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**fastfred****Guest**

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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