#### Example 1:  Exact hypothesis test:

# The exact binomial test:

binom.test(x=14, n=17, p=0.60, alternative="greater")


#### Example 2:  Exact hypothesis test:

# The exact binomial test:

binom.test(x=8, n=19, p=0.35, alternative="two.sided")



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### Finding Clopper-Pearson confidence intervals:


#### Example 2 again:  Exact CI:

# The Clopper-Pearson CI for p (note 95% is the default confidence level):

binom.test(x=8, n=19, alternative="two.sided")$conf.int

# Using a 98% confidence level:

binom.test(x=8, n=19, alternative="two.sided", conf.level=0.98)$conf.int

#### Example 1 again:  Exact CI:

# The Clopper-Pearson 90% CI for p:

binom.test(x=14, n=17, alternative="two.sided", conf.level=0.90)$conf.int

# A 90% lower confidence bound:

binom.test(x=14, n=17, alternative="greater", conf.level=0.90)$conf.int



#### An alternative CI method:  The Wilson score CI:

# Example 2:

prop.test(x=8, n=19, alternative="two.sided")$conf.int

# Using a 98% confidence level:

prop.test(x=8, n=19, alternative="two.sided", conf.level=0.98)$conf.int

## Which method produces shorter intervals in this example?