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Statistics for the Social Sciences Psychology 340 Spring 2010 Probability & The Normal Distribution PSY 340 Statistics for the Social Sciences Reminders • Quiz 2 due Thursday • Homework 3 due Tues, Feb 2 • Exam 1 Thurs Feb. 11 PSY 340 Statistics for the Social Sciences Basics of Probability Possible successful outcomes Probability All possible outcomes • Probability – Expected relative frequency of a particular outcome • Outcome – The result of an experiment PSY 340 Statistics for the Social Sciences Flipping a coin example What are the odds of getting a “heads”? n = 1 flip Possible successful outcomes Probability All possible outcomes One outcome classified as heads Total of two outcomes = 1 2 = 0.5 PSY 340 Statistics for the Social Sciences Flipping a coin example n=2 Number of heads 2 1 1 What are the odds of getting two “heads”? One 2 “heads” outcome Four total outcomes = 0.25 0 This situation is known as the binomial # of outcomes = 2n PSY 340 Statistics for the Social Sciences Flipping a coin example n=2 Number of heads 2 1 1 0 What are the odds of getting “at least one heads”? Three “at least one heads” outcome Four total outcomes = 0.75 PSY 340 Flipping a coin example Statistics for the Social Sciences n=3 3= n = 2 2 8 total outcomes HHH Number of heads 3 HHT 2 HTH 2 HTT 1 THH 2 THT 1 TTH 1 TTT 0 PSY 340 Flipping a coin example Statistics for the Social Sciences Number of heads 3 Distribution of possible outcomes probability (n = 3 flips) .4 .3 .2 .1 .125 .375 .375 .125 0 1 2 3 Number of heads 2 X f p 3 1 .125 2 2 1 3 3 .375 .375 1 0 1 .125 1 2 1 0 PSY 340 Flipping a coin example Statistics for the Social Sciences Distribution of possible outcomes probability (n = 3 flips) .4 .3 .2 .1 .125 .375 .375 .125 0 1 2 3 Number of heads Can make predictions about likelihood of outcomes based on this distribution. What’s the probability of flipping three heads in a row? p = 0.125 PSY 340 Flipping a coin example Statistics for the Social Sciences Distribution of possible outcomes probability (n = 3 flips) .4 .3 .2 .1 .125 .375 .375 .125 0 1 2 3 Number of heads Can make predictions about likelihood of outcomes based on this distribution. What’s the probability of flipping at least two heads in three tosses? p = 0.375 + 0.125 = 0.50 PSY 340 Flipping a coin example Statistics for the Social Sciences Distribution of possible outcomes probability (n = 3 flips) .4 .3 .2 .1 .125 .375 .375 .125 0 1 2 3 Number of heads Can make predictions about likelihood of outcomes based on this distribution. What’s the probability of flipping all heads or all tails in three tosses? p = 0.125 + 0.125 = 0.25 PSY 340 Statistics for the Social Sciences Hypothesis testing Distribution of possible outcomes (of a particular sample size, n) Can make predictions about likelihood of outcomes based on this distribution. • In hypothesis testing, we compare our observed samples with the distribution of possible samples (transformed into standardized distributions) • This distribution of possible outcomes is often Normally Distributed PSY 340 Statistics for the Social Sciences The Normal Distribution • The distribution of days before and after due date (bin width = 4 days). -14 0 14 Days before and after due date PSY 340 Statistics for the Social Sciences The Normal Distribution • Normal distribution PSY 340 The Normal Distribution Statistics for the Social Sciences • Normal distribution is a commonly found distribution that is symmetrical and unimodal. – Not all unimodal, symmetrical curves are Normal, so be careful with your descriptions • It is defined by the following equation: 1 2 2 Z-scores -3 -2 -1 0 1 2 3 e (X ) 2 / 2 2 Estimating Probabilities in a Normal Distribution PSY 340 Statistics for the Social Sciences probability Same logic as before . 4 . 3 . 2 . 1 50%-34%-14% rule .125 0 .375 .375 1 2 .125 3 50% Number of heads -3 -2 -1 0 1 2 3 PSY 340 Statistics for the Social Sciences Estimating Probabilities in a Normal Distribution 50%-34%-14% rule 50% 34.13% 13.59% -3 -2 -1 0 1 2 3 PSY 340 Statistics for the Social Sciences Estimating Probabilities in a Normal Distribution Similar to the 68%-95%-99% rule 34.13% 34.13% 2.28% -3 13.59% -2 -1 68% 0 2.28% 13.59% 1 2 3 PSY 340 Statistics for the Social Sciences Estimating Probabilities in a Normal Distribution Similar to the 68%-95%-99% rule 34.13% 34.13% 2.28% -3 13.59% -2 -1 95% 0 2.28% 13.59% 1 2 3 PSY 340 Statistics for the Social Sciences The Unit Normal Table Understand your table z 0 : : 1.00 : : 2.31 2.32 z 0 • The normal distribution is often transformed into z-scores. • Gives the precise proportion of scores (in z-scores) above or below a given score in a Normal distribution • There are many ways that this table gets organized • Learn to understand what is in the table • What do the numbers represent? PSY 340 Statistics for the Social Sciences The Unit Normal Table Understand your table Z 0.00 0.01 0.02 : : 1.0 : 1.3 : : 4.00 Prop in Body Prop in tail Prop btwn mean and z .5000 .5000 .0000 .5040 .4960 .0040 .5080 .4920 .0080 • : : : : : : .8413 .1587 .3413 : : : .9032 .0968 .4032 : : : : : : .99997 .00003 .49997 From the left side of the dist. In tail z 0 The normal distribution is often transformed into z-scores. – Contains the proportions of a Normal distribution – Proportion between the z-score and left side of the distribution – Proportion in the tail to the right of corresponding z-scores – Proportion between the z-score and the mean • Note: This means that this table lists only positive Z scores PSY 340 Statistics for the Social Sciences The Unit Normal Table Understand your table z .00 .01 0 : : 1.0 : : 2.3 2.4 : 0.5000 : : 0.1587 : : 0.0107 0.0082 : 0.4960 : : 0.1562 : : 0.0104 0.0080 : In tail z 0 • The normal distribution is often transformed into z-scores. – Contains the proportions in the tail to the left of corresponding z-scores of a Normal distribution • This means that the table lists only positive Z scores • The different columns give the second decimal place of the z-score The unit normal table I have provided online (see ‘statistical tables’ link at top of labs) PSY 340 Statistics for the Social Sciences The Unit Normal Table Understand your table Mean to Z z Mean to Z In tail 0 : : 1.00 : : 2.31 2.32 : 0.0000 : : 0.3413 : : 0.4896 0.4898 : 0.5000 : : 0.1587 : : 0.0104 0.0102 : In tail z 0 • The normal distribution is often transformed into z-scores. – Contains the proportions – Proportion between the z-score and the mean – Proportion in the tail to the left of corresponding z-scores of a Normal distribution • Note: This means that this table lists only positive Z scores PSY 340 Statistics for the Social Sciences The Unit Normal Table Understand your table z .00 .01 -3.4 -3.3 : : 0 : : 1.0 : : 3.3 3.4 0.0003 0.0005 : : 0.5000 : : 0.8413 : : 0.9995 0.9997 0.0003 0.0005 : : 0.5040 : : 0.8438 : : 0.9995 0.9997 From the left side of the dist. 0 z • The normal distribution is often transformed into z-scores. – Contains the proportions to the left of corresponding z-scores of a Normal distribution • This table lists both positive and negative Z scores Another common way the unit normal table is presented in textbooks PSY 340 Statistics for the Social Sciences Z 0.00 0.01 0.02 : : 1.0 : 1.3 : : 4.00 Using the Unit Normal Table Prop in Body Prop in tail Prop btwn mean and z .5000 .5000 .0000 .5040 .4960 .0040 .5080 .4920 .0080 : : : : : : .8413 .1587 .3413 : : : .9032 .0968 .4032 : : : : : : .99997 .00003 .49997 • Steps for figuring the percentage below a particular raw or Z score: 1. Convert raw score to Z score (if necessary) XM z SD 2. Draw normal curve, where the Z score falls on it, shade in the area for which you are finding the percentage 3. Make rough estimate of shaded area’s percentage (using 50%-34%-14% rule) PSY 340 Statistics for the Social Sciences Z 0.00 0.01 0.02 : : 1.0 : 1.3 : : 4.00 Using the Unit Normal Table Prop in Body Prop in tail Prop btwn mean and z .5000 .5000 .0000 .5040 .4960 .0040 .5080 .4920 .0080 : : : : : : .8413 .1587 .3413 : : : .9032 .0968 .4032 : : : : : : .99997 .00003 .49997 • Steps for figuring the percentage below a particular raw or Z score: 4. Find exact percentage using unit normal table – Use your sketch and understanding of the table 5. Check the exact percentage is within the range of the estimate from Step 3 PSY 340 SAT Example problems Statistics for the Social Sciences • The population parameters for the SAT are: μ = 500, σ = 100, and it is Normally distributed Suppose that you got a 630 on the SAT. What percent of the people who take the SAT get your score or worse? z X 630 500 From the table: 1.3 100 z(1.3) =.0968 So 90.32% got your score or worse -2 -1 That’s 9.68% above this score 1 2 PSY 340 Statistics for the Social Sciences The Normal Distribution • You can go in the other direction too – Steps for figuring Z scores and raw scores from percentages (or proportions): 1. Draw normal curve, shade in approximate area for the percentage (using the 50%-34%-14% rule) 2. Make rough estimate of the Z score where the shaded area starts 3. Find the exact Z score using the unit normal table - So now you’re looking for a percentage/proportion in the body of the table, and then looking to see what z-score it corresponds to 4. Check that your Z score is similar to the rough estimate from Step 2 5. If you want to find a raw score, change it from the Z score PSY 340 Statistics for the Social Sciences Testing Hypotheses • Looking ahead: How do we determine this? – Core logic of hypothesis testing • Considers the probability that the result of a study could have come about if the experimental procedure had no effect • “Studies” typically look not at single scores, but rather samples of scores. So we need to think about the probability of getting samples with particular characteristics (means). observed difference test statistic difference expected by chance Z (X X ) X Based on standard error or an • Next time: estimate of the standard error – The distribution of sample means