Inference on Population Proportions

  See chapter 18 of text for details.

  Obstructive sleep apnea is a sleep disorder that causes a person to
stop breathing momentarily and then awaken briefly. These sleep
interruptions, which may occur hundreds of times in a night, can
drastically reduce the quality of rest and cause fatigue during waking
hours. Researchers at Stanford University studied 159 commercial truck
drivers and found that 124 of them suffered from obstructive sleep apnea
(Chest, May 1995).

 Do the data suggest that more than 70% of truck drivers suffer from
obstructive sleep apnea?

 We have the choice of two functions in R for inference on
proportions: binom.test and prop.test.  binom.test is an exact
procedure, but it only works on inference of a single proportion.
prop.test is an approximate method, but it can compare several
proportions.

>binom.test(124,159,p=0.7,alternative="greater")

         Exact binomial test 

data:  124 out of 159 
number of successes = 124, n = 159, p-value = 0.01543 
alternative hypothesis: true probability of success is greater than 0.7 
95 percent confidence interval:
 0.718954 1.000000 
sample estimates:
probability of success 
             0.7798742

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> prop.test(124,159,p=.7,alternative="greater", correct=F)

	1-sample proportions test without continuity correction

data:  124 out of 159, null probability 0.7 
X-squared = 4.8305, df = 1, p-value = 0.01398
alternative hypothesis: true p is greater than 0.7 
95 percent confidence interval:
 0.7213938 1.0000000 
sample estimates:
        p 
0.7798742 

> sqrt(4.8305) = 2.19784, which equals the z test statistic. 

 Are the p-values of the 2 methods exactly the same?

 Scope of inference?

 Comparing Two Population Proportions:

    The Salk polio vaccine trials of 1954 included a double-blind
experiment in which elementary school children of consenting parents
were assigned at random to injection with the Salk vaccine or with a
placebo. Both treatment and control groups were set at 200,000 because
the target disease, infantile paralysis, was uncommon. The data are 
below:

                      Paralysis Victim

		   Yes      No        Total
		-----------------------------
 Placebo          142      199,858   200,000
 Vaccine           56      199,944   200,000

 Is the proportion of children contracting paralysis the same for the
placebo and vaccine groups? 

 For the test:  x1 = 142, x2 = 56, n1 = n2 = 200,000

> prop.test(c(142, 56), c(200000, 200000))
 
  2-sample test for equality of proportions with continuity correction

data:  c(142, 56) out of c(2e+05, 2e+05) 
X-squared = 36.508, df = 1, p-value = 1.520e-09
alternative hypothesis: two.sided 
95 percent confidence interval:
 0.0002871449 0.0005728551 
sample estimates:
 prop 1  prop 2 
0.00071 0.00028 
 
> sqrt(36.508)
[1] 6.042185. This is the z test-statistic.