prop_chisqtestlab10

Lab 10: Inference for proportions and chi-square tests using R STAT 630, Fall 2018 Case study data set : The General Social Survey (GSS) is a major survey that tracks American demographics, characteristics, and views on social and cultural issues. It is con- ducted by the National Opinion Research Center (NORC) at the University of Chicago. The 2002 GSS data can be accessed from the resampledata library. It contains the re- sponses of 2765 participants. library (resampledata) data ( "GSS2002" ) Variable descriptions: Variable Description Region Interview location Gender Gender of respondent Race Race of respondent Education Highest level of education Marital Marital status Religion Religious affiliation Happy General happiness Income Respondent’s income Politics Political views Marijuana Legalize marijuana? Death Penalty Death penalty for murder? OwnGun Have gun at home? GunLaw Require permit to buy a gun? Pres00 Whom did you vote for in the 2000 presidential election? Postlife Believe in life after death? Example: two proportion z-test and confidence interval Using this data, let’s test the hypothesis that men and women differ in their beliefs in an afterlife. Specifically, we are testing H 0 : p f = p m H A : p f 6 = p m where p f is the proportion of females that believe in an afterlife, and p m is the proportion of men that believe in an afterlife. 1

First, remove the missing data. index <- which ( is.na (GSS2002 $ Gender) | is.na (GSS2002 $ Postlife)) gender <- GSS2002 $ Gender[ - index] postlife <- GSS2002 $ Postlife[ - index] Next, we can examine some contingency tables: addmargins ( table (gender, postlife)) ## postlife ## gender No Yes Sum ## Female 98 550 648 ## Male 138 425 563 ## Sum 236 975 1211 prop.table ( table (gender, postlife), margin = 1 ) ## postlife ## gender No Yes ## Female 0.1512346 0.8487654 ## Male 0.2451155 0.7548845 The sample sizes are large enough in each cell (greater than 5), so the conditions for the z - test are satisfied. Based on the table of row proportions, it appears that a higher proportion of females believe in an afterlife than males. We can proceed with the hypothesis test to determine if this is statistically significant. n <- length (postlife) n_f <- sum (gender == 'Female' ) n_m <- sum (gender == 'Male' ) # sample proportion of females that believe in afterlife: phat_f <- sum (gender == 'Female' & postlife == 'Yes' ) / n_f # sample proportion of males that believe in afterlife: phat_m <- sum (gender == 'Male' & postlife == 'Yes' ) / n_m # pooled sample proportion phat <- sum (postlife == 'Yes' ) / n 2

Next, we examine some contingency tables: addmargins ( table (education, death_penalty)) ## death_penalty ## education Favor Oppose Sum ## Left HS 117 72 189 ## HS 511 200 711 ## Jr Col 71 16 87 ## Bachelors 135 71 206 ## Graduate 64 50 114 ## Sum 898 409 1307 prop.table ( table (education, death_penalty), margin = 1 ) ## death_penalty ## education Favor Oppose ## Left HS 0.6190476 0.3809524 ## HS 0.7187060 0.2812940 ## Jr Col 0.8160920 0.1839080 ## Bachelors 0.6553398 0.3446602 ## Graduate 0.5614035 0.4385965 We can use the function chisq.test() to implement the chi-square test for independence. chisq1 <- chisq.test (education, death_penalty) chisq1 ## ## Pearson's Chi-squared test ## ## data: education and death_penalty ## X-squared = 23.451, df = 4, p-value = 0.0001029 The chi-square test statistic is 23.45 with a p-value = 0 . 0001029. Since the p-value < 0 . 01 we reject H 0 , and conclude that the survey provides convincing evidence that education level and position on the death penalty are dependent. Note that the results from the function are the same as the manual computations on the lecture slides. 4

The object chisq1 also has attributes that we can extract by name. attributes (chisq1) ## $names ## [1] "statistic" "parameter" "p.value" "method" "data.name" "observed" ## [7] "expected" "residuals" "stdres" ## ## $class ## [1] "htest" # test statistic chisq1 $ statistic ## X-squared ## 23.45093 # p-value chisq1 $ p.value ## [1] 0.0001028891 # table of expected counts chisq1 $ expected ## death_penalty ## education Favor Oppose ## Left HS 129.85616 59.14384 ## HS 488.50650 222.49350 ## Jr Col 59.77506 27.22494 ## Bachelors 141.53634 64.46366 ## Graduate 78.32594 35.67406 5