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 "teh") or word errors that result in a real but incorrect word. Spell-checking software will catch nonword errors but not word errors. Human proofreaders catch 70% of word errors. You ask a fellow student to proofread an essay in which you have deliberately made 15 word errors. What is the smallest number of misses m with

P(X m)

no larger than 0.05? You might consider m or more misses as evidence that a proofreader actually catches fewer than 70% of word errors.

Explanation / Answer

here probability of miss =1-0.7 =0.3

and n=15

hence mean =np =0.3*15 =4.5

also std deviation =(np(1-p))1/2 =1.7748

hence from normal aapproxmiation of binomial distribution:

for no larger then 0.05 probabilty of m, given true that proofreader catch 70% errors

at 95 percentile ; z =1.6449

hence corresponding m =mean +z*std error =4.5+1.6449*1.7748=7.41

hence m=8

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