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Mathematics 243
Completely Randomized Designs
Day 13 - Fat Tuesday
1. Completely randomized design.
2. Four examples: dolphins, distracted drivers, yawning, First-year seminars
Improve
Not
Yawn
Not
Dolphins
10
5
Seed
10
24
Control
3
12
Missed
Made
None
4
12
Cell
7
17
Retained
Dropped
Passenger
2
22
FYS
212
84
None
113
49
3. General form
”Success”
”Failure”
Treatment A
a
c
Treatment B
b
d
Null Hypothesis: There is no relationship between success rate and treatment (the true success rate is independent
of treatment)
4. Exact test. The p value of the test is the probability of getting results at least as extreme (far away from the
expected) as these if the null hypothesis is true.
Equivalently, if a + b successes are distributed randomly to the two treatment groups A and B, how likly is it that
the result would be at least as extreme as a successes in group A?
5. The hypergeometric distribution revisited.
m white balls (successes)
n black balls (failures)
a simple random sample of k balls are chosen (treatment A)
x is the number of white balls in the sample (treatment A, successes)
function (& parameters)
explanation
dhyper(x,m,n,k)
returns P(X = x)
phyper(q,m,n,k)
returns P(X ≤ q)
rhyper(nn,m,n,k)
makes nn random draws of the random variable X and
returns them in a vector.
6. Dolphin example. m = 13, n = 17, k = 15
> 1-phyper(9,13,17,15)
[1] 0.01266384
> dolphins=matrix(c(10,3,5,12),nrow=2,byrow=T)
> fisher.test(dolphins,alt='greater')
Fisher's Exact Test for Count Data
data: dolphins
p-value = 0.01266
alternative hypothesis: true odds ratio is greater than 1
95 percent confidence interval:
1.531074
Inf
sample estimates:
odds ratio
7.375228