Statistics Lecture 23

zLast Day: Regression zToday: Finish Regression, Test for Independence (Section 13.4) zSuggested problems: 13.21, 13.23

Computer Output zWill not normally compute regression line, standard errors, … by hand zKey will be identifying what computer is giving you

SPSS Example

zWhat is the Coefficients Table?

zWhat is the Model Summary? zWhat is the ANOVA Table

Back to Probability zThe probability of an event, A, occurring can often be modified after observing whether or not another event, B, has taken place zExample: An urn contains 2 green balls and 3 red balls. Suppose 2 balls are selected at random one after another without replacement from the urn. yFind P(Green ball appears on the first draw) yFind P(Green ball appears on the second draw)

Conditional Probability zThe Conditional Probability of A given B :

zExample: An urn contains 2 green balls and 3 red balls. Suppose 2 balls are selected at random one after another without replacement from the urn. yA={Green ball appears on the second draw} yB= {Green ball appears on the first draw} yFind P(A|B) and P(A c |B)

Example: zRecords of student patients at a dentist’s office concerning fear of visiting the dentist suggest the following proportions zLet A={Fears Dentist}; B={Middle School} zFind P(A|B)

Conditional Probability and Independence zIf fearing the dentist does not depend on age or school level what would we expect the probability distribution in the previous example to look like? zWhat does this imply about P(A|B)? zIf A and B are independent, what form should the conditional probability take?

Summarizing Bivariate Categorical Data zHave studied bivariate continuous data (regression) zOften have two (or more) categorical measurements taken on the same sampling unit zData usually summarized in 2-way tables zOften called contingency tables

Test for Independence zSituation: We draw ONE random sample of predetermined size and record 2 categorical measurements zBecause we do not know in advance how many sampled units will fall into each category, neither the column totals nor the row totals are fixed

Example: zSurvey conducted by sampling 400 people who were questioned regarding union membership and attitude towards decreased spending on social programs zWould like to see if the distribution of union membership is independent of support for social programs

zIf the two distributions are independent, what does that say about the probability of a randomly selected individual falling into a particular category zWhat would the expected count be for each cell? zWhat test statistic could we use?

Formal Test zHypotheses: zTest Statistic: zP-Value:

Spurious Dependence zConsider admissions from a fictional university by gender zIs there evidence of discrimination?

zConsider same data, separated by schools applied to: zBusiness School: zLaw School:

zSimpson’s Paradox: Reversal of comparison due to aggregation zContradiction of initial finding because of presence of a lurking variable