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Chapter 6: Problem 65

Under what conditions is the normal distribution usually used as an approximation to the binomial distribution?

Short Answer

Expert verified
The normal distribution is usually used as an approximation to the binomial distribution when the number of trials (n) is sufficiently large, the probability of success (p) is not too close to 0 or 1, and both np and n(1 鈭� p) are greater than or equal to five.

Step by step solution

01

Describe the conditions for approximation

The normal distribution is generally used as an approximation to the binomial distribution under these conditions: (1) The number of trials (n) is sufficiently large. (2) The probability of success (p) is not too close to 0 or 1. (3) The np and n(1 鈭� p) values are both greater than or equal to five.
02

Explain the Central Limit Theorem

The Central Limit Theorem states that when the number of trials in an experiment is sufficiently large, the distribution of the sum of independent variables will approach a normal distribution regardless of the shape of the original distribution.
03

Discuss the Law of Large Numbers

The Law of Large Numbers states that as the number of trials of a random process increases, the mean outcome of the process approaches the expected or 'true' value. It helps in approximating the binomial distribution to normal when the number of trials is large.

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Most popular questions from this chapter

Fast Auto Service provides oil and lube service for cars. It is known that the mean time taken for oil and lube service at this garage is 15 minutes per car and the standard deviation is \(2.4\) minutes. The management wants to promote the business by guaranteeing a maximum waiting time for its customers. If a customer's car is not serviced within that period, the customer will receive a \(50 \%\) discount on the charges. The company wants to limit this discount to at most \(5 \%\) of the customers. What should the maximum guaranteed waiting time be? Assume that the times taken for oil and lube service for all cars have a normal distribution.

What is the difference between the probability distribution of a discrete random variable and that of a continuous random variable? Explain.

Fast Auto Service guarantees that the maximum waiting time for its customers is 20 minutes for oil and lube service on their cars. It also guarantees that any customer who has to wait longer than 20 minutes for this service will receive a \(50 \%\) discount on the charges. It is estimated that the mean time taken for oil and lube service at this garage is 15 minutes per car and the standard deviation is \(2.4\) minutes. Suppose the time taken for oil and lube service on a car follows a normal distribution. a. What percentage of the customers will receive the \(50 \%\) discount on their charges? b. Is it possible that a car may take longer than 25 minutes for oil and lube service? Explain

A charter bus company is advertising a singles outing on a bus that holds 60 passengers. The company has found that, on average, \(10 \%\) of ticket holders do not show up for such trips; hence, the company routinely overbooks such trips. Assume that passengers act independently of one another. a. If the company sells 65 tickets, what is the probability that the bus can hold all the ticket holders who actually show up? In other words, find the probability that 60 or fewer passengers show up. b. What is the largest number of tickets the company can sell and still be at least \(95 \%\) sure that the bus can hold all the ticket holders who actually show up?

Find the area under the standard normal curve a. between \(z=0\) and \(z=1.95\) b. between \(z=0\) and \(z=-2.05\) c. between \(z=1.15\) and \(z=2.37\) d. from \(z=-1.53\) to \(z=-2.88\) e. from \(z=-1.67\) to \(z=2.24\)
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