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.


Athletes on Social Issues In a survey of college athletes, 84% said they are willing to speak up and be more active in social issues. You randomly select 25 college athletes. Find the probability that the number who are willing to speak up and be more active in social issues is (c) between 18 and 22, 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 enough, and (2) both np and n(1-p) are greater than or equal to 5. Here, n = 25 and p = 0.84. Calculate np = 25 * 0.84 and n(1-p) = 25 * (1 - 0.84). 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 = sqrt(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 of athletes is between 18 and 22 (inclusive), adjust the range to 17.5 to 22.5 for the normal distribution.
4. Step 4: Standardize the values using the z-score formula: z = (x - μ) / σ. Compute the z-scores for x = 17.5 and x = 22.5 using the mean and standard deviation 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 cumulative probability at z = 17.5 from the cumulative probability at z = 22.5 to find the probability that the number of athletes is between 18 and 22, inclusive.

Key Concepts

Binomial Distribution

The 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 a college athlete willing to speak up on social issues.

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, we can check if the conditions are met to use the normal distribution for approximating the probability of athletes willing to speak up.

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 between 18 and 22 athletes are willing to speak up, which can be done using either the binomial formula or the normal approximation, depending on the conditions met.