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.


Social Media A survey of Americans found that 55% would be disappointed if Facebook disappeared. You randomly select 500 Americans and ask them whether they would be disappointed if Facebook disappeared. Find the probability that the number who say yes is (a) less than 250

Verified step by step guidance

1. Step 1: Verify if the normal approximation to the binomial distribution can be used. The conditions are: (1) The sample size (n) is large, and (2) both np and n(1-p) are greater than or equal to 5. Here, n = 500 and p = 0.55. Calculate np = 500 * 0.55 and n(1-p) = 500 * (1 - 0.55). Check if both values meet the condition.
2. 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)). Calculate these values.
3. Step 3: Apply the continuity correction for the normal approximation. Since the problem asks for the probability that the number is less than 250, adjust the value to 249.5 to account for the discrete-to-continuous transition.
4. Step 4: Standardize the value using the z-score formula: z = (x - μ) / σ, where x is the adjusted value (249.5), μ is the mean, and σ is the standard deviation. Compute the z-score.
5. Step 5: Use the standard normal distribution table (or a calculator) to find the cumulative probability corresponding to the calculated z-score. This will give the probability that the number of Americans who say yes is less than 250. Sketch the normal curve, marking the mean and the area corresponding to the probability.

Key Concepts

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, the survey of Americans represents a binomial scenario where each individual either expresses disappointment (success) or not (failure).

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, with n=500 and p=0.55, we can check these conditions to determine if the normal approximation is valid.

Unusual Events

An unusual event in statistics is typically defined as an outcome that has a low probability of occurring, often less than 5%. In the context of the binomial distribution, identifying unusual events helps in understanding the likelihood of certain outcomes, such as finding the probability of fewer than 250 people expressing disappointment. This concept is crucial for interpreting results and making informed decisions based on statistical data.