MATH 145 Activities for Mon., Nov. 4: What is the relationship between . . .
1. What is the relationship between the following, specialized to the case of a binomial
setting in which n = 30, p = 0.4?
(a) The tree diagram
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plotDist("binom", params=list(30, .4), cex=.4, pch=19)
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(b) The picture produced by
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MATH 145: What is the relationship between . . .
(c) The command dbinom(1, 30, .4)
(d) The sampling distribution for the count of successes X ∼ Binom(30, .4)
(e) The sampling distribution for p̂, which may be simulated using the commands
>
>
manyPHats = replicate(100000, mean(sample(c(0,1), 30, replace=T, prob=c(.6,.4))))
histogram(manyPHats, breaks=seq(-.5,30.5)/30)
Percent of Total
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manyPHats
(f) The normal distributions portrayed by
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plotDist("norm", params=list(12, sqrt(7.2)))
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and
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plotDist("norm", params=list(.4, sqrt(.008)))
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MATH 145: What is the relationship between . . .
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2. Why is it that categorical data, when broken into two categories success and failure
(generic words), acquired over a fixed number n of trials and counted for successes,
is treated as binomial? In particular, why is it treated that way even when we do not
sample with replacement?
3. What is an hypothesis test? What kind of hypotheses have we been able to test thus
far in the course? Give specific examples.
4. What is the null distribution? How does one obtain the null distribution in the various
cases of hypothesis tests you listed in the last problem?
5. Which of these phrases make sense?
• the sampling distribution for p
• the sampling distribution for p̂
• the sampling distribution for X, the count of successes in n independent trials
• the sampling distribution for x̄
• the sampling distribution for σ
• the sampling distribution for µ
• the sampling distribution for the sample median
Why must some of these be viewed as nonsensical? For those that make sense, which
are known to become increasingly normal as n increases, by the central limit theorem?
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