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 (c) between 240 and 280, inclusive.

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). Ensure 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 we are finding the probability that the number is between 240 and 280 (inclusive), adjust the range to 239.5 to 280.5 to account for the discrete-to-continuous transition.
4. Step 4: Standardize the values using the z-score formula: z = (x - μ) / σ. Compute the z-scores for x = 239.5 and x = 280.5 using the mean (μ) and standard deviation (σ) calculated earlier.
5. Step 5: Use the standard normal distribution table (or a calculator) to find the probabilities corresponding to the z-scores. Subtract the smaller probability from the larger probability to find the total probability for the range. Sketch the graph of the normal curve, shading the area between the two z-scores to represent 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 success is defined as an individual expressing disappointment if Facebook disappeared.

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 if this approximation is valid.

Probability Calculation

Probability calculation involves determining the likelihood of a specific outcome occurring within a defined set of possibilities. For the given problem, we need to calculate the probability that the number of Americans who would be disappointed falls between 240 and 280. This can be done using either the binomial formula or the normal approximation, depending on the validity of the approximation conditions.