##### Categorical dependent variables
##### 
##### Script: sca-52.R
##### 

## The purpose of this script is to show you how to perform
## hypothesis tests and calculate confidence intervals 
## concerning population proportions.
##
## While working through this script, keep the following
## workflow in mind:
##   I have to test the proportion of some samples
##   I need to assume the counts come from a Binomial distribution
##       1 proportion   1 population:  binom.test()
##       2 proportions  2 populations: prop.test()
##      >2 proportions  1 population:  chisq.test()
##





#####
## Example 1a: I would like to estimate the proportion of Oklahomans
##             who plan on voting for the Republican candidate. To
##             determine this, I ask a random sample of 100 people.
##             of those people, 60 state they will vote for the 
##             Republican candidate. 
##   
##             What is a central 95% confidence interval?

binom.test(60,100)

##             I am 95% confident that the true proportion of Oklahomans
##             planning on voting for the Republican candidate is between
##             49.7 and 69.7%.

## Example 1b: My friend states that the Democratic candidate will win
##             Oklahoma. Does the data support her?

binom.test(60,100, p=0.5,alternative="less")

##             Because p=0.9824 > 0.05=alpha, the data support the null
##             hypothesis, not my friend's research hypothesis.




#####
## Example 2:  I hypothesize that the proportion of males on the OSU-
##             Stillwater campus is the same as that on the OSU-Tulsa
##             campus. To test this, I sample 100 people from OSU-
##             Stillwater and 50 people from OSU-Tulsa. 50% of the 
##             OSU-Stillwater people were male, but only 40% of the
##             OSU-Tulsa people were.

prop.test( c(50,20), c(100,50) )

##             Because p=0.3253>0.05=alpha, we cannot reject the null
##             hypothesis. We conclude that there is not sufficient
##             evidence for concluding that the proportion of males
##             differs on the two compuses. 

binom.test(50,100)             

##             We are 95% sure that the proportion of males at OSU-
##             Stillwater is between 39.8 and 60.2%.

binom.test(20,50)

##             We are 95% sure that the proportion of males at OSU-
##             Tulsa is between 26.4 and 54.8%.

prop.test( c(50,20), c(100,50) )

##             The central 95% confidence interval for how much 
##             greater the proportion of males is at OSU-Stillwater 
##             than at OSU-Tulsa is from -8 to 28%.


















#####
## Example 3:  I hypothesize that my four-sided die is fair. To test
##             this, I will it 700 times and record my results in 
##             the following table:
##             
##             |   1   |   2   |   3   |   4   |
##             ---------------------------------
##             |  168  |  152  |  192  |  188  |
##             
##             Do the data support my hypothesis?

chisq.test( c(168,152,192,188) )

##             Because p = 0.1156>0.05=alpha, we cannot reject the
##             null hypothesis. We cannot conclude that the die
##             is unfair.


## More generally, the command would be

chisq.test( c(168,152,192,188), p=c(0.25,0.25,0.25,0.25) )

## However, since the *expected* proportions are equal (according
## to the null hypothesis), R does not require you to give the second 
## slot.
























#####
## Example 4:  M&Ms states that the distribution of colors is
##             
##             Blue    Brown    Green    Orange    Red    Yellow
##              24%      13%      16%       20%    13%       14%
##
##             To test this, I get a bag of 100 M&Ms, measuring the 
##             frequency of each color. The results are tabulated 
##             in this table
##             
##             Blue    Brown    Green    Orange    Red    Yellow
##               20       10       20        22     17        11
##
##             
##             Do the data support the corporate contention?


chisq.test( c(20,10,20,22,17,11), p=c(0.24,0.13,0.16,0.20,0.13,0.14) )

## Because p=0.489 > 0.05 = alpha, we fail to reject the null 
## hypothesis and conclude that there is not sufficient evidence
## for claiming that the corporate claim is incorrect. In fact,
## 95% confidence intervals for the proportion of each color are:

## Blue    0.126 to 0.292
## Brown   0.049 to 0.176
## Green   0.126 to 0.292
## Orange  0.143 to 0.314
## Red     0.102 to 0.258
## Yellow  0.056 to 0.188