22S:166
Solutions to practice problems for the final exam


1.

Courses file
   Course id  (unique primary key)
   Course number
   Course section
   Course semester

Textbook file
   Textbook ISBN  (unique primary key)
    Title
    First author
    Publisher
    Copyright date
    Selling price

Linking file
    Course id
    Textbook ISBN

    Note:  both of these fields are foreign keys. 
           Together they make up the primary key for the linking file.



2a.
> y <- rbinom( 1, 16, 0.5)
> y
[1] 9

2b.

> binom.test(y,16)

        Exact binomial test

data:  y and 16 
number of successes = 9, number of trials = 16, p-value = 0.8036
alternative hypothesis: true probability of success is not equal to 0.5 
95 percent confidence interval:
 0.2987769 0.8024659 
sample estimates:
probability of success 
                0.5625 


2c.
> binom.test(y,16)$p.value
[1] 0.8036194

We cannot reject the null hypothesis that p = 0.5 at the .05 significance
level, because the p-value is 0.804, which is larger than .05.


---------------------------------------------------------------------------
2d.


> binom.simstudy
function(reps, sampsize)
{
# reps: number of replicate samples
# sampsize: sample size of each sample

y <- rbinom(reps, sampsize, 0.5)
p.values <- matrix(0,nrow=reps, ncol=3)

for(i in 1:reps) {

        p.values[i,1] <- binom.test(y[i],sampsize, p=0.5)$p.value
        p.values[i,2] <- prop.test(y[i],16,p=0.5, correct=T)$p.value
        p.values[i,3] <- prop.test(y[i],16,p=0.5,correct=F)$p.value

}

reject <- matrix( p.values < 0.05, nrow=reps)
results <- matrix(apply(reject,2,mean),nrow=1)
results <- rbind( results, sqrt(results * (1-results)/reps) )

dimnames(results) <- list(c("size","std err"),
c("binom.test","prop.test, Yates corr","prop.test, no correction"))
results
}
-----------------------------------------------------------------------------

> binom.simstudy(20000,16)
          binom.test prop.test, Yates corr prop.test, no correction
size    0.0200000000          0.0200000000              0.075000000
std err 0.0009899495          0.0009899495              0.001862458