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Chapter 6: Q.91 (page 429)

Bag check Thousands of travelers pass through the airport in Guadalajara,

Mexico, each day. Before leaving the airport, each passenger must go through the customs inspection area. Customs agents want to be sure that passengers do not bring illegal items into the country. But they do not have time to search every traveler's luggage. Instead, they require each person to press a button. Either a red or a green bulb lights up. If the red light flashes, the passenger will be searched by customs agents. A green light means "go ahead." Customs agents claim that the light has probability 0.30of showing red on any push of the button. Assume for now that this claim is true. Suppose we watch 20 passengers press the button. Let R=the number who get a red light. Here is a histogram of the probability distribution of R:

a. What probability distribution does R have? Justify your answer.

b. Describe the shape of the probability distribution.

Short Answer

Expert verified

(a)All the binomial conditions are fulfilled here. Thus, it could be said that R follows the normal distribution.

(b) The distribution of R does not depart majorly from normality.

Step by step solution

01

Part (a) Step 1: Given Information

Given the graph,

02

Part (a) Step2: Simplification

Following requiremens are fulfilled by the random variable R.

1. Probability of success, that is, probability of getting red light on any press of the buttton ( p) which is equal to 0.30is fixed.

2. Number of press are fixed.

3. Trials of getting red light are independent of each other.

4. There are two outcomes, that is getting red light and not getting red light.

All the binomial conditions are fullfilled here. Thus, it could be said that R follows the binomial distribution.

03

Part (b) Step 1: Given information

Given the graph is,

04

Part (b) Step 2: Simplification

The probability distribution shows that the shape of the histogram is roughly bell-shaped. As a result, the distribution of R does not deviate significantly from normality.

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Most popular questions from this chapter

Exercises 21 and 22 examine how Benford鈥檚 law (Exercise 9) can be used to detect fraud.

Benford鈥檚 law and fraud A not-so-clever employee decided to fake his monthly expense report. He believed that the first digits of his expense amounts should be equally likely to be any of the numbers from 1 to 9. In that case, the first digit Yof a randomly selected expense amount would have the probability distribution shown in the histogram.

(a) What鈥檚 P(Y<6)? According to Benford鈥檚 law (see Exercise 9), what proportion of first digits in the employee鈥檚 expense amounts should be greater than 6? How could this information be used to detect a fake expense report?

(b) Explain why the mean of the random variable Yis located at the solid red line in the figure.

(c) According to Benford鈥檚 law, the expected value of the first digit is 渭X=3.441. Explain how this information could be used to detect a fake expense report.

Give me some sugar! Machines that fill bags with powdered sugar are supposed to

dispense 32ounces of powdered sugar into each bag. Let x=the weight (in ounces) of the

powdered sugar dispensed into a randomly selected bag. Suppose that xcan be modeled

by a Normal distribution with mean 32ounces and standard deviation 0.6ounce. Find P(x鈮�31). Interpret this value.

Life insurance If four 21-year-old men are insured, the insurer鈥檚 average income is

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where Xiis the income from insuring one man. Assuming that the amount of income earned on individual policies is independent, find the mean and standard deviation of V. (If you compare with the results of Exercise 57, you should see that averaging over more insured individuals reduces risk.)

Take a spin An online spinner has two colored regions-blue and yellow. According to the website, the probability that the spinner lands in the blue region on any spin is 0.80. Assume for now that this claim is correct. Suppose we spin the spinner12 times and let X= the number of times it lands in the blue region. Make a graph of the probability distribution of X. Describe its shape.

Let Y denote the number of broken eggs in a randomly selected carton of one dozen 鈥渟tore brand鈥� eggs at a local supermarket. Suppose that the probability distribution of Y is as follows.

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a. What is the probability that at least 10 eggs in a randomly selected carton are unbroken?

b. Calculate and interpret 渭Y.

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