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 (b) at least 300

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) should be large, and (2) both np and n(1-p) should be greater than or equal to 5. Here, n = 500 and p = 0.55. Calculate np = 500 × 0.55 and n(1-p) = 500 × (1 - 0.55).
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 'at least 300,' this corresponds to P(X ≥ 300). Using the continuity correction, this becomes P(X ≥ 299.5).
4. Step 4: Standardize the value using the z-score formula: z = (X - μ) / σ. Substitute X = 299.5, μ, and σ into the formula to calculate the z-score.
5. Step 5: Use the standard normal distribution table (or a calculator) to find the probability corresponding to the calculated z-score. This will give you the probability that at least 300 Americans would say they are disappointed if Facebook disappeared.

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. In this context, the success is defined as an individual expressing disappointment if Facebook disappeared. The parameters of a binomial distribution are the number of trials (n) and the probability of success (p), which in this case is 0.55.

Normal Approximation to the Binomial

The normal approximation to the binomial distribution can be used when the number of trials is large, typically when 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 scenario, with n=500 and p=0.55, we can check if this 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 this problem, determining whether the event of at least 300 people expressing disappointment is unusual involves calculating the probability using either the binomial or normal approximation and comparing it to this threshold.