Some commercial airplanes recirculate approximately 50% of the cabin air in orde

Question

Some commercial airplanes recirculate approximately 50% of the cabin air in order to increase fuel efficiency. The researchers studied 1102 airline passengers, among which some traveled on airplanes that recirculated air and others traveled on planes that did not recirculate air. Of the 519 passengers who flew on planes that did not recirculate air, 108 reported post-flight respiratory symptoms, while 113 of the 583 passengers on planes that did recirculate air reported such symptoms. Is there sufficient evidence to conclude that the proportion of passengers with post-flight respiratory symptoms differs for planes that do and do not recirculate air? Test the appropriate hypotheses using = 0.05. You may assume that it is reasonable to regard these two samples as being independently selected and as representative of the two populations of interest. (Use a statistical computer package to calculate the P-value. Use pdo not recirculate pdo recirculate. Round your test statistic to two decimal places and your P-value to four decimal places.)

Explanation / Answer

as it is mentioned that we need to use a statistical package , hence we shall use the open source statistical package R to answer this , the complete R snippet is as follows

x<- c(108,113) ## number of successes
n<- c(519,583) ## number of trials

## z test for proportions

prop.test(x, n, p = NULL, alternative = "two.sided",
correct = FALSE)

it is a two sided test as we are interested in the "difference" part only

prop.test(x, n, p = NULL, alternative = "two.sided",
+ correct = FALSE)

   2-sample test for equality of proportions without continuity correction

data: x out of n
X-squared = 0.34861, df = 1, p-value = 0.5549
alternative hypothesis: two.sided
95 percent confidence interval:
-0.03315950 0.06169438
sample estimates:
prop 1 prop 2
0.2080925 0.1938250

## z stat is
z.prop = function(x1,x2,n1,n2){
numerator = (x1/n1) - (x2/n2)
p.common = (x1+x2) / (n1+n2)
denominator = sqrt(p.common * (1-p.common) * (1/n1 + 1/n2))
z.prop.ris = numerator / denominator
return(z.prop.ris)
}

z.prop(108,113,519,583)

the result is

z.prop(108,113,519,583)
[1] 0.5904332

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