PSYCH 210 Lecture 7 Outline of Last Lecture I. Continue variabilitya. Disadvantages of SIQb. Variance and standard deviationOutline of Current Lecture I. Effects of Adding, Subtracting, Multiplying and Dividing ConstantsII. Descriptive Statisticsa. Z-scoresIII. Inferential Statistics Previewa. ProbabilityCurrent LectureI. Formulas will not be on exama. Adding Constanti. Mnew = Mold + Cii. s2new = s2old1. Only shifted data; variance remains sameiii. snew = soldb. Subtracting Constanti. Mnew = Mold – Cii. s2new = s2oldiii. snew = soldc. Multiplying by a Constanti. Μnew = Mold (C)ii. s2new = s2old(C2)iii. snew = sold (C)1. Graph: Shifting up, wider and shorterd. Dividing By a Constanti. Mnew = Mold/CThese notes represent a detailed interpretation of the professor’s lecture. GradeBuddy is best used as a supplement to your own notes, not as a substitute.ii. S2new = S2old/Ciii. Snew = Sold/C1. Graph: Shifting back, narrower and tallerII. Descriptive Statisticsa. Z-Scoresi. Ex) Imagine Jif = $4.50 for 28 oz., Skippy is $5.72 for 40 oz.1. Which is better buy?a. 4.50/1oz = .1607/ox - Jifb. .1430/0z - Skippy2. Standardizing scores!ii. Ex) History Exam: 75, Law Exam: 681. Which subject is he better at?2. We don’t know if we can directly compare these! (Same scale?)a. Helpful Information:i. Mean scores (or μ)ii. Percentileiii. Variabilityiv. Total scoreb. History Exam: μ = 100, σ = 10c. Law: μ = 60, σ = 4iii. A descriptive Statistic that allows us to standardize scores based on distributioniv. How to Calculate z-scores?1. z= (x-μ)/σa. Def: Tells us how far away a particular score falls away from the mean, in standard deviation units2. History z = (75-100)/10 = -2.5a. Negative Sign: falls below the mean3. Law z = (68-60)/4 = 2a. Positive Sign: falls above the meanv. Important notes about z-scores1. Converting raw scores into z-scores does NOT change the shape ofa distribution!a. Only renaming values, but values retain relative positionsb. You can convert any population to z-scores (doesn’t matterwhat shape it was originally)2. The mean of a distribution of zs always = 0a. Phrasing values based on how far they are from mean!3. The SD of a distribution of zs = 1vi. Converting FROM z TO x (Opposite Direction)1. x = μ + zσ2. z=1.2, μ=50, σ=7x=50 + 1.2 (7) = 58.4a. ‘Eyeball’ method to check for error!vii. Z-Scores as Conversion Factors1. Convert scores from one distribution into a new distribution with a new μ and σ2. Ex) UW: μ=50, σ=5, xUW = 62U of Indiana: μ=100, σ=10a. What score would you have if your team were playing in U of Indiana Basketball league?i. Calculate z-score using ‘old’ (starting distribution) valuesz= 62-50/5 = 2.4ii. Convert z-score back to raw score using ‘new’ values (new distribution)x= 100 + 2.4(10) = 124 III. Inferential Statistics Previewa. The ‘Mozart Effect’i. Experiment1. Mozarta. M = 115, s = 62. White noisea. M = 102, s = 4ii. Did exposure to Mozart make kids smarter?b. Basic question we always ask in Inferential Statistics: What is the cause of the difference between groups?i. Two Choices:1. Difference is due to the experimental treatment/manipulation2. Difference is due to chanceii. What allows us to choose between Option 1 and Option 2?1. ProbabilityIV. Probabilitya. 2 Basic Typesi. A priori (‘ahead of time’)1. Basic Formula: p(event) = f(event)/N(# of events)2. Event: rolling a 5, drawing a 10, flipping a heads, etc.3. Ex) 1 flip of coinp(heads) = ½ 4. Ex) Roll of 6-sided diep(5) = 1/6ii. Empirical1. Our focus for Inferential Statistics2. 2 Requirements must be met:a. Each event in the population must have an equal chance ofbeing selected1. Condition met with RANDOM SAMPLING.b. There must be a constant probability for each and every selectioni. Ex using a priori) p(heart) = f/N = 13/52 = ¼2nd draw: p(heart) = 12/511. Condition met by SAMPLING WITH REPLACEMENT3. Problem: In real Psychological Experiments, cannot sample with replacement (must debrief participant)a. Ex) Pop = 29 yr olds, Sample = 1,000,000p(particular 20 yr. old) = 1/1,000,000p(particular other 20 yr. old) = 1/(1,000,000-1)b. Cheat: underlying probability essentially same because difference is so