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 (c) between 380 and 390 inclusive.

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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 = √(np(1-p)). Compute these values using the given n and p.

Step 3: Apply the continuity correction for the range 380 to 390 inclusive. Adjust the range to 379.5 to 390.5 to account for the discrete nature of the binomial distribution when approximating with a continuous normal distribution.

Step 4: Standardize the values 379.5 and 390.5 using the z-score formula: z = (x - μ) / σ. Compute the z-scores for both 379.5 and 390.5 using the mean (μ) and standard deviation (σ) calculated earlier.

Step 5: Use the standard normal distribution table (or a calculator) to find the probabilities corresponding to the z-scores. Subtract the cumulative probability at the lower z-score from the cumulative probability at the upper z-score to find the probability that the number of adults willing to share their health information is between 380 and 390 inclusive.

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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 specific number of individuals willing to share their health information.

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 and enabling the use of z-scores for probability determination.

Probability Calculation

Probability calculation involves determining the likelihood of a specific event occurring within a defined sample space. In this scenario, it requires calculating the probability of a certain number of adults (between 380 and 390) being willing to share their health information, using either the binomial formula or the normal approximation, depending on the conditions met.