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 12 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

Solution:

To me this question provides evidence of why it is so important to first identify your random variable. Notice that the constant m is the number of misses, this means that the variable X, is not the same as in question 7.3. Let us define the random variable Y as the number of misses out of 20 attempts. So this means that p = 0.3.

We want the sum of the individual probabilities to be at most 0.5 = P(Y = m) + P(Y = m+1) + …P(Y = 20).
Let us try and see what P(Y = 15) gives us.

P(Y = 15) = 0.000037 which is very small, thus m must be smaller than 15.
P(Y = 11) = 0.0120 which is still less than 0.05.

But the question is what value of m produces a probability sum equal to at most 0.05 when we consider values greater than or equal to m. Below is the distribution of Y.
I can see by inspecting the table and estimating the sums, that the value of m that would give P(Y m) 0.05, is for m to equal 10.

Let me verify by calculating P(Y 10) = 1 – P(Y 9)
= 1 – binomdist(9,20,0.3, true)
= 0.047962. Thus P(Y 10) = 0.047962 < 0.05.

Y P(Y = y) Y P(Y = y)
0 0.000798 11 0.0120
1 0.006839 12 0.0039
2 0.027846 13 0.0010
3 0.071604 14 0.0002
4 0.130421 15 0.0000
5 0.178863 16 0.0000
6 0.191639 17 0.0000
7 0.164262 18 0.0000
8 0.114397 19 0.0000
9 0.06537 20 0.0000
10 0.030817

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