Random Thoughts 2012 (COMP 066) Jan-Michael Frahm Jared Heinly source: fivethirtyeight.com

Election Polls Virginia at 8:30 pm was 58% Romney and 41% Obama with 12% of the polls in That is a poll of 971, 000 people Why did Obama win it now? 2

Election Polls Why is Florida still not having a projected winner? Why does Ohio already have a projected winner with the same percentage of polls in? 3

Statistic of Support for Candidates USAToday: “Romney leads in states with most American cars”  eight of the top 10 states for registration of new american build cars are Romney supporters  two others are Swing states (Iowa & Michigan)  In fact the next four states are also Romney supporters  Obama has solid support for 9 of the 10 states with the most foreign registrations Is this statement true? How do we compute if its true? 4

Hypothesis Testing What we want is to test a hypothesis H 0 Hypothesis is usually a number to characterize a population  percentage of cancer in population  average size of a person in the US  …. In hypothesis testing we also need an alternative hypothesis that we pick to make a statement if H 0 is rejected 5

Alternative Hypothesis Alternative Hypothesis H a  selected to support the rejection statement  typical choices:  H a < H 0 is less is the desired statement if H 0 is rejected  H a <> H 0 is different is the desired statement if H 0 is rejected (H 0 is false)  H a < H 0 is larger is the desired statement if H 0 is rejected  H a often called research hypothesis How to select what is H 0 and what is H a ?  H 0 is typically the statement you want to verify  H 0 is typically assumed to be true unless there is strong evidence its wrong (similar to jury trial) 6

Find a Sample to Test Hypothesis Select a sample of size N to test the hypothesis  all rules of good samples apply that we have seen before for polls.  sample size still is influencing your certainty of the decision/estimate Compute the desired value  e.g. average height of males) What does that value tell us?  its only the characteristic of our sample set  we will need to extract its characteristics for the hypothesis evidence 7

Standardizing the Sample Value Convert to standard score (probability of the result) 1.Take out the Null Hypothesis H 0 (value from sample – H 0 )  if small this indicates you are close to H 0, if far H 0 is less likely 2.divide by standard error of the statistic s  this normalizes the distance to equalize close and far in 1) to what the deviation of the value is. What distance is good to reject or not reject? 8

How to reject H 0 Previous normalization brings value into standard value distribution  called Z-distribution or Normal distribution Test for value being likely or unlikely given the distribution  if within likely region keep H 0  if unlikely reject H 0 9

Z-Distribution (Normal distribution) 10

How to reject H 0 Previous normalization brings value into standard value distribution  called Z-distribution or Normal distribution Test for value being likely or unlikely given the distribution  if within likely region keep H 0  if unlikely reject H 0 Note that if the value is not rejected that does not mean its accepted either! Only means there is not enough evidence to reject 11

Finding the Likelihood Value is called p value Can be looked up in reference tables EXCEL: v p NORM.S.DIST(value.TRUE) For alternative hypothesis being:  less than p=v p  not equal p=2 v p  larger than p = 1- v p 12

Interpreting p-value set your cutoff called α (e.g. α = 0.05) if the p-value is:  less than 0.01 result is considered highly statistically significant reject null hypothesis  if between α and 0.01 (not close to α) result is statistically significant reject null hypothesis  if close to α result is marginally statistically significant either way is fine for rejection or not rejection  if greater than α don’t reject Always ask for p-value and α to make up your own mind 13

Testing for Proportion of Population Again for proportions we need to test differently 1.Compute proportion of population that is positive  regular percentage calculation 2.Subtract proportion that is claimed 3.Calculate standard error 4.divide step 2 by the standard error 14

Statistic of Support for Candidates USAToday: “Romney leads in states with most American cars”  eight of the top 10 states for registration of new american build cars are Romney supporters  two others are Swing states (Iowa & Michigan)  In fact the next four states are also Romney supporters  Obama has solid support for 9 of the 10 states with the most foreign registrations Is this statement true? How do we compute if its true? 15

Small Samples Use t-distribution 16

T-distribution 17 source: Wikipedia

Small Samples Use t-distribution Accounts for the sample size Value can be found in tables Excel: T.DIST 18

Errors Error Type 1: Wrong rejection Error Type 2: Missed rejection 19