PSYCH 210 Lecture 9 Outline of Last Lecture I. Continue probabilityII. Probability and z-scoresa. How to use Unit Normal Tablei. Type Iii. Type IIiii. Type IIIOutline of Current Lecture I. Binomial distributionsII. Sampling Distributionsa. Not on first examCurrent LectureI. Answering questions about z-scoresa. Percentile and percentile rank problems (with a normal distribution)i. What is the percentile rank of x?1. Type I2. Recall that a percentile is a raw (x) score associated with a given percentage of scores falling at or below itII. Binomial distributiona. The normal distribution represents continuous measuresi. What if we have discrete data with only two outcomes?1. Special case of probability b. Examplesi. Coin flip (heads or tails)ii. True/false questions (correct or incorrect)iii. Rolling one die (desired or undesired outcome)c. TerminologyThese 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.i. Only 2 outcomes (A and B)1. p = p(A)2. q = p(B)3. p + q = 1.00ii. number of events = nd. Binomial Distribution is a unique distributioni. BUT normal distribution is a good approximation WHEN we have a LARGE N1. Will approximate the normal distribution if pn and qn = or >10a. If so, then:i. μ = pnii. σ = square root of npqe. Ex) Coin Flip 4 timesi. n = 4ii. x = headsiii. A = headsiv. B = tailsv. p = p(heads) = 0.5vi. q = p(tails) = 0.51. (Based on ONE coin flip!)vii. p = f/Nviii. Probabilities add up to 1, mirror normal distribution (if N was much larger)f. Ex) T/F quiz: 25 MC questionsi. What is the probability of getting exactly 16 (true/false) questions right if you guessed on every question?1. p(x=16)?2. pn, qn ≥ 10a. n= 25b. p= 0.5i. p = p(correct)c. q= 0.5i. q = p(incorrect)d. pn = 0.5(25) = 12.5 ≥ 10 ✓e. qn = 0.5(25) = 12.5 ≥ 10 ✓f. μ= pn = 12.5g. σ = √npq = √25(.5)(.5) = 2.5i. p(x=16)?1. Retranslate questions: p(15.5<x<16.5)?2. (Only translate into real limits for binomial distributions!)h. Type III z-score problemi. Starts like Type I1.2. Convert x’s to z’sa. z=(15.5-12.5)/2.5 = 1.2b. z = (16.5 – 12.5) / 2.5 = 1.63. Find probability by adding or subtracting areasa. Tail p for z = 1.2 = 0.1151b. Tail p for z = 1.6 = 0.0548c. .1151 – 0.0548 = 0.0603g. p and q are not always 0.5i. MC quiz (a,b,c,d)ii. p = p(correct) = 0.25iii. p = p(incorrect) = 0.751. (Based on ONE question!)2. p = f/Nh. Types of binomial questionsi. p(x = …)ii. p(x>...)1. p(x>20)?2. Translate:a. Sketch:b. p(x>20.5) iii. p(x<…)1. p(x<18)2. Translate:a. Sketchb. p(x<17.5)i. Practice Problem Set distributed in classi. Answers onlineIII. Transition to Inferential statisticsa. Following material NOT on first exam!b. So far, we know how to…i. Find relative locations of raw scores in a distribution (percentiles)ii. Standardize scores (z-scores)iii. Find probability associated with particular z-scores using normal distributionc. For inferential statistics, we’ll need to:i. Compare means of different groupsii. Find out whether the difference between them is meaningful1. What is the cause of the difference btwn groups?a. Difference is due to experiential treatment/manipulationb. Difference is due to chanceiii. So, we need to ask questions about means rather than about raw scoresiv. How to use probability to help answer these questions1. Drawing samples from populations:a. Statistics calculated from a sample will never be identical to parameters from the populationb. Sampling error inevitablec. BUT:i. If we draw repeated samples from the same population, we can get a good idea of the nature ofthe population1. = A sampling distributiona. Def: A distribution of statistics obtained by selecting all possible random samples of specific size (n) from a populationb. Usually we will make a distribution of sample