Approximating a Binomial Distribution In Exercises 17 and 18, a binomial experiment is given. Determine whether you can use a normal distribution to approximate the binomial distribution. If you can, find the mean and standard deviation. If you cannot, explain why.

Bachelor’s Degrees Twenty-two percent of adults over 18 years of age have a bachelor’s degree. You randomly select 20 adults over 18 years of age and ask whether they have a bachelor’s degree.

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Step 1: Determine if the normal approximation to the binomial distribution can be used. The rule of thumb is that the normal approximation is appropriate if both np ≥ 5 and n(1-p) ≥ 5, where n is the number of trials and p is the probability of success.

Step 2: Calculate np, where n = 20 (number of trials) and p = 0.22 (probability of success). Use the formula np = n × p.

Step 3: Calculate n(1-p), where n = 20 and p = 0.22. Use the formula n(1-p) = n × (1-p).

Step 4: Check the conditions np ≥ 5 and n(1-p) ≥ 5. If both conditions are satisfied, the normal approximation can be used. If not, explain why the approximation is not valid.

Step 5: If the normal approximation is valid, calculate the mean (μ) and standard deviation (σ) of the binomial distribution. Use the formulas μ = n × p and σ = √(n × p × (1-p)).

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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: n (the number of trials) and p (the probability of success on each trial). In this context, the success is defined as an adult having a bachelor's degree.

Normal Approximation to the Binomial

The normal approximation to the binomial distribution can be used when certain conditions are met, specifically when both np and n(1-p) are greater than or equal to 5. This allows for the use of the normal distribution to estimate probabilities and calculate the mean and standard deviation, simplifying analysis for larger sample sizes.

Mean and Standard Deviation of a Binomial Distribution

The mean (μ) of a binomial distribution is calculated as μ = np, while the standard deviation (σ) is given by σ = √(np(1-p)). These formulas provide essential measures of central tendency and variability, which are crucial for understanding the distribution of successes in the context of the given problem.