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Chapter 6: Problem 45

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 both \(np\) and \(n(1-p)\) are greater than or equal to 5 (Rule of 5) or 10 (Rule of 10), where \(n\) is the number of trials and \(p\) is the probability of success in a single trial.

Step by step solution

01

Understanding of Normal Distribution and Binomial Distribution

The binomial distribution is discrete, meaning it only takes on integer values, and it's often used to model the number of successes in a fixed number of Bernoulli trials (trials with exactly two outcomes, often labeled as success or failure). The normal distribution, on the other hand, is a continuous distribution, often used when we're dealing with large populations or samples.
02

The Rule of 5

The first rule for using a normal distribution to approximate a binomial distribution is usually mentioned as the Rule of 5. The normal distribution is a good approximation of the binomial distribution if both \(np \geq 5\) and \(n(1-p) \geq 5\), where \(n\) is the number of trials and \(p\) is the probability of success in a single trial.
03

The Rule of 10

The Rule of 10 is a more conservative rule that states the normal distribution is a reasonable approximation to the binomial distribution if both \(np \geq 10\) and \(n(1-p) \geq 10\). With these thresholds, the approximation tends to be better (though it is hard to quantify precisely what is meant by 'better' in this context).

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Most popular questions from this chapter

A construction zone on a highway has a posted speed limit of 40 miles per hour. The speeds of vehicles passing through this construction zone are normally distributed with a mean of 46 miles per hour and a standard deviation of 4 miles per hour. Find the percentage of vehicles passing through this construction zone that are a. exceeding the posted speed limit b. traveling at speeds between 50 and 57 miles per hour

Jenn Bard, who lives in the San Francisco Bay area, commutes by car from home to work. She knows that it takes her an average of 28 minutes for this commute in the morning. However, due to the variability in the traffic situation every morning, the standard deviation of these commutes is 5 minutes. Suppose the population of her morning commute times has a normal distribution with a mean of 28 minutes and a standard deviation of 5 minutes. Jenn has to be at work by \(8: 30\) A.M. every morning. By what time must she leave home in the morning so that she is late for work at most \(1 \%\) of the time?

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鈥檚 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.

Determine the area under a normal distribution curve with \(\mu=55\) and \(\sigma=7\) a. to the right of \(x=58\) b. to the right of \(x=43\) c. to the left of \(x=68\) d. to the left of \(x=22\)

Determine the following probabilities for the standard normal distribution. a. \(P(-1.83 \leq z \leq 2.57)\) b. \(P(0 \leq z \leq 2.02)\) c. \(P(-1.99 \leq z \leq 0)\) d. \(P(z \geq 1.48)\)
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