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Q. 100

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The Practice of Statistics for AP

Found in: Page 406

Book edition 4th

Author(s) David Moore,Daren Starnes,Dan Yates

Pages 809 pages

ISBN 9781319113339

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Short Answer

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 $40^{t h}$ 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 $10.3093 i n v o i c e s .$

b. The invoice amounts are not genuine.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Part (a) Step 1: Given Information

The probability $= 0.097$

Randomly selected invoices from a vendor until you find one whose amount begins $= 8 o r 9 .$

Part (a) Step 2: Explanation

Given:

$p = 0.097$

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

$μ = \frac{1}{p}$

$= \frac{1}{0.097}$

$\approx 10.3093$

Part (b) Step 1: Given Information

The probability $= 0.097$

Randomly selected invoices from a vendor until you find one whose amount begins $= 8 o r 9 .$

Part (b) Step 2: Explanation

Given:

$p = 0.097$

Geometric probability formula:

$P (X = k) = q^{k - 1} p = (1 - p)^{k - 1} p$

Find the probability using the complement rule:

$P (X \geq 40) = 1 - P (X < 40) = 1 - P (X = 1) - P (X = 2) - \dots - P (X = 39) = 1 - 0.9813 \approx 0.0187 = 1.87 %$

Most popular questions for Math Textbooks

Suppose you roll a pair of fair, six-sided dice until you get doubles. Let $T =$ the number of rolls it takes.

Find $P (T = 3)$ . Interpret this result in context.

Explain whether the given random variable has a binomial distribution

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In which of the following situations would it be appropriate to use a Normal distribution to approximate probabilities for a binomial distribution with the given values of n and p?

(a) $n = 10, p = 0.5$

(b) $n = 40, p = 0.88$

(c) $n = 100, p = 0.2$

(d) $n = 100, p = 0.99$

(e) $n = 1000, p = 0.003$

Ms. Hall gave her class a $10$ -question multiple-choice quiz. Let $X =$ the number of questions that a randomly selected student in the class answered correctly. The computer output below gives information about the probability distribution of $X$ . To determine each student’s grade on the quiz (out of $100$ ), Ms. Hall will multiply his or her number of correct answers by $10$ . Let $G =$ the grade of a randomly chosen student in the class.

$\begin{array}{l} N & Mean & Median & StDev & Min & Max & Q_{1} & Q_{3} \\ 30 & 7.6 & 8.5 & 1.32 & 4 & 10 & 8 & 9 \end{array}$

(a) Find the median of $G$ . Show your method.

(b) Find the IQR of $G$ . Show your method.

(c) What shape would the probability distribution of $G$ have? Justify your answer

Most states and Canadian provinces have government-sponsored lotteries. Here is a simple lottery wager, from the Tri-State Pick $3$ game that New Hampshire shares with Maine and Vermont. You choose a number with $3$ digits from $0$ to $9$ ; the state chooses a three-digit winning number at random and pays you $$ 500$ if your number is chosen. Because there are $1000$ numbers with three digits, you have a probability of $1 / 1000$ of winning. Taking $X$ to be the amount your ticket pays you, the probability distribution of $X$ is

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