Question 1085822: Auditors at a small community bank randomly sample 100 withdrawal transactions made during the week at an ATM machine located near the bank’s main branch. Over the past 2 years, the average withdrawal amount has been $50 with a standard deviation of $40. Since audit investigations are typically expensive, the auditors decide to not initiate further investigations if the mean transaction amount of the sample is between $45 and $55. What is the probability that in any given week, there will be an investigation?
Options:
1.25%
B. 2.5%
C. 10.55%
D. 21.1%
E. 50%
Its related to sampling distribution
I came up with 21.1%
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! We're in the xbar distribution (sample mean distribution, or distribution of sample means). In this case the center is at mu = 50 and the standard error is SE = s/sqrt(n) = 40/sqrt(100) = 40/10 = 4
This distribution is normally distributed because of the central limit theorem. The fact that n = 100 makes n > 30 true indicates that we can use this idea.
Using a calculator, the value of P(45 < x < 55) is roughly 0.7887
Subtracting 1 from that value gives 1-0.7887 = 0.2113 which converts to 21.13%
That rounds to 21.1% so you have the correct answer. Nice work.
You can use an online tool such as this one to check your work.
That specific calculator also provides the sketch of the shaded area.
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