Psyc 235: Introduction to Statistics 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)# 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!

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)# Outcomes3

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)# Outcomes3 73

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 = =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!!