Unit 1: Introduction to data To do: 3. Introduction to statistical inference ▶ Performance Assessment 1: by 11:59pm on Friday. ▶ Problem Set 3: by next Thursday before class. Sta 101 - Spring 2015 ▶ Lab 2: by next Wednesday. ▶ Readiness Assessment 2: next Tuesday in class. Duke University, Department of Statistical Science January 22, 2015 My OH today : 4:30-6pm. Dr. Windle Slides posted at http://bitly.com/windle2 1 Themes/Ideas/Tools of the day Definition of probability Today: probability fundamentals and their connection to inference. Ideas: 1. Proportion and probability are connected. Probability: 2. A bucket of cards. 1. The chance that something will happen. 3. Larger random samples provide more reliable information. 2. A branch of mathematics that studies chance. 4. In statistics, you often need to think about how you would 3. The long-run relative frequency of an event. act/decide before seeing the actual data. Tools: 1. probability distribution / probability mass function 2. bar plot 3. h-plot 2 3
For categorical or discrete numerical variables: Idea : Proportion and probability are intimately connected. Tool : A “probability distribution” or a “probability mass function” is a table that describes the possible outcomes of a Examples: random process and the probabilities of those possible ▶ A coin outcomes. ▶ A die Rolling a die possible outcomes 1 2 3 4 5 6 probability 1/6 1/6 1/6 1/6 1/6 1/6 4 5 Visualizations Visualizations 6 7
Probability and proportion Visualizations Rolling a die possible outcomes 1 2 3 4 5 6 probability 1/6 1/6 1/6 1/6 1/6 1/6 relative frequency table Number of dots 1 2 3 4 5 6 Relative Frequency 1/6 1/6 1/6 1/6 1/6 1/6 8 9 Visualizations Cards analogy Clicker question What proportion of cards are red? (b) 1/6 (b) 13/52 (b) 4/52 (b) 1/2 10 11
Cards analogy Question: If we think of this deck of cards as a population what are the attributes/variables of interest? Case Suit Number Color 1 Clubs Ace Black 2 Clubs 2 Black 3 Clubs 3 Black . . . . . . . . . . . . 51 Diamonds Queen Red Clicker question 52 Diamonds King Red What is the probability of drawing a face card? (a) 12/52 (c) 26/52 (d) 13/52 (b) 16/52 12 13 Sample size demo Rolling two dice 1 2 3 4 5 6 1 1,1 1,2 1,3 1,4 1,5 1,6 2 2,1 2,2 2,3 2,4 2,5 2,6 Idea : Lots of things in statistics can be thought of as a deck of 3 3,1 3,2 3,3 3,4 3,5 3,6 4 4,1 4,2 4,3 4,4 4,5 4,6 cards / a bucket of cards. 5 5,1 5,2 5,3 5,4 5,5 5,6 6 6,1 6,2 6,3 6,4 6,5 6,6 Example: Duke students. Rolling two dice and summing 1 2 3 4 5 6 1 2 3 4 5 6 7 2 3 4 5 6 7 8 3 4 5 6 7 8 9 4 5 6 7 8 9 10 5 6 7 8 9 10 11 6 7 8 9 10 11 12 14 15
Asch conformity experiments Application Exercise 1.4b: bitly.com/windle2 Idea : In statistics, you often need to think about how you would act before seeing the actual data. 16 17 Asch conformity experiments Asch conformity experiments Null hypothesis: there is a little conformity. Alt. hypothesis: there is more than a little conformity. 18 19
Asch conformity experiments Asch conformity experiments H 0 : proportion of the population that will conform is 0.1. H 0 : proportion of the population that will conform is 0.1. H A : proportion of the population that will conform is > 0 . 1 . H A : proportion of the population that will conform is 0.3. 20 21 Asch conformity experiments Asch conformity experiments Question: based on these plots, how many conformers do you need to see before you reject the null? Often we ask ourselves: How can I avoid making the wrong decision in the case that the null hypothesis is true? 22 23
We decide to reject based on seeing a relatively unlikely observation. This is ONE WAY to come up with a PROCEDURE for deciding to stick with/reject the null. 24
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