery APSA Conference, Sept

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Draft Lottery APSA Conference, Sept

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Morals: Statistics matters! Summaries can illuminate Randomization can be difficult APSA Conference, Sept

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Activity 11: Televisions and Life Expectancy Is there an association between the two variables? So sending televisions to countries with lower life expectancies would cause their inhabitants to live longer? r =.743 APSA Conference, Sept

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Morals: Confounding Don’t jump to conclusions from observational studies The association is real but consider carefully the interpretation of graph and wording of conclusions (and headlines) APSA Conference, Sept

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Activity 6 Revisited (Memorizing Letters) Produce, interpret graphical displays to compare performance of two groups Does research hypothesis appear to be supported? Any unusual features in distributions? APSA Conference, Sept

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Lunch! Questions? Write down and submit any questions you have thus far on the statistical or pedagogical content… APSA Conference, Sept

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Exploring Randomness Activity 12: Random Babies Last NamesFirst Names JonesJerry MillerMarvin SmithSam WilliamsWilly APSA Conference, Sept

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Random Babies Last NamesFirst Names JonesMarvin Miller Smith Williams APSA Conference, Sept

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Random Babies Last NamesFirst Names JonesMarvin MillerWilly Smith Williams APSA Conference, Sept

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Random Babies Last NamesFirst Names JonesMarvin MillerWilly SmithSam Williams APSA Conference, Sept

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Last NamesFirst Names JonesMarvin MillerWilly SmithSam WilliamsJerry Random Babies APSA Conference, Sept

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Last NamesFirst Names JonesMarvin MillerWilly SmithSam 1 match WilliamsJerry Random Babies APSA Conference, Sept

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Random Babies Long-run relative frequency Applet: “Random Babies” APSA Conference, Sept

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Random Babies: Mathematical Analysis APSA Conference, Sept

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Random Babies APSA Conference, Sept

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0 matches: 9/24=3/8 1 match: 8/24=1/3 2 matches: 6/24=1/4 3 matches: 0 4 matches: 1/24 Random Babies APSA Conference, Sept

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Goal: Interpretation in terms of long-run relative frequency, average value 30% chance of rain… First simulate, then do theoretical analysis Able to list sample space Short cuts when are actually equally likely Simple, fun applications of basic probability Morals: Treatment of Probability APSA Conference, Sept

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ELISA test used to screen blood for the AIDS virus Sensitivity: P(+|AIDS)=.977 Specificity: P(-|no AIDS)=.926 Base rate: P(AIDS)=.005 Find P(AIDS|+) Initial guess? Bayes’ theorem? Construct a two-way table for hypothetical population Activity 13: AIDS Testing APSA Conference, Sept

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Positive Negative Total AIDS No AIDS Total 1,000,000 AIDS Testing APSA Conference, Sept

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Positive Negative Total AIDS 5,000 No AIDS 995,000 Total 1,000,000 AIDS Testing APSA Conference, Sept

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Positive Negative Total AIDS ,000 No AIDS 995,000 Total 1,000,000 AIDS Testing APSA Conference, Sept

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Positive Negative Total AIDS ,000 No AIDS73,630921, ,000 Total 1,000,000 AIDS Testing APSA Conference, Sept

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Positive Negative Total AIDS ,000 No AIDS73,630921, ,000 Total78,515921,485 1,000,000 AIDS Testing APSA Conference, Sept

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Positive Negative Total AIDS ,000 No AIDS73,630921, ,000 Total78,515921,485 1,000,000 P(AIDS|+) = 4885/78,515=.062 AIDS Testing APSA Conference, Sept

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Positive Negative Total AIDS ,000 No AIDS73,630921, ,000 Total78,515921,485 1,000,000 P(AIDS|+) = 4885/78,515=.062 P(No AIDS|-) = 921,370/921,485 = AIDS Testing APSA Conference, Sept

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Intuition about conditional probability can be very faulty Confront misconception head-on Conditional probability can be explored through two-way tables Treatment of formal probability can be minimized Morals: Surprise Students! APSA Conference, Sept

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Activity 14: Reese’s Pieces APSA Conference, Sept

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Take sample of 25 candies Sort by color Calculate the proportion of orange candies in your sample Construct a dotplot of the distribution of sample proportions Reese’s Pieces APSA Conference, Sept

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Turn over to technology Reeses Pieces applet Reeses Pieces applet (www.rossmanchance.com/applets/) Reese’s Pieces APSA Conference, Sept

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Study randomness to develop intuition for statistical ideas Not probability for its own sake Always precede technology simulations with physical ones Apply more than derive formulas Morals: Sampling Distributions APSA Conference, Sept

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Left FrontRight Front Left RearRight Rear Activity 15: Which Tire? APSA Conference, Sept

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People tend to pick “right front” more than ¼ of the time Variable = which tire pick Categorical (binary) How often would we get data like this by chance alone? Determine the probability of obtaining at least as many “successes” as we did if there were nothing special about this particular tire. Which Tire? APSA Conference, Sept

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Let = proportion of all … who pick right front H 0 : =.25 H a : >.25 Test statistic z = p-value = Pr(Z>z) How does this depend on n? Test of Significance Calculator Which Tire? APSA Conference, Sept

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nz-statisticp-value … Which Tire? APSA Conference, Sept

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Fun simple data collection Effect of sample size hard to establish result with small samples Never “accept” null hypothesis Morals: Formal Statistical Inference APSA Conference, Sept

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Activity 16: Kissing the Right Way Biopsychology observational study Güntürkün (2003) recorded the direction turned by kissing couples to see if there was also a right- sided dominance. APSA Conference, Sept

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Kissing the Right Way Is 1/2 a plausible value for the probability a kissing couple turns right? Coin Tossing applet Is 2/3 a plausible value for the probability a kissing couple turns right? Is the observed result in the tail of the “what if” distribution? APSA Conference, Sept

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Kissing the Right Way Determine the plausible values for the probability a kissing couple turns right… The values that produce an approximate p- value greater than.05 are not rejected and are therefore considered plausible values of the parameter. The interval of plausible values is sometimes called a confidence interval for the parameter. APSA Conference, Sept

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Kissing the Right Way How does this compare to estimate + margin of error? Or the even simpler approximation? APSA Conference, Sept

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Morals: Kissing the Right Way Interval estimation as (more?) important as significance Confidence interval as set of plausible (not rejected) values Interpretation of margin-of-error APSA Conference, Sept

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Activity 17: Reese’s Pieces Revisited Calculate 95% confidence interval for from your sample proportion of orange Does everyone have same interval? Does every interval necessarily capture ? What proportion of class intervals would you expect? Simulating Confidence Intervals applet What percentage of intervals succeed? Change confidence level, sample size APSA Conference, Sept

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Morals: Reese’s Pieces Revisited Interpretation of confidence level In terms of long-run results from taking many samples Effects of confidence level, sample size on confidence interval APSA Conference, Sept

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78 Example 18: Dolphin Therapy Subjects who suffer from mild to moderate depression were flown to Honduras, randomly assigned to a treatment APSA Conference, Sept

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Dolphin Therapy Is dolphin therapy more effective than control? Core question of inference: Is such an extreme difference unlikely to occur by chance (random assignment) alone (if there were no treatment effect)? APSA Conference, Sept

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80 Some approaches Could calculate test statistic, p-value from approximate sampling distribution (z, chi-square) But it’s approximate But conditions might not hold But how does this relate to what “significance” means? Could conduct Fisher’s Exact Test But there’s a lot of mathematical start-up required But that’s still not closely tied to what “significance” means Even though this is a randomization test APSA Conference, Sept

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81 3S Approach Simulate random assignment process many times, see how often such an extreme result occurs Assume no treatment effect (null model) Re-randomize 30 subjects to two groups (using cards) Assuming 13 improvers, 17 non-improvers regardless Determine number of improvers in dolphin group Or, equivalently, difference in improvement proportions Repeat large number of times (turn to computer) Ask whether observed result is in tail of what if distribution Indicating saw a surprising result under null model Providing evidence that dolphin therapy is more effective APSA Conference, Sept

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82 Analysis Dolphin Study applet APSA Conference, Sept

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83 Conclusion Experimental result is statistically significant And what is the logic behind that? Observed result very unlikely to occur by chance (random assignment) alone (if dolphin therapy was not effective) APSA Conference, Sept

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Morals Re-emphasize meaning of significance and p-value Use of randomness in study Focus on statistical process, scope of conclusions APSA Conference, Sept

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85 Activity 19: Sleep Deprivation Does sleep deprivation have harmful effects on cognitive functioning three days later? 21 subjects; random assignment Core question of inference: Is such an extreme difference unlikely to occur by chance (random assignment) alone (if there were no treatment effect)? APSA Conference, Sept

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86 Sleep Deprivation Simulate randomization process many times under null model, see how often such an extreme result (difference in group medians or means) occurs Start with tactile simulation using index cards Write each “score” on a card Shuffle the cards Randomly deal out 11 for deprived group, 10 for unrestricted group Calculate difference in group medians (or means) Repeat many times (Randomization Tests applet) APSA Conference, Sept

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Sleep Deprivation Conclusion: Fairly strong evidence that sleep deprivation produces lower improvements, on average, even three days later Justification: Experimental results as extreme as those in the actual study would be quite unlikely to occur by chance alone, if there were no effect of the sleep deprivation APSA Conference, Sept

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Exact randomization distribution Exact p-value 2533/ =.0072 (for difference in means) APSA Conference, Sept

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Morals: Randomizations Tests Emphasizes core logic of inference Takes advantage of modern computing power Easy to generalize to other statistics APSA Conference, Sept

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Activity 6 Revisited (Memorizing Letters) Conduct randomization test to assess strength of evidence in support of research hypothesis Enter data into applet Summarize conclusion and reasoning process behind it Does non-significant result indicate that grouping of letters has no effect? APSA Conference, Sept

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Activity 20: Cat Households 47,000 American households (2007) 32.4% owned a pet cat or the other way around! test statistic: z=-4.29 p-value: virtually zero 99% CI for (.31844,.32956) APSA Conference, Sept

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Morals: Limits of statistical significance Statistical significance is not practical significance Especially with large sample sizes Accompany significant tests with confidence intervals whenever possible APSA Conference, Sept

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Activity 21: Female Senators 17 women, 83 men in % CI for : = = (.096,.244) APSA Conference, Sept

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Always consider sampling procedure Randomness is key assumption Garbage in, garbage out Inference is not always appropriate! Sample = population here Morals: Limitations of Inference APSA Conference, Sept

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Activity 22: Game Show Prices Sample of 208 prizes from The Price is Right Examine a histogram 99% confidence interval for the mean Technical conditions? What percentage of the prizes fall in this interval? Why is this not close to 99%? APSA Conference, Sept

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Morals: Cautions/Limitations Prediction intervals vs. confidence intervals Constant attention to what the “it” is APSA Conference, Sept

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Activity 23: Government Spending 2004 General Social Survey: Is there an association between American adults’ opinion on federal government spending on the environment and political inclinations? APSA Conference, Sept

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Government Spending Descriptive analysis APSA Conference, Sept LiberalModerateConservativeTotal Too Much About Right Too Little Total

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Government Spending Inferential analysis – 3S approach 1. Chi-square statistic 2. Simulate sampling distribution of chi-square test statistic under null hypothesis of no association Randomly mix up political inclinations, determine “could have been” table Repeat many times and examine “what if” distribution of chi-square values under null hypothesis APSA Conference, Sept

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Government Spending 3. Strength of evidence Is observed chi-square value in tail of distribution? Summarize: What conclusions should be drawn? Very statistically significant Not cause and effect Ok to generalize to adult Americans APSA Conference, Sept

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Government Spending What about federal spending on the space program? APSA Conference, Sept More or less evidence of association? Larger or smaller p-value?

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General Advice Emphasize the process of statistical investigations, from posing questions to collecting data to analyzing data to drawing inferences to communicating findings Use simulation, both tactile and technology-based, to explore concepts of inference and randomness Draw connections between how data are collected (e.g., random assignment, random sampling) and scope of conclusions to be drawn (e.g., causation, generalizability) Use real data from genuine studies, as well as data collected on students themselves Present important studies (e.g., draft lottery) and frivolous ones (e.g., flat tires) and especially studies of issues that are directly relevant to students (e.g., sleep deprivation) APSA Conference, Sept

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General Advice (cont.) Lead students to “discover” and tell you important principles (e.g., association does not imply causation) Keep in mind the research question when analyzing data Graphical displays can be very useful Summary statistics (measures of center and spread) are helpful but don’t tell whole story; consider entire distribution Develop graph-sense, number-sense by always thinking about context Use technology to reduce the burden of rote calculations, both for analyzing data and exploring concepts Emphasize cautions and limitations with regard to inference procedures APSA Conference, Sept

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Implementation Suggestions Take control of the course Collect data from students Encourage predictions from students Allow students to discover/tell you findings Precede technology simulations with tactile Promote collaborative learning Provide lots of feedback Follow activities with related assessments Intermix lectures with activities Don’t underestimate ability of activities to teach materials Have fun! APSA Conference, Sept

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Suggestion #1 Take control of the course Not “control” in usual sense of standing at front dispensing information But still need to establish structure, inspire confidence that activities, self-discovery will work Be pro-active in approaching students Don’t wait for students to ask questions of you Ask them to defend their answers Be encouraging Instructor as facilitator/manager APSA Conference, Sept

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Suggestion #2 Collect data from students Leads them to personally identify with data, analysis; gives them ownership Collect anonymously Can do out-of-class E.g., matching variables to graphs APSA Conference, Sept

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Suggestion #3 Encourage predictions from students Fine (better…) to guess wrong, but important to take stake in some position Directly confront common misconceptions Have to “convince” them they are wrong (e.g., Gettysburg address) before they will change their way of thinking E.g., AIDS Testing APSA Conference, Sept

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Suggestion #4 Allow students to discover, tell you findings E.g., Televisions and life expectancy “I hear, I forget. I see, I remember. I do, I understand.” -- Chinese proverb APSA Conference, Sept

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Suggestion #5 Precede technology simulations with tactile/ concrete/hands-on simulations Enables students to understand process being simulated Prevents technology from coming across as mysterious “black box” E.g., Gettysburg Address (actual before applet) APSA Conference, Sept

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Suggestion #6 Promote collaborative learning Students can learn from each other Better yet from “arguing” with each other Students bring different background knowledge E.g., Matching variables to graphs APSA Conference, Sept

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Suggestion #7 Provide lots of feedback Danger of “discovering” wrong things Provide access to “model” answers after the fact Could write “answers” on board Could lead discussion/debriefing afterward APSA Conference, Sept

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Suggestion #8 Follow activities with related assessments Or could be perceived as “fun and games” only Require summary paragraphs in their own words Clarify early (e.g., quizzes) that they will be responsible for the knowledge Assessments encourage students to grasp concept Can also help them to understand concept E.g., fill in the blank p-value interpretation APSA Conference, Sept

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Suggestion #9 Inter-mix lectures with activities One approach: Lecture on a topic after students have performed activity Students better able to process, learn from lecture having grappled with issues themselves first Another approach: Engage in activities toward end of class period Often hard to re-capture students’ attention afterward Need frequent variety APSA Conference, Sept

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Suggestion #10 Do not under-estimate ability of activities to “teach” material No dichotomy between “content” and “activities” Some activities address many ideas E.g. “Gettysburg Address” activity Population vs. sample, parameter vs. statistic Bias, variability, precision Random sampling, effect of sample/population size Sampling variability, sampling distribution, Central Limit Theorem (consequences and applicability) APSA Conference, Sept

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Suggestion #11 Have fun! APSA Conference, Sept

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Assessment Advice Two sample final exams Carefully match the course goals Be cognizant of any review materials you have given the students Use real data and genuine studies Provide students with guidance for how long they should spend per problem Use multiple parts to one context but aim for independent parts (if a student cannot answer part (a) they may still be able to answer part (b)) Use open-ended questions requiring written explanation Aim for at least 50% conceptual questions rather than pure calculation questions (Occasionally) Expect students to think, integrate, apply beyond what they have learned. Sample guidelines for student projects APSA Conference, Sept

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Promoting Student Progress Document and enhance student learning Element of instruction Interactive feedback loop Diagnostic with indicators for change Throughout the course To student and instructor Encourage self-evaluation Multiple indicators APSA Conference, Sept

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Student Projects Best way to demonstrate to students the practice of statistics Experience the fine points of research Experience the “messiness” of data From beginning to end Formulation and Explanation Constant Reference statweb.calpoly.edu/bchance/stat217/projects.html APSA Conference, Sept

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Resources APSA Conference, Sept

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Resources GAISE reports APSA Conference, Sept

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Resources TeachingWithData.org APSA Conference, Sept

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Resources Inter-University Consortium for Political and Social Research (ICPSR) APSA Conference, Sept

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Resources APSA Conference, Sept

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Resources https://app.gen.umn.edu/artist/ APSA Conference, Sept

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Resources /jse_data_archive.html APSA Conference, Sept

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Background Readings Guidelines for teaching introductory statistics Reflections on what distinguishes statistical content and statistical thinking Educational research findings and suggestions related to teaching statistics Collections of resources and ideas for teaching statistics Suggestions and resources for assessing student learning in statistics APSA Conference, Sept

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Thanks very much! Questions, comments? APSA Conference, Sept

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My Syllabus Briefly W1: Collecting Data W2: Graphical/Numerical W3: NormalProject 1 W4: Exam 1 Project 2 W5: Probability W6: Sampling Distributions W7: Inference W8: Inference APSA Conference, Sept

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My Syllabus Briefly W9: Two Samples W10: Exam IIProject 3 W11: Two variables W12: Inference for Regression W13: Two-way TablesProject 4 W14: ANOVA W15: Presentations APSA Conference, Sept

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Non-simulation approach Exact randomization distribution Hypergeometric distribution Fisher’s Exact Test p-value = =.0127 APSA Conference, Sept

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