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4. A tax inspector is conducting random spot checks on tax returns for small
businesses. A sample of small businesses is randomly chosen by computer and
then their tax records are checked by several different criteria. If the records
fail any of the criteria then the records are forwarded on to be audited in detail.
Assume that the time taken to carry out a spot check on a small businesses tax
return is approximately normally distributed with a mean of 15.8 minutes and
a standard deviations of 3.4 minutes.
a. The following question was posed to a group of students:
What are the expected value and standard deviation for the
total time taken to spot check 17 randomly selected businesses
accounts?
One of the students answered as follows:
Let C be the time taken to spot check one business accounts.
So that C ~ Normal(μ =15.8 minutes, σ = 3.4 minutes)
Let T be the total time taken to spot check 17 accounts.
So that T =17×C. Hence, E[T] = 17×E[C] = 17×15.8 = 268.6
and sd[T] = 17×sd[C] = 17×3.4 = 57.8.
What incorrect assumption underlies this working? Redo the
calculations using the correct assumptions.
b. The inspector sets aside 5 hours of each day to do spot checks.
i. On what proportion of days could the inspector complete 17
spot checks in the allocated time?
ii. Let R be the time remaining from the five hours after carrying
out 17 spot checks. Give the distribution (with values of
parameters) for R.
iii. The inspector decides that after carrying out 17 spot checks, if
there is more than 12 minutes time remaining, she will start
another spot check, otherwise she will start on other work.
Using this strategy, what is the probability of the inspector
starting an 18th spot check?
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