##### SCA-12
##### 
##### One-Sample Proportions Tests
##### 

### This gives a few examples of the analysis process for testing
### for a population proportion

#   Note: This is not the procedure used by the book. It uses something
#   akin to our "proportions test." That procedure is approsimate. This 
#   is exact.





### Preamble

# Import extra functionality
source("http://rfs.kvasaheim.com/stat200.R")






### Example I: The Proportion of Juniors at Knox
#
#   I would like to estimate the number of Juniors at
#   Knox. To do this, I randomly sample 100 students
#   around campus. 
#
#   In that sample, the number of Juniors was 31. 
#

binom.test(x=31, n=100)

binom.plot(x=31, n=100)

#   Conclusion: We are 95% confident that the proportion of Juniors
#   at Knox College is between 22.1 and 41.0%, with a point estimate 
#   of 31%.






### Example II: Expensive Gas Stations
#
#   I would like to estimate the proportion of gas stations
#   in Galesburg that have gas prices greater than $2.50 per 
#   gallon.
#
#   To do this, I randomly select 25 from the phone book, call
#   them, and record if their current price for regular gas
#   exceeds $2.50 per gallon.
#
#   In my sample, 11 had gas more expensive than $2.50.
#

binom.test(x=11, n=25)

binom.plot(x=11, n=25)

#   Conclusion: I am 95% confident that the proportion of gas
#   stations with gasoline more expensive than $2.50 is 
#   between 24.4% and 65.1%, with a point estimate of 44%.





### Example III: Soccer
#
#   I would like to estimate the proportion of shots on goal that
#   result in a score for the best soccer team on the planet: The
#   Portland Timbers.
#
#   To estimate this, I randomly select 10 games and measuure the 
#   total number of shots on goal and the total number of points 
#   scored.
#
#   In those 10 games, there were 19 shots on goal and 8 points
#   scored.
#

binom.test(x=8, n=19)

#   Conclusion: A 95% confidence interval for the proportion of shots
#   on goal that result in a score is between 20.3% and 66.5%, with a
#   point estimate of 42.1%.





### Example IV: The Central Limit Theorem
#
#   I would like to estimate the proportion of my statistics students
#   who know that the Central Limit Theorem makes no conclusions about
#   the distribution of the data, only about the distribution of the
#   sample means (or sample totals).
#
#   To do this, I place the following statement in the True/False
#   part of the final examination:
#
#      The Central Limit Theorem states that as the sample size increases, 
#      the distribution of the sample approaches the Normal distribution 
#      (assuming the variance is finite).
#
#   In a sample of 30 students, 5 correctly circled F.
#

binom.test(x=5, n=30) 

#   Conclusion: I am 95% confident that the proportion of my students who 
#   understand the Central Limit Theorem is between 5.6% and 34.7%, with
#   a point estimate of 16.7%.





### Example V: Fifteen Hours
#
#   I would like to estimate the proportion of my statistics students
#   who actually spend 15 hours or more) on statistics. To check, I 
#   randomly sampled 30 of them and asked if they spent 15 hours or
#   more working on statistics last week.
#
#   Of those 30, none said yes.
#

binom.test(x=0, n=30)

#   Conclusion: I am 95% confident that between 0% and 11.6% of my
#   students spend 15 hours or more on statistics each week, with
#   a point estimate of 0%.