Question

Recall the report On the Front Line: The Work of First Responders in a Post-9/11 World. We will use data from this report to investigate the question: Are alcohol-related problems among New York firefighters associated with participation in the 9/11 rescue?

Here again are our observed data:

No risk. for alcohol problems Moderate to Severe risk for alcohol problems 793 309 1102 Participated in 911 rescue 441 110 551 Did Not Participate in 911 rescue 419 1653 1234

State the appropriate hypotheses for the chi-square test for independence in this case. Now you will check whether the conditions for the chi-square test are met. In order to do this, you’ll need to first launch the actual research report and read the last paragraph on page iii of the introduction (starting with the “The study was fully funded…”)

The study was fully funded by the Smithers Institute at the School of Industrial and Labor Relations. The findings presented in this report are based on survey data collected from New York City firefighters and officers during the summer and fall of 2003. In total, out of a target sample of some 3,600 firefighters and officers, data were collected from 1,653 firefighters and 379 officers; approximately 2/3 of whom were on the scene at the World Trade Center on 9/11. The target sample represents one third of all NYC firefighters. Rather than identifying a random sample of firefighters, we drew a stratified, random sample of 144 fire companies (out of approximately 350 FDNY companies) and targeted all of the members of these companies (typically between 20-30 firefighters and officers per company) for data collection. Stratification was based on the relative work intensity of each company (i.e., average daily runs). We used the FDNY’s three-category system to differentiate between more and less active firehouses, randomly sampling an equal number of the firehouses from each of these three categories (i.e., 48 from highly active, 48 iv from moderately active and 48 from relatively inactive) for inclusion in the study. As might be expected in this highly male-dominated workforce, 99 percent of those included in our sample were men.

Answer 1

The hypothesis for the test is,

Ho : The alcohol related problems among New York firefighters is not associated with participation in 9/11 rescue.

Vs

Ha : The alcohol related problems among New York firefighters is associated with participation in 9/11 rescue.

The chi-square test statistic is,

$\chi ^{2}=\sum \frac{(Oi - Ei)^{2}}{Ei}$

Where, Oi is observed frequencies.

Ei is expected frequencies.

To find expected frequencies we use the formula,

Ei = (Row total * Column total)/ Overall total

$E11 = \frac{1102*1234}{1653}=822.6667$

$E12 = \frac{1102*419}{1653}=279.3333$

$E21= \frac{551*1234}{1653}=411.3333$

$E22= \frac{551*419}{1653}=139.6667$

Hence the test statistic is,

$\chi ^{2}=\frac{(793-822.6667)^{2}}{822.6667}+\frac{(309-279.3333)^{2}}{279.3333}+\frac{(441-411.3333)^{2}}{411.3333}+\frac{(110-139.6667)^{2}}{139.6667}$$\chi ^{2}=12.6617$

Let, level of significance $\alpha$ = 0.05

Also, df = (r-1)*(c-1)

df = 1

Hence using table of critical value for chi square distribution the critical value is 3.841

Here, $\chi ^{2} > \chi ^{2}crit$

Hence we reject the null hypothesis Ho.

Therefore there is sufficient evidence to say that the alcohol related problems among New York firefighters is associated with participation in 9/11 rescue.