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Psyc 235: Introduction to Statistics http://www.psych.uiuc.edu/~jrfinley/p235/ To get credit for attending this lecture: SIGN THE SIGN-IN SHEET To-Do • ALEKS: aim for 18 hours spent by the end of this week • Jan 30th Target Date for Descriptive Statistics • Watch videos: 1. Picturing Distributions 2. Describing Distributions 3. Normal Distributions Quiz 1 • NOT GRADED • available starting 8am Thurs Jan 31st, through Friday • can do on ALEKS from home, etc • No access to any other learning or reviewing materials until either they finish the quizzes or after Friday • 3.5 hour time limit Review: (2 Steps Forward and 1 Step Back) • Distribution For a given variable: the possible numerical values & the number of times they occur in the data Many ways to represent visually Summarizing Distributions • Descriptive Measures of Data Measures of C_nt__l T__d__cy Measures of D__p_rs__n Central Tendency • Mean, Median, Mode Mean vs Median & outliers (Bill Gates example) skewed distributions Standard Deviation • Conceptually: about how far, generally, each datum is from the mean 2 formulas?? Population vs Sample • In Psychology: Population: hypothetical, unobservable not just all humans who ARE, but all humans who COULD BE. must estimate mean, standard deviation, from: Sample is the only thing we ever have Descriptive -> Inferential? • How can we make inferences about a population if we just have data from a sample? • How can we evaluate how good our estimate is? • “Do these sample data really reflect what’s going on in the population, or are they maybe just due to chance?” PROBABILITY • The tool that will allow us to bridge the gap from descriptive to inferential • we’ll start by using simple problems, in which probability can be calculated by merely COUNTING Flipping a Coin • Say I flip a coin... OMG Heads!!!! Do you care? Why Not? • Sample Space: (draw on board) collection of all possible outcomes for a given phenomenon coin toss: {H,T} mutually exclusive: either one happens, or the other Flipping a Coin • Probability(Heads)? • So.... must the next one be Tails? • No! Independent trials Random Phenomenon: can’t predict individual outcome can predict pattern in the LONG RUN • Probability: relative # times something happens in the long run 2 Coin Flips • OMG 2 Heads! impressed yet? • Sample space (draw on board) Prob(2 Heads): 1/4 outcome: single observation • OMG 2 of same! Prob(2 Heads OR 2 Tails): event: subset of the sample space made of 1 or more possible outcomes Larger Point • OMG 30 Heads in a row! NOW maybe you’re finally interested... • OMG drew 3 yellow cars! interesting? boring? can’t tell! • Descriptive Stats: measuring & summarizing outcomes • Inferential Stats: to understand some outcome, must consider it in context of all possible outcomes that could’ve occurred (sample space) Counting Rules • Count up the possible outcomes that is: define the sample space • 2 Main ways to do this: Permutations when order matters Combinations when order doesn’t matter Permutation: Ordered Arrangement • Example used: Horse Race... • MUTANT HORSE RACE! Permutation: Ordered Arrangement “HorseFace McBusterWorthy wins 1st place!!” ...in a one-horse race! # Horses (n)# Winning Places (r) 1 3 3 1 1 3 # Outcomes Permutation: Ordered Arrangement • For n objects, when taking all of them (r=n), there are n! possible permutations. • 3 horses (n) & 3 winning places (r) --> 3*2*1=6 possible outcomes • For n objects taken r at a time: n! (n-r)! • 7 horses & 3 winning places?... Combination: Unordered Arrangement • Example used: Combo Plate! QuickTime™ and a decompressor are needed to see this picture. Combination: Unordered Arrangement • Mexican restaurant’s menu: taco, burrito, enchilada • How many different 3-item combos can you get? # Menu Items (n) Combo Size (r) 3 3 # Outcomes Combination: Unordered Arrangement • Mexican restaurant’s menu: taco, burrito, enchilada tamale, quesadilla, taquito, chimichanga • How many different 3-item combos can you get? # Menu Items (n) Combo Size (r) 3 7 3 3 # Outcomes Combination: Unordered Arrangement • For n objects, when taking all of them (r=n), there is 1 combination • For n objects taken r at a time: n! r!(n-r)! Multiplication Principle (a.k.a. Fundamental Counting Principle) • For 2 independent phenomenon, how many different ways are there for them to happen together? # possible joint outcomes? • Simply multiply the # possible outcomes for the two individual phenomena • Example: flip coin & roll die • 2*6=12 Multiplication Principle (a.k.a. Fundamental Counting Principle) • Can be used with Permutations &/or Combinations • Ex: Lunch at the Racetrack 7 horses racing 7 items on the cafe menu I see the results of the race (1st, 2nd, 3rd) and order a 3-item combo plate. How many different ways can this happen? Calculating Probabilities • Counting rules (Permutation, Combination, Multiplication): Define sample space (# possible outcomes) • Probability of a specific outcome: 1 sample space • Probability of an event? event: subset of sample space made of 1 or more possible outcomes Calculating Probabilities • Sample Space: 7 Micro Machines (3 yellow, 4 red) • Outcome: draw the yellow corvette Probability = 1/7 • Event: draw any yellow car there are 3 outcomes that could satisfy this event: yellow corvetter, yellow pickup, yellow taxi Probability = 3/7 Probability of Draws w/ Replacement • Replacement: resetting the sample space each time --> independent phenomena so use multiplication principle • Ex: 3 draws with replacement Event: drawing a red car all 3 times Probability: 4/7 * 4/7 * 4/7 = 64/343 = 0.187 =18.7% Probability of Draws w/o Replacement 1. Use counting rules to define sample space 2. Use counting rules to figure out how many possible outcomes satisfy the event 3. divide #2 by #1. Probability of Draws w/o Replacement • Ex: Drawing 3 cars w/o replacement Event: drawing 2 red & 1 yellow (don’t care about order) --> use Combinations Define Sample space: Count outcomes that satisfy event treat red & yellow as independent use combinations, then multiplication principle Divide Recap • Today: Probability is the tool we’ll use to make inferences about a population, from a sample Counting rules: define sample space for simple phenomena Intro to calculating probability • Next time: Probability rules, more about events, Venn diagrams Remember • • • • Quiz 1 starting Thursday Office hours Thursday Lab Put your ALEKS hours in!!