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Published byMartha Wattley Modified over 4 years ago

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**5.5 Normal Approximations to Binomial Distributions**

Decide when a normal distribution can approximate a binomial distribution Find the continuity correction Use a normal distribution to approximate binomial probabilities

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**Normal approximation to a binomial distribution**

If np>5 and nq>5, then the binomial random variable x is approximately normally distributed, with mean μ=np and standard deviation σ=√npq where n is the number of independent trials, p is the probability of success in a single trial, and q is the probability of failure in a single trial.

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**Normal approximation to a binomial distribution**

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**Try it yourself 1 Approximating a Binomial Distribution**

Consider the following binomial experiment. Decide whether you can use the normal distribution to approximate x, the number of people who reply yes. If you can, find the mean and standard deviation. If you cannot, explain why.

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Try it yourself 1 Five percent of adults in the United States are planning to purchase a 3D TV in the next two years. You randomly select 125 adults in the United States and ask them if they are planning to purchase a 3D TV in the next two years. n = 125, p = 0.05, q = 0.95 Mean: Standard deviation: Normal distribution can be used

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**Try it yourself 2 56.5<x<83.5 x<54.5**

Using a Continuity Correction Use a continuity correction to convert each of the following binomial intervals to a normal distribution interval. The probability of getting between 57 and 83 successes, inclusive. The probability of getting at most 54 successes. 56.5<x<83.5 x<54.5

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**Try it yourself 3 0.0918 Approximating a Binomial Probability**

Five percent of adults in the United States are planning to purchase a 3D TV in the next two years. You randomly select 125 adults in the United States and ask them if they are planning to purchase a 3D TV in the next two years. What is the probability that more than 9 respond yes? (See Try it yourself 1) 0.0918

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**Try it yourself 4 0.0132 Approximating a Binomial Probability**

Fifty-eight percent of adults say that they never wear a helmet when riding a bicycle. You randomly select 200 adults in the United States and ask them if they wear a helmet when riding a bicycle. What is the probability that at most 100 adults will say they never wear a helmet when riding a bicycle? 0.0132

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**Try it yourself 5 0.0177 Approximating a Binomial Probability**

A survey reports that 24% of Internet users use Mozilla® Firefox® as their browser. You randomly select 150 Internet users and ask them whether they use Firefox® as their browser. What is the probability that exactly 27 will say yes? 0.0177

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