Approximating Binomial Probabilities In Exercises 19–26, determine whether you can use a normal distribution to approximate the binomial distribution. If you can, use the normal distribution to approximate the indicated probabilities and sketch their graphs. If you cannot, explain why and use a binomial distribution to find the indicated probabilities. Identify any unusual events. Explain.

Advancing Research In a survey of U.S. adults, 77% said are willing to share their personal health information to advance medical research. You randomly select 500 U.S. adults. Find the probability that the number who are willing to share their personal health information to advance medical research is (b) more than 360

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Step 1: Verify if the normal approximation to the binomial distribution can be used. Check the conditions: (1) The sample size (n) should be large, and (2) both np and n(1-p) should be greater than or equal to 5. Here, n = 500 and p = 0.77. Calculate np = 500 * 0.77 and n(1-p) = 500 * (1 - 0.77).

Step 2: If the conditions are satisfied, approximate the binomial distribution using a normal distribution. The mean (μ) and standard deviation (σ) of the binomial distribution are given by μ = np and σ = sqrt(np(1-p)). Calculate these values.

Step 3: Apply the continuity correction for the normal approximation. Since the problem asks for the probability that the number is more than 360, adjust the value to 360.5 (to include the continuity correction).

Step 4: Standardize the value using the z-score formula: z = (x - μ) / σ, where x is the adjusted value (360.5), μ is the mean, and σ is the standard deviation. Compute the z-score.

Step 5: Use the standard normal distribution table or a statistical software to find the probability corresponding to the calculated z-score. Subtract this probability from 1 to find the probability that the number is more than 360. Sketch the graph of the normal curve, marking the mean, the z-score, and the area representing the probability.

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주요 개념

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Binomial Distribution

A binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. It is characterized by two parameters: the number of trials (n) and the probability of success (p). In this context, it helps determine the likelihood of a certain number of individuals (e.g., those willing to share health information) in a sample.

Normal Approximation to the Binomial

The normal approximation to the binomial distribution is applicable when the number of trials is large, and both np and n(1-p) are greater than 5. This allows us to use the normal distribution to estimate probabilities for binomial outcomes, simplifying calculations. In this case, it can be used to approximate the probability of more than 360 individuals willing to share their information.

Unusual Events

An unusual event in statistics is typically defined as one that has a probability of occurring less than 5%. Identifying unusual events helps in understanding the significance of results. In this scenario, determining whether the probability of more than 360 individuals willing to share their information is unusual can provide insights into the general willingness of the population regarding health information sharing.