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Expert-verified Found in: Page 406 ### The Practice of Statistics for AP

Book edition 4th
Author(s) David Moore,Daren Starnes,Dan Yates
Pages 809 pages
ISBN 9781319113339 # Using Benford's law According to Benford's law (Exercise $5$, page $353$ ), the probability that the first digit of the amount of a randomly chosen invoice is an $8$ or a $9$ is $0.097$. Suppose you examine randomly selected invoices from a vendor until you find one whose amount begins with an $8$ or a $9$.(a) How many invoices do you expect to examine until you get one that begins with an $8$ or $9$? Justify your answer.(b) In fact, you don't get an amount starting with an $8$ or $9$ until the ${40}^{th}$ invoice. Do you suspect that the invoice amounts are not genuine? Compute an appropriate probability to support your answer.

a. The number of invoices expected to examine until begins with an $8$ or $9$ is $10.3093invoices.$

b. The invoice amounts are not genuine.

See the step by step solution

## Part (a) Step 1: Given Information

The probability$=0.097$

Randomly selected invoices from a vendor until you find one whose amount begins$=8or9.$

## Part (a) Step 2: Explanation

Given:

$p=0.097$

Probability (or mean) of a geometric variable is the reciprocal of the expected number:

$\mu =\frac{1}{p}$

$=\frac{1}{0.097}$

$\approx 10.3093$

## Part (b) Step 1: Given Information

The probability $=0.097$

Randomly selected invoices from a vendor until you find one whose amount begins $=8or9.$

## Part (b) Step 2: Explanation

Given:

$p=0.097$

Geometric probability formula:

$P\left(X=k\right)={q}^{k-1}p\phantom{\rule{0ex}{0ex}}=\left(1-p{\right)}^{k-1}p$

Find the probability using the complement rule:

$P\left(X\ge 40\right)=1-P\left(X<40\right)\phantom{\rule{0ex}{0ex}}=1-P\left(X=1\right)-P\left(X=2\right)-\dots -P\left(X=39\right)\phantom{\rule{0ex}{0ex}}=1-0.9813\phantom{\rule{0ex}{0ex}}\approx 0.0187\phantom{\rule{0ex}{0ex}}=1.87\mathrm{%}$ ### Want to see more solutions like these? 