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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

Life insurance A life insurance company sells a term insurance policy to 21-year-old males that pays \(100,000 if the insured dies within the next 5 years. The probability that a randomly chosen male will die each year can be found in mortality tables. The company collects a premium of \)250 each year as payment for the insurance. The amount Y that the company earns on a randomly selected policy of this type is \(250 per year, less the \)100,000 that it must pay if the insured dies. Here is the probability distribution of Y:

(a) Explain why the company suffers a loss of $98,750 on such a policy if a client dies at age 25.

(b) Calculate the expected value of Y. Explain what this result means for the insurance company.

(c) Calculate the standard deviation of Y. Explain what this result means for the insurance company.

Skee Ball Refer to Exercise 4. Find the mean of X. Interpret this value.

Spoofing (4.2) To collect information such as passwords, online criminals use "spoofing" to direct Internet users to fraudulent websites. In one study of Internet fraud, students were warned about spoofing and then asked to log into their university account starting from the university's home page. In some cases, the log-in link led to the genuine dialog box. In others, the box looked genuine but, in fact, was linked to a different site that recorded the ID and password the student entered. The box that appeared for each student was determined at random. An alert student could detect the fraud by looking at the true Internet address displayed in the browser status bar, but most just entered their ID and password.

a. Is this an observational study or an experiment? Justify your answer.

b. What are the explanatory and response variables? Identify each variable as categorical or quantitative.

Easy-start mower? A company has developed an "easy-start" mower that cranks the engine with the push of a button. The company claims that the probability the mower will start on any push of the button is 0.9. Assume for now that this claim is true. On the next 30 uses of the mower, let T=the number of times it starts on the first push of the button. Here is a histogram of the probability distribution of T :

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

b. Describe the shape of the probability distribution.

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
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