Typographic errors in a text are either nonword errors (as when \"the\" is typed

Question

Typographic errors in a text are either nonword errors (as when "the" is typed as "the") or word errors that result in a real but incorrect word. Spellchecking software will catch nonword errors but not word errors. Suppose human proofreaders catch 68% of nonword errors. You ask a fellow student to proofread an essay in which you have deliberately made 28 nonword errors. (a) What is the mean number of errors caught? What is the mean number of errors missed? You see that these two means must add to 28, the total number of errors. (b) What is the standard deviation sigma of the number of errors caught? (c) Suppose that a proofreader catches 89.1% of nonword errors, so that p = 0.891. What is sigma in this case? What is sigma if p = 0.973? (d) What happens to the standard deviation of a binomial distribution as the probabilityof a success gets close to 1? sigma increase towards 1 sigma increase towards 0 sigma decrease towards 0 sigma decrease towards 10 sigma remains constant

Explanation / Answer

a)

Note that the probability of being caught is p = 0.68, while the probability of not being caught is 1 - p = 0.32.

Hence,

u(caught) = n p = 28*0.68 = 19.04 [ANSWER]

u(missed) = n(1-p) = 28*(1-0.68) = 8.96 [ANSWER]

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b)

s = standard deviation = sqrt(np(1-p)) =    2.468359779 [ANSWER]

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c)

If p = 0.891:

s = standard deviation = sqrt(np(1-p)) =    1.649039721 [ANSWER]

If p = 0.973:

s = standard deviation = sqrt(np(1-p)) =    0.85766427 [ANSWER]

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d)

As there is a factor of 1 - p, then

OPTION C: sigma decreases towards 0 [ANSWER]

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